Near-rings : the theory and its applications, Volume 23 (North-Holland Mathematics Studies)

NEAR-RINGS

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NORTH-HOLLAND MATHEMATICS STUDIES

Near-hgs The Theory and its Applications GUNTERPILZ lnstitut fur Mathematik Johannes-Kepler- Universitat L inz Linz, Austria

Revised edition

NORTH-HOLLAND PUBLlSHlNG COMPANY AMSTERDAM. NEW YORK.OXFORD

23

Q

North-Holland Publishing Company. 1983

All rights reserved. N o part of this publication may he reproduced, srored in a retrieval system, or rransmitred, in any form or by any means. eleclronic. mechanical, photocopying, recording or otherwise, wirhour the prlorpermission of the copyright owner.

ISBN: 0 7204 0.566 I

First printing 1977 Revised edition 198.3

Piihhshers . NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK . OXFORD

Sole distrrbutorc for rhe U S A rind Canado ELSEVIER SCIENCE PUBLISHING COMPANY, I N C 52 VANDERBILT AVENUE. NEW YORK. N Y 10017

PRINTED IN THE NETHERLANDS

TO MY BELOVED WIFE GERTl

vi

INTERDEPENDENCE G U I D E T h e n u m b e r s i n d i c a t e t h e o n e s o f t h e p a r a o r a p h e s ; 7a is 5 7 , s e c t i o n a), a n d s o o n . F u l l l i n e s m e a n h e a v y , d o t t e d l i n e s s l i g h t dependencies.(59j is a m e r e c o l l e c t i o n o f results.)

vii

PREFACE TO THE SECOND EDITION

Since t h e appeahence 0 6 t h e 6 h t edition 0 6 t k i b book, a bubbtantiae nwnbm 06 papeu and hedlLet6 on neah-hingb came o u t and new p w ~ 2 0 6 t h e theohy wme bohn. Hence 1 wh vehy pleased when Nohth-Holland o66med me t h e p o b b i b U y t o phoduce an updated, h e v h e d , comeoted and extended vetldion 06 tk ib book on n m - h i n g b , which iA b-tiee t h e o d y one i n t k i b d i d d ( b u t t~ exceUent o t h m t e x ~ %me i n pheparration) At that h e , I ' d i d n ' t know t h e ammount 0 6 wohk I had accepted. T k i s edition contains a fiemendoub numbm 0 6 minoh additions and comections. A@m pm6o~m.h~thede changed, 1 h e a l l y know t h e b i g di,f6mence behueen "countable i n ~ i n i t e "and "uncountable binite" now. In d a d , mobt 0 6 t h e hedURtd edition ahe i n dome way incohpolrated o h at least didcovmed a6tm t h e touched i n tkid edition. A h a , {OWL mote c h a p t w wme added. They c o n c m hegutm nearr-hingb, h e neah-hingb , b i c e w z m neah-hingb and t h e connections b&en neah-hingb and automata. The chapten on polynomial neah-hingb wh bubbtantiaeey eneahged. The .fAt 0 6 n e a h - h g b 06 mall ohdm u ~ bextended by adding b.iY~uOtuhaeindotm d o n s and bome neah-hingb on t h e non-cyclic abeLian ghDupb 06 ohdm b and on Ad. A h a , t k i s edition contains 2 2 2 hemahkable (counteh)examples 0 6 n m h g b . Mobt extensions 0 6 tkis edition me ~aiheuwoven i n t o t h e t e x t ; hence 7 do hope that t k i d hecond edition iA not j u b t a b p U extension 06 t h e 6 h . t editio n The " h e L i g i o u h wah" , i d h i g h t oh t e 6 t n e a h - h g b ahe "b&m", iA 6 a unb a W e d . 1 do t h n k thlLt h i g h t nem-hingb me h i g h t , b u t t h e book en& uLith a con&atoh.q chaptm ubing le6t neah-hingb. Many t h a n k go t o NofLth-Hotland doh the.& o66m and t h e m o d t pleasant coopeh~ak, oLLt .tion. Many coUeagueb conthibuted by m a k i n g .impohtant a e n ~ ~ pointing e h h o u and heading p a h t d 0 6 t h e new manuhchipt. Majoh conhibutionh came ( i n d p h a b d c ohdet) &om G. Behch (Tubingen, Gmmany), J.U. P.M&&um (Edinbwrgh, S c o f h n d ) , S.V.ScoM (Auckland, New Z e d a n d ) , Y.S.So (Taichung, Taiwan) and H.J.Wain& (Clauhthae, G m a n y ) . A b o , many thank go t o G.KoUm and A. Kutzlm dolr t h e i h Q.xc&.&~~ typing j o b . They incohpolrated t h e additions 6 0 bhil@k%j i n t o t h e t e x t t h a t nobody,who only w a d t o head thebe a d d i t i o n s , can didcovm them. L a s t , b u t not at l e a s t , I deeply thank my wide G W . She hdped me mobt.

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ix

FROM THE PREFACE TO THE FtRST EMTiON a "mhg ahe g e n w z e d hingd. Roughly dpoken, a nem-hing ( N , + , . I , &ehe + i.6 not nece.6bafdi.g a b d i a n and wikh o n t y one dinMblLtive

Nm-hhgd

bw,,.

Neah-hingd ahide i n a natuhae way: &tabe t h e be2 Mlr) 0 6 att M p p h ~ g d 06 a gmup (r,+l i n t o &e..t.ed, de6ine additian + point-whely and o a6 cornpabLthn. Then (M(r1 ,+, 0 ) i.6 a n m - h i n g . Even i 6 r h a b e l h n , o n l y one disthibutive &UI iA aeUayb 6utd.ieeed: ( i + g ) q h= 6oh+goh holds by t h e de6inhXon 06 6+g white doh 6o(g+h)= dog+deh we mutd have to adbwne t h a t 4 i.6 a homomohphibm. Anothm cxrunpte i.6 d U p p ! i d by t h e polynonciaed w . 8 . t . addLtion and hub6.titLLtion. A WeU-known tlebUet i n hing theotly dayb that e v a y hing can be embedded into t h e hing Elr) 06 a l l endomotlpL.bm 06 borne a b U n gmup r. Foh n m h g b we pmve ( 1.161 that e v a y nem-&g can be embedded into M(r) 60% dame gmup r. Hmce one migkt v i w hing theohy ab t h e " f i n m t h w h y 04 gmup mappingb", while n m - h i n g b pmvide t h e "non-.&npAIL thwty". SUILpILi6ingly, a t o t 06 ".&nm tle6uLtb" can be L t a n b 6 r n e d ~ t ot h e gmnehae c a e aiteh dLLitabte changes. Foh i n b m c e , t h e "aXomb1' 06 hing theohy, t h e phinritive h i n g d , m e d a d b e d by t h e ~auwusd e n b d y theohem 06 N . 3acobbon (doh h g s w i t h minimUn condition: W e d d m b m - M h t h w h m on bimpte h i n g d ) . foh nean-hiftgb bAni.kb tlebbcLetb cancaning phimit.iue neat-hingd wme 0 M a i n e d v i a t h e u m k 06 6QvQAd aLLthohd ( b u t t h e pmo$b aht to&d..ly d L 6 6 a ~ n t )t:h e hole a$ ffo%(V,Y) doh aingb iA played by M(r) oh dome hekzted t y p e s i n the n m - h h g m e (4.52, 4.54).

t d i s t O h i d y , t h e 6 h t 6tep tolllaha neah-hingb wab an a c O d C h e d Q W h done by V i c h o n i n 1905. ffe dhowed tfmt thehe do ex#t ' ' ~ i c t d bw i t h onty one d i s ~ % M v e W ' ( = n m - 6 i d d 6 ) . Some ylatch thebe nearr-dieeds & h o d up again and phDued t o be uae6LLe i n c o o k d i d z i n g cehtain impohtant daAbeA 06 g w m a c p . h e d ( h e m that lk6cahte'b method 06 coohdinatizing t h e " u b d " pkkne by t h e d i d d 06 heal numbm wa6 one 0 6 t h e m o d t ducceb6~u.t d t e p b .in geomeZq). It wa6 Zabbenhaus who wa6 abte t o d & W e all ~i.n..Lte n m - 6 i e C d ~ ( 1 . 3 4 ) . ~ U m d a y b ,n m - 4 i e l d b ahe a nrigkty tool h~chahactdz&g doubly thanbLtive gtloupd ( 8 . 4 4 1 , inincidence ghoups (8.6%) and Fmb&us gmupb ( 8 . 8 1 ) . Since t h e dwn 06 A3uo endomohppkibm 06 a n o n - a b m gmup ( r , + )i.6 not an endomotlpki6m i n genehat, t h e d e t d E ( r ) 06 aU 6 W e 4wn6 and di6?4menceb 06 endomohphh6 0 6 r m e consklehed. hebpect t o addition

Preface

X

and compobLtion, t h u e E ( r ) '6 ahe neah-hingb belonging t o t h e C h b 06 t h e "dib.thibU.t.iw&y gwnehated" neah-hingb Many p#u2 0 6 t h e we,U-ehtnblhhed fheohy 06 hingb wehe tRand5eNred t o n m hGzgd and n w neat-hing-bpecidic ~ e a t w uw4he dincovetled, bLLieding up a theoay 0 6 nu&-hingb b t e p b y b t e p . Up t o now, about 550 p a p m on Wah-hingb (and n u - d i d d s ) w i t h about 8000 pageb appeahed i n phint, b u t t h e m exha2 no book on t k i s b u b j e o t . Tkid book .thieb t o unidy t h e theohy and L t h feminology and t o g i v e a bgbtematic and w&-asbohted account 06 t h e phuent n t a t e 06 t h e t h e o q . Some aemahn am .to be made: ( a ) GenmaUy, 7 avoided t o g i v e ph006b doh t h e o h m which ahe &hm not &ng t h e main b . & m 06 dibcudb-ion oh a h long ~ ones which contain bpeciae mcthoh seemingly a,oflicable o d y i n .thLh context, cannot be b h p t 4 i e d b y pewhiow hunuets and invo.P,ve many o t h m ( e . g . geornethicat) dC?.tdied, b u t ahe heaiLiey accubib& in t h e f.Ltaatwre. ( b l Sevmak? hulLetd 6oUowing dhom uvLiwm& d g e b m oh &om t h e t h e o h y 0 4 ghoup with mubkple opnehatou ahe c i t e d , bu.t not p v e d i n a d m to dev,i.de hes1Letb which ahe bp&fiic doh n m - h i n g b and Rhobe which ahe

.

not. @L m y 6~wmbeing a m a e coUecLion 0 5 - t & w i d hesuets concaning dome "pathologiCae" b y b t m tU.ithout any application t o o t h m bzmches 06 mathematicn. A p a h t dotun t h e appfic&onb concaning axiomaticb and geomethy mentioned above, bpeciae & b b u 06 d i n i t e n u - h i n g b ( t h e AinLte "plans" nea-hingb) g i v e nw and highey eddi&nt C h b u 06 balanced p a h a w e A m ( 8 . 1 1 7 - 8 . 1 2 4 ) . Moheovm, incomplete block debigm aeheady [uith b& these p l m neah-hingb can be uaed t o chaaactmize Fhobg m u p , hence a h o 6.inite ghouph w i t h d i w e d - p o i n t - d u e a&tomohpkism ghoupb ( 8 . 9 6 , 8.971. 16 r i d a &inite, inwaAhaXy a h p l e n0n-abcL.h gmup, E ( r ) h "phimctive" and t h ee 6 o h e ewmy belb-map 06 r &ixing z m h .thk " b u m 0 6 endomohpkism" [exact @munLLeation i n 7 . 4 7 ) . Anotha v m i o n 06 t h e d e n b a y theohem 4.52 b h o u t h a t t h e d m a y p m p e h t y h ( i n t h e n e a - h i n g - c a e ) bometking Like an i n t m p o l a t i o n phopehty,giwing t h e hesuet that i d a neah-king N (with nome addition& p k o p u d i u l 06 rmppingb on a ghoup r "intehpokk..tes" at zmo and at &a o t h m p k c u t h e n N "intmpakk..tudl &eady at a k b & w u j ( ~ . i n i t & q )many p o d ( 4 . 6 5 ) . m o , n u - h i n g b n i g k t be t h e appkophiate t o o l t o d e w h p a "non-ab&i.an homobgiCae algebha" ( 9 . 2 6 4 ) and dhow up again i n &geb&c topology ( 9 . 2 6 2 1 , 6unctiovd a d y b h ( 9 . 2 6 1 ) and i n

Nem-hing Zheohy h

Preface

xi

categohies xkth ghoup objeotn ( 9 . 2 6 5 ) . FinaUy, t h e a u t b h hopa that neathingo and " n m - k i n g m o d d a b t ( = N-ghOup6) Wiee pmue ;to be u&zdd doh Q n m b m 06 theohiG which .thy t o genehdeize "fineat" hedu&% t o t h e "YPOMf i n e a t cahe", doh . i n h t ~ n ~i en t h e t h e o k i a 06 automata and dynamicat bybtemb (bee § 9 ill, t o make t h e phovehb "16 you Xhy t o non-fineahize, you luiee dind t h e neah-hingb nice" come thue. Fmm t h e h i n g - t h e o h d c d point 06 view, many b i z m e bitua,t.bnts m i h e i n neah-hingb. f o h example, not evehy Le6.t i d e d .ih a hubneah-hing. Howeveh, thehe me bevehd bpLLcLte appUcaChm 06 nm-hingn 2O hi4 t h e o w (doh t h e fim-Rhtg-hehuk% dhow what h con&ined 2O hingn and whdt , .ih n o t ) and t o d v m d dgebha (becawe a h i g h pacentage 06 dedinitiom and ha& 06 t h e neah-hing theohy c a y oveh t o u n b ~ ddg e b h a ) . On quotations: Redehenced t o o t h m b e o t i o n h ahe done e.g. b y " Z d ) " meaning " 5 2 , beckLon d ) " oh by " 2 d 3 ) " abbhevhting "52, section d ) , numbeh 3 ) " , Numbm 6ok'lowing nama od authom hedeh t o t h e bibi!.iogmphy at t h e end 06 given, a.U p a p e a 06 t k i b authoh t h e book. 16 o n l y t h e authoh'b name c i t e d i n t h e bib&ogmphy ahe meant. T k i e bibfiagtraphy a b d d be ~ a i h a y cornplWe ah A a h ah nm-hingb ahe concehned. F O h nem-dieL& and hdaZed bubjectb we o n l y U t .thohe papeh?, which dihec;teY in@uenced t h e mat&L i n t k i b book. T k i n bibfioghaphy ~ a compiled 6 i n domm yea^ by J.C&y, G.B&ch, J.Mdone, H.UeatheMy and t h e authoh. Nama i n b u c k & hedeh t o t h e l?,ibt 06 "Supp&nwUa~y WOht2A11which c o n t a - h t h e non-neaA-h&g-papm oited tkio book. Sevehd hebu&h i n t k i n monaghaph ahe n w oh i n a new (and hope&Uy hnphoved) &hm UlithoLLt b p e c i d noLice. In t h e beginning 06 p o o 6 b thehe h no hepetition 06 t h e ahbump-tiom ( t o have space). l I * I t and l t e f I mean t h a t t h e diheotion indicated .ih h e a t e d at moment

(in p o o i b 06 equivdenced).

It 0 a pLmwle t o thank Utr. E . Fhedh.ibbon 06 t h e ed.Ltohid btadd 06 t h e Nohth-HuLlund Pubuhing Company and t h e heviweh doh a pLeaant cuopehhation and a L o t oh uhadul buggebXionh. Many thank4 go &a t o h. Hobpodah doh

xi i

Preface

he& excelLent typing j o b and t o G . ReLbch, Y.-S.

So, H.E. ReRC, J.D.P.

Meldhum and t o M . L . Holcombe d o h heading p a & 06 t h e manwchipt and ptouiding wed& kid and &po.ILtant commenL5. h(o4.t 06 CLee 1 haue t o thank my wide doh h a pcLtience and endutance i n fiuing w d h an aboevLt-mindedhunbund i n t h e p a t ye&. And now good Puck and much dun w L t h nem-hingb!

Remahh: A "Nedt~-h.CngNe~14leeMeh" c o m a ouA once oh twice a yeat, c o n t a i n i n g in@~ma-tiona b o u t t h e hecent deuelvpmed i n t h e theo.ty 04 n e m hingb. 16 you want t v o b t a i n c o p i e ~ ,w h i t e t o A. Obwaed o h t o t h e aLLthoh 06 tkin book.

xi ii

CONTENTS

..................................... ............................. ..................... .........................................

Interdependence guide Preface t o t h e second e d i t i o n From t h e p r e f a c e t o t h e f i r s t e d i t i o n

5 0 PREREQUISITES . PART I :

4

NEAR-RINGS

................... ............. ...................................... ........................................ ................................... ........... .................................... ............................... ...................................... .... ......................... .................... ......................................... ............................... ....................................... ...................... ........................... .............................. I D E A L THEORY .......................................... a ) Sums ............................................... 1 ) Sums a n d d i r e c t sums ............................ 2 ) D i s t r i b u t i v e sums ............................... b ) C h a i n c o n d i t i o n s ................................... c ) D e c o m p o s i t i o n t h e o r e m s ............................. d ) P r i m e i d e a l s ....................................... 1 ) P r o d u c t s o f s u b s e t s ............................. 2 ) P r i m e i d e a l s .................................... 3 ) S e m i p r i m e i d e a l s ................................ e ) N i l a n d n i l p o t e n t .................................. a ) Fundamental d e f i n i t i o n s and p r o p e r t i e s 1 ) Near-rings 2 ) N-groups 3) Substructures 4 ) Homomorphisms and i d e a l - l i k e c o n c e p t s 5) Annihilators 6 ) Generated o b j e c t s b ) Constructions 1 ) P r o d u c t s . d i r e c t sums a n d s u b d i r e c t p r o d u c t s 2 ) Near-rings o f quotients 3 ) F r e e n e a r - r i n g s and N - g r o u p s c ) Embeddings 1 ) Embedding i n M(T) 2 ) More beds d ) Some a x i o m a t i c c o n s i d e r a t i o n s 1 ) Miscellaneous r e s u l t s 2 ) Related structures

. P A R T I1 :

4

1

FOR B___EGINNERS

I T H E E L E M E N T A R Y THEORY OF N E A R - R I N G S

h 2

vi vii i x

6 7 7 13 14 15 20 23 24 24 26 29 33 33 37 38 38 41 43 44 44 49 50

53 61 61 62 66

69

STRUCTURE T H E O-~ RY

...................... T y p e s o f N - g r o u p s .................................. C h a n g e o f t h e n e a r - r i n g ............................ M o d u l a r i t y ......................................... Q u a s i r e g u l a r i t y .................................... I d e m p o t e n t s ........................................

3 ELEMENTS OF .

THE STRUCTURE THEORY

a) b) c) d) e) f ) More o n m i n i m a l i t y

.................................

74

75 81 84 89 91 95

xiv

Contents

................................. a ) G e n e r a l ........................................... 1 ) Definitions a n d elementary results . . . . . . . . . . . . . 2 ) T h e centralizer ................................ 3 ) I n d e p e n d e n c e a n d density ....................... b ) 0-primitive n e a r - r i n g s ............................ c ) I-primitive near-rings ............................ d ) 2-primitive near-rings ............................ 1 ) 2-primitive n e a r - r i n g s ......................... 2 ) 2-primitive near-rings with identity . . . . . . . . . . . 3 ) 2-primitive zero-symmetric n e a r - r i n g s with identity a n d a m i n i m a l l e f t i d e a l .............. 4 ) 2-primitive n e a r - r i n g s with identity a n d m i n i m u m condition .............................. 5 ) A n application to interpolation theory . . . . . . . . . R A D I C A L T H E O R Y .......................................

6 4 P R I M I T I V E NEAR-RINGS

6 5

..................... ...... ...

102 103 103 106 110 115 120 124 124 126 130 131 133

135

.....................................

Jacobson-type 1 ) Definitions a n d characterizations o f the r a d i c a l s 1 3 6 near-rings ....................

.................................................... Jacobson-type ..................................... go 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J 1 ............................................... ................................................... ..................................................... .. ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................................

P A R T 111:

0

S P E C I A L CLASSES O F N E A R - R I N G S

.................. Elementary ........................................ Some axiomatics ................................... Constructions o f d . g . n e a r - r i n g s . . . . . . . . . . . . . . . . . . Distributively generated n e a r - r i n g s with finiteness conditions ........................................ " F r e e " distributively generated near-rings . . . . . . . . 0 - g r o u p s a n d ( N . D ) . g r o u p s .........................

6 DISTRIBUTIVELY GENERATED NEAR-RINGS

a) b)

c) d)

e)

f ) 9 ) Structure theory

0

7

.................................. T R A N S F O R M A T I O N N E A R - R I N G S ............................ a ) M:(r) ............................................. b ) M(r) a n d Mo(r) ....................................

............................... ............................. functions ........... ........................................... ........................................... ........................... ...........................................

E(r). A(r) and I ( r ) d ) P o l y n o m i a l near-rings 1 ) Polynomials a n d polynomial 2 ) R[xl 3) P(R) 4 ) I d e a l theory i n R [ x l 5) F[xl c)

170

171 174 176 178 180 182 184

188 189 197 206 215 215 218 219 220 223

Contents

6 ) r[xl and

P ( r ) ...................................

7 ) P o l y n o m i a l s o v e r i ; - g r o u p s ....................... 8) C o n c l u d i n g r e m a r k s ..............................

5 8

5

9

.

230 233 244

..................... 248 a ) N e a r - f i e l d s ........................................ 249 1 ) C o n d i t i o n s t o be a n e a r - f i e l d . . . . . . . . . . . . . . . . . . . 2 4 9 2 ) T h e a d d i t i v e group of a n e a r - f i e l d .............. 2 5 1 3 ) The c e n t e r a n d t h e k e r n e l o f a n e a r - f i e l d ....... 2 5 3 4 ) D i c k s o n n e a r - f i e l d s ............................. 254 5 ) N e a r - f i e l d s a n d d o u b l y t r a n s i t i v e g r o u p s . . . . . . . . 258 6 ) Normal n e a r - f i e l d s a n d i n c i d e n c e g r o u p s . . . . . . . . . 260 7 ) P l a n a r n e a r - f i e l d s .............................. 265 b ) P l a n a r n e a r - r i n g s .................................. 268 1 ) T h e s t r u c t u r e o f p l a n a r n e a r - r i n g s . . . . . . . . . . . . . . 268 2 ) P l a n a r n e a r - r i n g s a n d BIB-designs ............... 276 M O R E CLASSES OF N E A R - R I N G S ............................ 287 a ) I F P - n e a r - r i n g s ..................................... 288 1 ) I F P - n e a r - r i n g s .................................. 288 2 ) p - n e a r - r i n g s .................................... 298 3 ) B o o l e a n n e a r - r i n g s .............................. 300 b ) N e a r - r i n g s w i t h o u t ................................. 301 1 ) N e a r - r i n g s w i t h o u t n i l p o t e n t e l e m e n t s . . . . . . . . . . . 301 2 ) N e a r - r i n g s w i t h o u t z e r o d i v i s o r s ................ 305 313 c ) A f f i n e n e a r - r i n g s .................................. d ) N e a r - r i n g s on g i v e n g r o u p s ......................... 321 1 ) M u l t i p l i c a t i o n s o n a g r o u p ...................... 321 2 ) N e a r - r i n g s o n s i i i i p l e a n d o n c y c l i c g r o u p s . . . . . . . 325 3 ) N e a r - r i n g s w i t h i d e n t i t i e s o n g i v e n g r o u p s . . . . . . 327 4 ) N e a r - r i n g s w i t h o t h e r p r o p e r t i e s o n g i v e n g r o u p s . 330 333 e ) O r d e r e d n e a r - r i n g s ................................. 345 f ) R e g u l a r n e a r - r i n g s ................................. g ) Tame n e a r - r i n g s .................................... 350 h ) B i c e n t r a l i z e r n e a r - r i n g s ........................... 361 i ) N e a r - r i n g s a n d a u t o m a t a ............................ 378 j) M i s c e l l a n e o u s ...................................... 392 N E A R - F I E L D S A N D P L A N A R NEAR-RINGS

APPENDIX

.

xv

..................................................

N e a r - r i n g s o f low o r d e r ............................... 2 2 2 r e m a r k a b l e e x a m p l e s and c o u n t e r e x a m p l e s . . . . . . . . . . . L i s t o f some open p r o b l e m s ............................ B i b l i o g r a p h y .......................................... S u p p l e m e n t a r y works ................................... L i s t o f s y m b o l s and a b b r e v i a t i o n s ..................... I n d e x .................................................

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1

80

PREREQUISITES

For t h e c o n c e p t of s e t s we can use any one of t h e usual s e t t h e o r i e s w i t h t h e axiom o f c h o i c e and u s i n g c l a s s e s . I n o r d e r t o a v o i d l o g i c a l d i f f i c u l t i e s a s much a s p o s s i b l e , we u s e s t a t e m e n t s a b o u t c l a s s e s only a s a b b r e v i a t i o n s of " l e s s obscure o n e s " . F o r i n s t a n c e , i f 7 d e n o t e s t h e c l a s s of a l l f i n i t e s e t s , "FE$" i s only a n a b b r e v i a t i o n f o r "F i s a f i n i t e s e t " . " ~ x E A " stands f o r " t h e r e e x i s t s a n X E A " , "3X E A " f o r " t h e r e e x i s t s e x a c t l y one X E A " and " t / x ~ A " f o r " f o r a l l X E A " .

I n c l u s i o n w i l l be d e n o t e d by c , s t r i c t i n c l u s i o n by =. 0 w i l l d e n o t e t h e ( a n ) empty s e t a n d ilA t h e power s e t o f A ; i f A i ( i E I ) i s a c o l l e c t i o n o f s e t s , we w i l l w r i t e t h e e l e m e n t s o f Ai:=A X A i a s ( . . . , a i , . . . ) , where a i € A i . I f a l l A i 5 A t h e n i€ 1 i EB and a l s o fl A i : = A . A \ B i s t h e s e t - t h e o r e t i c d i f f e r e n c e . I f i €0 % i s a n equivalence r e l a t i o n on the s e t A , A / % w i l l be t h e f a c t o r s e t of A w . r . t . 2, a n d n:A -* A/?. w i l l be the canonical projection.

x

The s e t s o f a l l n a t u r a l numbers w i l l be d e n o t e d by I N , t h e n a t u r a l numbers t o g e t h e r w i t h 0 by INo , t h e prime numbers by IP, t h e i n t e g e r s by Z, t h e r a t i o n a l s by Q , t h e r e a l s by lR a n d t h e complex numbers by b .

I f f i s a f u n c t i o n from A t o B a n d i f A l c A

then

f/A1 be t h e r e s t r i c t i o n o f f t o A1 a n d f ( A 1 ) w i l l d e n o t e t h e imaqe o f A 1 under f . B A w i l l be t h e s e t o f a l l maps from A t o B . I f BSA, i : B + A w i l l be r e s e r v e d f o r t h e i n c l u s i o n map. I f A i s any s e t c o n t a i n i n q something l i k e a " z e r o e l e m e n t " 0 , lN, w h i l e P * i s n o t d e f i n e d . A * w i l l d e n o t e A \ { O ) . S o e . g . =:"

$ 0 PREREQUISITES

2

" F i e l d " w i l l a l w a y s mean " s k e w - f i e l d " . The s y r f l n i e t r i c ( a l t e r n a t i n g ) g r o u p on n s y m b o l s w i l l be d e n o t e d by S n ( A n , r e s p e c t i v e l y ) . The i n t e g e r s modulo n w i l l be w r i t t e n a s 1, a n d r e p r e s e n n-1). t e d by Zn = { O y l ¶ . . . ,

We need an a b s t r a c t v e r s i o n o f " g e n e r a t e d o b j e c t s " :

A52A

0.1 DEFINITION i s c a l l e d a Moore-system (Dubreil-Dub r e i l - J a c o t i n ) on A i f (a) A E ~ . (b) i s closed w.r.t. a r b i t r a r y i n t e r s e c t i o n s . 0.2 PROPOSITION If [B]&

:=

fl M MEM M 'B

&

i s a M o o r e - s y s t e m on A a n d i f BGA t h e n i s t h e s m a l l e s t e l e m e n t o f /CC(w.r.t. 5 ) c o n -

t a i n i n g 6.

0 . 3 DEFINITION L e t t h e n o t a t i o n be a s a b o v e . ( a ) [ElA i s c a l l e d t h e element o f A which i s a e n e r a t e d

by 6.. i s called f i n i t e l y oenerable (f.q.) f i n i t e subset B o f A w i t h [B]& = A.

( b ) AE,~!

i f there i s a

A

0.4 D E F I N I T I O N A Moore-system i s called inductive i f H c o n t a i n s t h e union of e v e r y c h a i n of elements o f A .

0 . 5 EXAMPLES ( a ) Z A i s a n i n d u c t i v e Y o o r e - s y s t e m on A . ( b ) The s e t o f a l l s u b q r o u p s o f a q r o u p r i s a n i n d u c t i v e M o o r e - s y s t e m on r . ( c ) The s e t o f a l l c l o s e d s u b s e t s o f a t o p o l o o i c a l s p a c e T i s a M o o r e - s y s t e m on T w h i c h i s n o t i n d u c t i v e i n q e n e ral. We now turn t o c h a i n c o n d i t i o n s .

3

$ 0 PREREQUISITES

0.6 DEFINITION ----fulfill

A ( p a r t i a l l y ) ordered set

(A,<)

i s said t o

t h e minimum c o n d i t i o n i f e v e r y non-empty

subset

c o n t a i n s ( a t l e a s t ) one m i n i m a l e l e m e n t .

0 . 7 PROPOSITION F o r a p a r t i a l l y o r d e r e d s e t

(A,s), t h e f o l l o -

wing conditions are equivalent: ( a ) The minimum c o n d i t i o n . ( b ) The d e s c e n d i n o c h a i n c o n d i t i o n (DCC): e v e r y s t r i c t chain

a2 >

al

...

o f elements o f A terminates

a f t e r f i n i t e l y many s t e p s 2 a2 2

al

chain

...

3

(or,

nEIN :

equivalently, a n = an+l

=

f o r each

...

).

0 . 8 DEFINITION L i n e a r l y o r d e r e d s e t s w i t h t h e minimum c o n d i t i o n a r e c a l l e d we1 1 - o r d e r e d .

3.9 R E M A R K I n r e p l a c i n g

,

by 2

5

"maximum c o n d i t i o n " ,

we g e t t h e c o n c e p t s o f

"ascendins chain covdi t i o n "

( A.- C C )

a n d " i n v e r s e we1 1 - o r d e r " . E v e r y non-empty s u b s e t o f an o r d e r e d s e t w i t h t h e minimum ( m a x i m u m ) c o n d i t i o n h a s t h e same p r o p e r t y . be a Moore-system o n t h e s e t A .

0 . 1 0 PROPOSITION L e t

( h ! , ~f u) l f i l l s Ifk

t h e A C C =>

i s inductive,

Proof. Let

(4,~ h a v)e

i s n o t f.g. trary cess,

[{bl}lA

B2:=

=:B1

Now l e t

/cI

ment M o f

Ml=M2=M3=

A,

[{bl,b2}]Ac

BcA.

ME&

T a k e some a r b i -

B1=B2cB3=

...

of

a contradiction..

be i n d u c t i v e and suppose t h a t e v e r y e l e -

A ...

i s f.g..

U Mi E A b e g e n e r a t e d b y ( s a y ) icLN B u t t h e r e i s some kEIN w i t h t h e p r o p e r -

A?. L e t M:=

{al,

...,a n ) . {al,

Assume m o r e o v e r t h a t

i s a s t r i c t i n f i n i t e chain o f elements

of

ty t h a t

i s f.g..

$. M . T a k e some b2El'r\B1 M. C o n t i n u i n g t h i s p r o -

one g e t s an i n f i n i t e c h a i n

elements o f

A

t h e A C C a n d a s s u m e t h a t some

and g e n e r a t e d b y

blEB.

and f o r m

every element o f

t h e converse also holds.

...,a n I C M k ,

again a contradiction.

s o we g e t

Mk = M, w h i c h i s

4

80 PREREQUISITES

F i n a l l y , i t s h o u l d be remarked t h a t i n g e n e r a l we use small l e t t e r s f o r e l e m e n t s , c a p i t a l s f o r s e t s and s c r i p t l e t t e r s f o r c o l l e c t i o n s of s e t s .

PART I NEAR-RINGS FOR BEGINNERS

31

THE ELEMENTARY THEORY OF NEAR-RINGS

3 2 IDEAL THEORY

6

§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS

a ) F U N D A M E N T A L DEFINITIONS A N D PROPERTIES

N e a r - r i n g s a r e g e n e r a l i z e d r i n q s : a d d i t i o n n e e d s n o t be comm u t a t i v e and ( m o r e i m p o r t a n t ) o n l y o n e d i s t r i b u t i v e l a w i s p o s t u l ated. Examples o f n e a r - r i n g s a r e (a) the set M(T) o f a l l m a p p i n g s on a n ( a d d i t i v e l y w r i t t e n ) g r o u p r w i t h p o i n t w i s e a d d i t i o n and c o m p o s i t i o n ; ( b ) t h e polynomials R[X] ( R a commutative r i n g w i t h i d e n t i t y ) under a d d i t i o n a n d s u b s t i t u t i o n ; ( c ) an a r b i t r a r y a d d i t i v e l y w r i t t e n q r o u p w i t h z e r o m u l t i p l i c a tion; its w e l l a s many o t h e r s . S i m i l a r t o r i n g t h e o r y , " m o d u l e s o v e r a n e a r - r i n q PI" ( ' ' n e a r m o d u l e s " or ' I N - g r o u p s " ) w i l l be i n t r o d u c e d . They p l a y an i m portant r 6 l e in t h e theory o f near-rings. This s e c t i o n contains t h e b a s i c d e f i n i t i o n s , examples a n d p r o p e r t i e s o f n e a r - r i n q s and N - g r o u p s , a n d o f s u b s t r u c t u r e s a n d i d e a l - l i k e o b j e c t s i n these kinds o f a l q e b r a s . S i n c e n e a r - r i n g s a n d Pi-groups ( w i t h a z e r o - s y m m e t r i c N ) a r e s p e c i a l c l a s s e s o f R - q r o u p s ( q r o u p s w i t h mu1 t i p l e o p e r a t o r s ) , a w h o l e bunch o f c o n c e p t s a n d r e s u l t s i s " a p r i o r i " a v a i l a b l e . Compared w i t h r i n g t h e o r y , some c o m p l i c a t i o n s a r i s e : a n e l e m e n t m u l t i p l i e d by 0 i s n o t 0 i n g e n e r a l , t h e c h a r a c t e r i z a t i o n o f i d e a l s i s a l i t t l e b i t more c o m p l i c a t e d , i d e a l s a r e n o t a l w a y s s u b a l g e b r a s , and so o n .

7

l a Fundamental definitions and properties

1.)

N E A R - R I NGS

1.1 DEFINITION A n e a r - r i n o i s a s e t N t o q e t h e r w i t h t w o b i n a r y operations

and

"+I'

such t h a t

'I.''

(a)

(N,+)

i s a group (not necessarily abelian)

(b)

(N,.)

i s a semiqroup

(c)

( n + n ) . n 3 = nl.n3+n2.n3 1 2 tributive law").

1.2 REMARKS I n v i e w o f ( c ) , " r i g h t near-ri no". (cl)

("riqht dis-

n1,n2,n3€N:

V

Postulating

+ n ) = nl.n2+nl.n3 2 3 one g e t s " l e f t n e a r - r i n n s " .

n1,n2,n3cN:

instead o f (c),

one speaks more p r e c i s e l y o f a

nl.(n

The t h e o r y

runs completely p a r a l l e l i n both cases, o f course;

so one

can d e c i d e t o use j u s t one v e r s i o n . Although l e f t n e a r - r i n q s a r e more f r e q u e n t l y used i n t h e l i t e r a t u r e u p t o now,

we w i l l u s e r i g h t n e a r - r i n a s :

* T h e l e f t d i s t r i b u t i v e l a w i s i n some way u n n a t u r a l i n n e a r - r i n g s o f f u n c t i o n s ( t h e most i m p o r t a n t examples) and e s p e c i a l l y unmotivated i n n e a r - r i n q s o f p o l y n o m i a l s and formal

power s e r i e s .

*An a d - h o c - t e s t

d o n e b y t h e a u t h o r showed

80% o f t h e books i n which rinq-modules r 6 l e use left-modules,

t h a t about

p l a y an important

w h i c h a r e a l s o more f a m i l i a r f r o m

t h e t h e o r y o f v e c t o r spaces.

I n 1.18,

we w i l l s e e t h a t

c h o o s i n g l e f t N-groups f o r c e s one t o use r i q h t n e a r - r i n q s . *The r i g h t d i s t r i b u t i v e law i s e x c l u s i v e l y used i n papers

on t h e c l o s e l y r e l a t e d c o n c e p t o f c o m p o s i t i o n r i n q s (which were s y s t e m a t i c a l l y s t u d i e d p r i o r t o n e a r - r i n q s ! ) . 1.3

NOTATION N e a r - r i n g s w i l l u s u a l l y b e d e n o t e d b y N , N ' , N 1 s i m i l a r symbols. on w i l l write

We a b b r e v i a t e ( N , + , . )

by

N.

i n most cases be i n d i c a t e d by j u x t a p o s i t i o n ; nln2

instead o f

n1.n2.

or

Multiplicati-

s o we

I n dealinq w i t h general

n e a r - r i n g s t h s n e u t r a l e l e m e n t o f (N,+)

w i l l be denoted

b y O . I N 1 w i l l be t h e o r d e r o f t h e n e a r - r i n g N.

THE ELEMENTARY THEORY OF NEAR-RINGS

01

8

The t e r m " n e a r - r i n g "

will b e d e n o t e d a near-ring,

by

-

w i l l o f t e n be a b b r e v i a t e d b y " n r . " .

T h r o u g h o u t t h i s monograph,

n. I f

the class o f a l l near-rings

"N" a p p e a r s ,

it

will a l w a y s b e

without further notice.

1 . 4 EXAMPLES (a) Let

r

be an a d d i t i v e l y w r i t t e n ( b u t n o t n e c e s s a r i l y

abelian)

group w i t h zero

o ("omykron").

f o l l o w i n g s e t s o f mappings f r o m

into

r

are nr.'s

substitution:

under p o i n t w i s e a d d i t i o n and

M(r):= { f : r + r } =rr . Fio(r):= { f : r + r l f ( o )

r

Then t h e

= 01.

If:r+r(f i s constant].

Mc(r):=

M0c ( r ) : = { f 6 : r + r 1 6 E r A f 6 ( y ) = (Evidently,r , Mc(r) a n d M:(r)

{:

i ff y+o i y=o I . are isomorphic

groups). Mcont(r):=

{f:r+I'lf i s c o n t i n u o u s ]

(r

a topological

group). A n o t h e r r e l a t e d example i s

I

Mdiff(IR):= (f:IR+ IR f i s differentiable), while the r e a l f u n c t i o n s h a v i n g an i n d e f i n i t e i n t e g r a l do n o t form a near-ring (they are n o t closed w . r . t .

composi-

tion). For

SsEnd(r)

MS(r):=

{f:r+rl

define

WS~S:

where 6 These (b) Let

M(r)

=

E v i d e n t l y , MIid1(r)

f o s = sOf1.

and

M{6](r)

= Mo(r),

i s t h e z e r o endomorphism.

M S ( r ) ' s w i l l become v e r y i m p o r t a n t i n 9 4 .

r

be as above.

N e a r - r i n g s on

r

a r e e.g.

(I-,+,*)

with

yi-6 = o

for all

y,6EI';

(r,+,+)

with

y+6 = y

for all

y,6cr.

M o r e g e n e r a l l y , t a k e some s u b s e t A o f r a n d d e f i n e A T h e n (r,+.*,,) i s a n e a r - r i n g i f y'A6:= ii ff 66 E4 A '

{;

04A.

(Multiplications o f t h i s type are called the " t r i v i a l ones" i n Malone ( 3 ) .

because t h e y a r e e x a c t l y t h o s e

ones w h i c h can be d e f i n e d o n any group, group i n t o a near-rinq.)

makinq t h i s

9

l a Fundamental definitions and properties Now l e t G b e a f i x e d - p o i n t - f r e e

r

(i.e.

Choose a n y s u b s e t

{Bi(iE.Il

z e r o o r b i t s o f G on

automorphism group on

g ( y ) = y -=> (y = o v q = i d ) ) .

bgEGbyEr:

r

o f t h e s e t o f a l l non-

(Betsch c a l l e d these o r b i t s

" 1 - o r b i t s " and t h e o t h e r ones " 0 - o r b i t s " ) ; m o r e o v e r , choose any s e t o f r e p r e s e n t a t i v e s { b i ~ B i \ i E I l = : B and d e f i n e

yoB6 t o b e o i f

64

u

Bi

and t o b e

=g6(y)

i E I

i f 6 i s i n some Bi, w h e r e g 6 i s t h e u n i q u e a u t o m o r p h i s m i n G s e n d i n q bi i n t o 6 :

Then

(r,t,=B)

i s a n e a r - r i n g as one sees b y l o o k i n g

a t t h e d i f f e r e n t p o s s i b l e c a s e s . These t y p e s o f n r . ' s were i n t r o d l i c e d by F e r r e r o ( 5 ) and w i l l p r o v e u s e f u l i n t h e t h e o r y o f p l a n a r and i n t e g r a l n e a r - r i n g s . Anyhow,

o n e s e e s t h a t e v e r y g r o u p c a n b e made i n t o a

n e a r - r i n g i n v a r i o u s w a y s . See a l s o O l i v i e r ( 2 ) . ( c ) L e t V b e a v e c t o r s p a c e o v e r some f i e l d F. C a l l a s u s u a l a map V + V a n a f f i n e map i f i t i s t h e s u m o f a l i n e a r and a c o n s t a n t one. The s e t M a f f ( V ) of all I

a f f i n e maps i s a g a i n a n e a r - r i n g ( o p e r a t i o n s a s i n (a)). ( d ) L e t R be a c o m m u t a t i v e r i n g w i t h i d e n t i t y .

are

(R[x].t,O)

and

(R[[~]],t,o),

where

Near-rings O

means

s u b s t i t u t i o n . Another n e a r - r i n g i s formed by t h e s e t P ( R ) o f a l l polynomial

f u n c t i o n s on R w i t h t h e opera-

t i o n s as i n ( a ) ( s e e 5 7 d ) ) .

( e ) O f course, every r i n g i s a near-rinq. 1.5 P R O P O S I T I O N

bn,n'EN:

Proof: as for r i n g s .

On = 0 A ( - n ) n '

= -nn'.

51

10

THE ELEMENTARY THEORY OF NEAR-RINGS

1 . 6 REMARK A s most o f o u r examples s h o w , n O = 0 and n ( - n l ) = -nn' d o n o t h o l d i n g e n e r a l . F o r i n s t a n c e , i n M(r) f o O = 0 means t h a t " f g o e s t h r o u g h t h e o r i g i n " a n d f o ( - f ' ) = - f o f ' means t h a t ' I f i s a n o d d f u n c t i o n " .

=

I _

One t h e r e f o r e d e f i n e s f o r a n e a r - r i n g N : 1 . 7 DEFINITION rt ( a ) N o : = I n E N l n O = 0 1 i s c a l l e d t h e z e r o - s y m m e t r i c p a-

of N. ( n E N l n O = n ) = (ncNlbl the constant p a r t of N .

( b ) Nc:=

N o and N C a r e i t s e l f n e a r - r i n g s 1.8 EXAMPLES

(M(r))o

= M0(r);

n'EN:

nn' = n)

i s called

(see 1.22 ( a ) ) . (M(I'))c = Mc(r).

1 . 9 DEFINITION N E ? ~ i s c a l l e d z e r o s y m m e t r i c

(constant) i f

N = No

( N = Nc, respectively). stand f o r t h e c l a s s e s of a l l zerosymmetric (constant) near-rings.

11,

(qc)

37,

a r e ( n o t a t i o n as i n 1 . 4 ) M o ( r ) , MS(r) i f ~ E S , every r i n g . M C ( r ) c n c , w h i l e M(r) o r R [ x ] a r e n e i t h e r i n 9, n o r i n 3,. C f . A d l e r ( I ) , p . 6 1 0 .

1.10 E X A M P L E S E l e m e n t s o f

1.11 DEFINITIONS The f o l l o w i n g c o n c e p t s a r e d e f i n e d a s i n r i n g

theory: l e f t ( r i g h t , - ) i d e n t i t i e s , l e f t ( r i g h t , - ) invertible elements, l e f t ( r i g h t , - ) cancellable elements, l e f t ( r i g h t , - ) z e r o d i v i s o r s , idempotent and n i l p o t e n t e l e m e n t s . M o r e o v e r , c a l l dcN d i s t r i b u t i v e i f n,n'EN: d(n+n') = dntdn'. Let N d : = { d E N / d i s d i s t r i butive}. Let 9,be t h e c l a s s o f a l l n e a r - r i n g s w i t h i d e n t i t y ( u s u a l l y d e n o t e d by 1 ) .

v

l a Fundamental definitions and properties

11

1 . 1 2 EXAMPLES The i d e n t i t y f u n c t i o n s e r v e s a s a n i d e n t i t y i n M(r) a n d Mo(r). I n v e r t i b l e i n t h e s e n e a r - r i n g s a r e e x a c t l y t h e b i j e c t i v e f u n c t i o n s . 2 x i s an example o f a n i l p o t e n t eleinent i n Z4[x]. Cartan ( 1 ) c h a r a c t e L i z e d a l l i n v e r t i b l e elements i n ( F [ [ x ] ! ) o , F a field: 1 aixi i=l h a s an i n v e r s e i n ( F [ [ X ] ] ) ~ ( w . r . t . 0 ) i f f al=/=O. I f N = M a f f ( V ) t h e n N d = HomF(V,V). I f N i s a r i n g t h e n N = N d . I t i s c l e a r t h a t NdcNo. I f N h a s a n i d e n t i t y 1 then l € N 0 . The n e x t a s s e r t i o n s t e m s f r o m B e r m a n - S i l v e r m a n ( 1 ) . G e n e r a l i z a t i o n s c a n be f o u n d i n K a a r l i ( 4 ) , L y o n s (4), M i r o n - S t e f a n e s c u ( I ) , R a m a k o t a i a h - R e d d y ( I ) , Zand ( l ) , ( Z ) . 1 . 1 3 PROPOSITION I f eEN i s i d e m p o t e n t t h e n we g e t J " P e i r c e decomposition": n = x 0+x 1' ncN ~ x o E I : x E N I x e = O ly x l € N e : Taking e = 0 one g e t s n = no+n C tf n E N % " E N O 3 n C € N c : Hence ( N , + ) = ( N o , + ) + ( N c , + ) a n d N o n N c = lo!. P r o o f . n = ( n - n e ) + n e w i l l do t h e d e c o m p o s i t i o n j o b . I f n = x o t x l = x 0' + x i w i t h x o , x ; ~ { x ~ N [ x e = O ) a n d x1 = y l e , = y i e ENe t h e n n e = x e = x ' e . B u t 1 1 x l e = y 1e e = y e = x 1 a n d x i e = I t follows t h a t x1 = a n d x 0 = x;.

.

xi

xi

xi.

1 . 1 4 DEFINITIONS Let N be a n e a r - r i n g . I f ( N , + ) i s a b e l i a n we c a l l N an a b e l i a n n e a r - r i n a ; i f ( N , . ) i s c o m m u t a t i v e we c a l l N i t s e l f a c o m m u t a t i v e n e a r r i n g . I f N = N d , N i s s a i d t o be d i s t r i b u t i v e . I f a l l nonz e r o e l e m e n t s o f N a r e l e f t ( r i g h t , - ) c a n c e l l a b l e , we s a y t h a t N f u l f i l l s the l e f t ( r i q h t , - ) c a n c e l l a t i o n law. N i s i n t e g r a l i f N h a s no n o n - z e r o d i v i s o r s o f z e r o . I f (N*=Ff\{OI,.) i s a g r o u p , N i s c a l l e d a n e a r - f i e l d ( a b b r e v i a t i o n : 6) A n. e a r - r i n g which i s n o t a r i n ? w i l l be r e f e r r e d t o a s a n o n - r i n q . S i m i l a r l y , a n o n - f i e l d i s a n f . w h i c h i s no f i e l d . A n e a r - r i n g w i t h t h e p r o p e r t y t h a t Nd generates (N,+) i s called a d i s t r i b u t i v e l y qenerated near-rinq ( d q n r . ) .

8 1 T H E E L E M E N T A R Y T H E O R Y OF N E A R - R I N G S

12

M(r)

1.15 EXAMPLES ( N o t a t i o n a s i n 1.4) abelian.

(r,+,o)

i s abelian iff

r

i s

s e r v e s as a n example o f a c o m m u t a t i v e

and d i s t r i b u t i v e n o n - r i n g ,

(r,+,*)

while

i s integral.

(r,t,mB) i s i n t e g r a l i f f a l l

I n the language o f 1.4(b),

non-zero o r b i t s are "1-orbits".

(Z2,t)

with

0 - 0 = 0.1

= 0,

1.0 = 1.1 = 1 i s a n f . A l l o t h e r n f ' s a r e z e r o - s y m m e t r i c . L e t r b e a g r o u p . I f r i s n o t a b e l i a n , t h e sum o f t w o endomorphisms i s n o t n e c e s s a r i l y an endomorphism any more. B u t t h e s e t o f a l l ( f i n i t e ) sums a n d d i f f e r q n c e s o f e n d o morphisms of r i s c l o s e d under a d d i t i o n and c o m p o s i t i o n and forms a dgnr.

E(r).

1.16 H I S T O R I C REMARKS N e a r - f i e l d s w e r e t h e f i r s t n r ' s c o n s i d e r e d i n the literature.

I n 1905, Dickson (1),(2)

changed t h e

m u l t i p l i c a t i o n i n a f i e l d i n o r d e r t o g e t examples o f "one-sided d i s t r i b u t i v e f i e l d s "

(= nf's)

showing t h a t t h e

second d i s t r i b u t i v e l a w does n o t f o l l o w f r o m t h e r e m a i n i n g axioms f o r a ( s k e w - ) f i e l d . I'

H i s "changed f i e l d s " a r e c a l l e d

D i c k s o n n f ' s " ( s e e $8 ( a ) 4 )).

A c o u p l e o f y e a r s l a t e r Veblen and Wedderburn

started

t o use n f ' s t o c o o r d i n a t i z e c e r t a i n k i n d s o f geometric planes

.

I n 1936, Zassenhaus (1) d e t e r m i n e d a l l f i n i t e n f ' s : have o r d e r

p n ( P E P , nEIN)

cases) Dickson n f ' s .

they

and a r e ( u p t o 7 e x c e p t i o n a l

I n ( 2 ) he showed u p t h e c o n n e c t i o n

between n f ' s and f i x e d - p o i n t

f r e e permutation groups.

O r e (l) F, urtwangler-Taussky

(1) and T a u s s k y (1) s t a r t e d

a x i o m a t i c c o n s i d e r a t i o n s i n t h e t h i r t i e s f o r w h a t we n o w c a l l near-rings.

A f i r s t name f o r t h e s e s t r u c t u r e s was p r o p o s e d i n 1 9 3 8 b y G I i e l a n d t ( 1 ) : "Stamm"

( = t r i b e ) ("stem"

i s s t i l l used

i n t h e I t a l i a n l i t e r a t u r e ) . W i e l a n d t a l s o announced s t r u c t u r e - t h e o r e t i c r e s u l t s i n t h i s note. The f i r s t o n e s t o u s e t h e n a m e l ' n e a r - r i n g " w e r e

i n 1936 and B l a c k e t t and P . J o r d a n

Zassenhaus

i n 1950.

I n 1 9 3 2 F i t t i n g (1) c h a r a c t e r i z e d t h o s e a u t o m o r p h i s m s o f ( n o n - a b e l i a n ) g r o u p s , w h o s e sum i s a n a u t o m o r p h i s m , thereby i m p l i c i t e l y s t a r t i n a t o consider dgnr's.

too,

13

l a Fundamental definitions and properties

Finally,

the fifties

b r o u g h t t h e s t a r t of a r a p i d

development o f t h e t h e o r y o f n e a r - r i n g s . Now we a r e g o i n g t o d e f i n e t h e a n a l o g u e o f t h e c o n c e p t o f a module i n r i n g t h e o r y :

2.) 1.17

c e r t a i n o p e r a t o r groups.

N-GROUPS DEFINITIONS L e t

.

(rs+)

be a g r o u p w i t h z e r o o and l e t

NE~.

Nxr-r ( r , u ) i s c a l l e d an N-group ( n 3Y )+nY ("near-module o v e r N" ( b u t c f . t h e d i f f e r e n t meaning e.g.

Let

p:

(1) ) ) i f

i n Karzel-Pieper

V

W

YEr

: ( n t n ' ) y = ny+n'y A (nn')y

n,n'EN

Nr f o r

t h e N-group

t h e c l a s s o f a l l N-groups.

To s i m p l i f y

If t h e m e a n i n g o f p i s c l e a r we w r i t e above.

Let

Ng be Nr

the notation,

further notice.

= n(n'y).

stands f o r N-groups throughout,

without

See a l s o K u z ' m i n ( 1 ) .

1.18 EXAMPLES ( a ) l e t N be a n r . Then an N-group,

u:

NxN-N (n,n')+nn'

makes

(N,+)

into

denoted by

( b ) Each ( l e f t ) module M o v e r a r i n g R i s an R - g r o u p . (c) Let

r

with

b e a g r o u p . Then p:

r

i s an M ( r ) - g r o u p

M(r)xr+r (fsY)

+

1.19 P R O P O S I T I O N T a k e Nr E

oY

f(Y)

Ng.

= 0;

(a)

WyEr;

(b)

b(ycr

b'ncN:

(c)

QnEN,:

no =

(d)

VyEr

(-n)y

= -ny;

0;

bnENc: ny = n o . ..

( a ) a n d (b): a s f o r ( r i n g - ) m o d u l e s . ( c ) : n o = nOo = 00 = o . ( d ) : ny = nOy = n o .

Proof.

M(T) I-

01

14

1.20

THE ELEMENTARY THEORY OF NEAR-RINGS

DEFINITION NT

W

E

Ng i s

called unitary i f NE

n,

and

ycr: iy = y.

n,

mc,

no, a n d a l l Ng a r e v a r i e t i e s i n t h e s e n s e o f Since u n i v e r s a l a l g e b r a i t makes s e n s e t o s p e a k a b o u t a l o t o f t h i n g s ( s e e a l s o Prehn ( 1 ) - ( 3 ) ) : 3 . ) SUBSTRUCTURES

1.21 DEFINITION ( a ) A subgroup M o f a nr. N w i t h subnear-rinq of N ( n o t a t i o n :

M.Mr M

( b ) A s u b g r 0 u p . A o f Nr w i t h N A c A +) N - s u b g r o u p of r (ASNr). 1 . 2 2 EXAMPLES ( a ) No and N c a r e s u b n e a r - r i n g s is a from 1.13 t h a t ( N , + ) i t s s u b g r o u p s ( N o , + ) and f o r the c o n v e r s e problem o f o u t o f a z e r o - s y m m e t r i c and

is called a

MSN).

i s s a i d t o be a n

o f N . Hence i t f a l l o w s split extension o f (Nc,+). See P i l z ( 9 ) , ( 1 0 ) constructing oear-rings a constant one.

( b ) I f N r i s a ( r i n g - ) module t h e n t h e N - s u b g r o u p s a r e j u s t t h e submodules o f r . L a t e r on we w i l l s e e t h a t t h e s u b n e a r - r i n g s o f t h e M ( r ) ' s a r e i n a c e r t a i n s e n s e a l r e a d y a l l n e a r - r i n g s . We know a l r e a d y o n e p r o c e d u r e t o g e t s u b n e a r - r i n g s o f M(r): t h e M S ( r ) ' s o f 1.4. Two more m e t h o d s a r e : 1.23 EXAMPLES ( a ) Take a s u b g r o u p A o f r . a subnear-rinq of M(r).

MA(r ): = { f c M ( r ) I f ( A )

( b ) T a k e a normal s u b g r o u p A o f r . Mr,*(T): = ( f c M ( T ) ( t / YET: f ( y + A ) C f ( y ) + A l s u b n e a r - r i n g o f M(r) ( c f . B e t s c h ( 3 ) ) . + ) T h e term "N-subRroup

of N" refers t o

".

E

i s a

A)

i s

15

l a Fundamental definitions and properties

1.24 REMARK W i e l a n d t ( 3 )

proposed a c o n s t r u c t i o n method f o r

o f M ( T ) w h i c h g i v e s t h e 3 k i n d s o f scrbnear-

subnear-rings

r i n g s m e n t i o n e d above as s p e c i a l cases. The m e t h o d i s a s f o l l o w s :

ra

Take a n y c a r d i n a l number a , f o r m t h e d i r e c t p r o d u c t

ra .

and a subgroup A o f t o be Let

Ma,A(r):

( a ) MA(r)

Each

fEM(r)

can be c o n s i d e r e d

i f i t i s d e f i n e d component-wise.

cM(ra)

= IfEM(r)[f(A)

E

A1

5

M(r).

Then

= Ml,A(r)

4 . ) HOMOMORPHISMS A N D I D E A L - L I K E SUBSETS 1 . 2 5 D E F I N I T I O N L e t FL,N'

N+N'

( a ) h:

m,nEN: (b) h:

NryNr'EN9.

i s c a l l e d a ( n e a r - r i n g ) homomorphism h(m+n)

Nr+Nr'

'j y , 6 E r

b e ~ r a)n d

h(m)

+

h ( n ) A h(mn) = h ( m ) h ( n ) .

i s c a l l e d an N-homomorphism

b

ncN:

h(y+6) = h(y)

+

if

h(6) A h(ny) = nh(y).

T h e r e seems t o b e n o n e e d f o r e x p l i c i t

definitions of

nr.-monomorphisms

HomN(r,r'),

K e r h,

I m h,

(Nw N ' ) ,

if

Hom(N,N'),

r 2 Nl"y

I f t h e r e e x i s t s a r n o n o m o r p h i s m Nw N '

and so on.

we s a y t h a t N i s e n b e d d a b l e i n N '

and w r i t e N C

N'. A s i m i l a r

c o n v e n t i o n a p p l i e s t o N-groups. 1 . 2 6 EXAMPLE

For a l l

y ~ ~ hy: r : N+r

E

HomN(N,r)

n+ny 1.27 DEFINITION L e t N

E

a~ n d

NrENq.

( a ) A normal subgroup I o f ( I 9 N) i f a ) IN E I

B)

tl

(N,t)

i s c a l l e d i d e a l of

N

n,n'~N WiEI: n(n'+i)-nn'EI. (N,+) with a ) are called r i q h t Normal subgroups R o f

01

16

T H E E L E M E N T A R Y T H E O R Y OF N E A R - R I N G S

i d e a l s o f N ( R 4, N ) , w h i l e normal s u b g r o u p s L o f ( N , t ) w i t h B ) a r e s a i d t o be l e f t i d e a l s ( L N). ( b ) A n o r m a l s u b g r o u p A o f r i s c a l l e d i d e a l o f Nr ( A dN r ) i f W Y E r W ~ E A W n E N : n(y+6)-nyEA O t h e r n a m e s : N - k e r n e l o r s u b m o d u l e ( c f . 1.33!). The t e r m " i d e a l " i s m o t i v a t e d by ( K u r o s h ) a n d i s v e r y handy i n f o r m u l a t i n g s i m u l t a n e o u s s t a t e m e n t s a b o u t N - g r o u p s arid n e a r - r i n g s

.

.

".

1 . 2 8 R E M A R K S The l e f t i d e a l s o f N c o i n c i d e w i t h t h e i d e a l s o f M o r e o v e r , o n e e a s i l y s e e s t h a t a s u b g r o u p I o f N (A o f r ) i s an i d e a l i f f n1 E n i (mod I ) n 2 E n; (mod I ) => n l t n 2 E n i + n i (mod I ) A n 1 n 2 z n i n h ( m o d I ) y 1 E y i (mod A ) ( y 2 z y i (mod A) A

WnEN:

nyl

Z

=>

y 1 t y 2 z y i t y i (mod A ) A

nyi(mod A ) ,

respectively).

S o : (mod I o r mod A) i s a " c o n g r u e n c e r e l a t i o n " ( c f . ( G r a t z e r ) ) i f I ( A ) i s a n i d e a l . I f I 9 N and I N, we w r i t e I 4 N , e t c . I n 1 . 2 7 , ( a ) B ) a n d ( b ) c a n a l s o be w r i t t e n a s vn,n'EN WisI : n ( i t n ' ) - n n ' ~ I and iycr W ~ E A VncN : n(&+y)-nycA

.

F a c t o r n r ' s N/I ( I sl N) a n d f a c t o r 14-aroups r / A ( A gN r ) a r e d e f i n e d a s u s u a l ( c f . a n y book on u n i v e r s a l a l g e b r a ) . I f L dQ N , t h e n N / L i s m e a n t i n t h e s e n s e o f N-groups. C l e a r l y I01 and N a r e i d e a l s o f N a s w e l l a s { o l and r a r e ones o f N T . These i d e a l s a r e c a l l e d t h e t r i v i a l i d e a l s .

17

l a Fundamental definitions and properties

1 . 2 9 THEOREM ("Homomorphism t h e o r e m " ) . (a) If

I 9N

t h e n t h e c a n o n i c a l map So N / I

nc-epimorphism. (b) Conversely, Ker h 9 N

if

h:

NU"

N/Ker

and

TI:

N+N/I

i s a

i s a h o m o m o r p h i c i m a g e o f N. i s an epimorphism t h e n % =

N'

.

The c o r r e s p o n d i n g s t a t e m e n t s h o l d f o r N-groups. The p r o o f i s analogous t o t h e one f o r g r o u p s , r i n q s o r universal algebras,

and hence o m i t t e d .

So i d e a l s a r e j u s t t h e k e r n e l s o f ( N - )

homomorphisms.

As u s u a l f o r " s o p h i s t i c a t e d " a l g e b r a i c s t r u c t u r e s we g e t w i t h t h e usual p r o o f :

1 . 3 0 THEOREM ( s o - c a l l e d " Z n d i s o m o r p h i s m t h e o r e m " ) N+N' b e a n e p i m o r p h i s m . T h e n h i n d u c e s a 1-1L e t h: c o r r e s p o n d e 11 c e b e t w e e n t h e subnear-rings

(ideal s) o f

N containing Ker h the subnear-rings

and (ideals)

N) * h(A):

o f N' by

A(E

Moreover,

f o r a l l i d e a l s I o f N containing Ker h

we g e t

N/I 2 h(N)/h(I).

If

n : N+N/I

i s t h e canonical epimorphism,

we t h e r e f o r e

get f o r a l l ideals J o f N containing I

Again t h e analogous statements h o l d f o r N-groups.

Observe

i n t h i s c a s e t h a t f o r t h e l a s t f o r m u l a we h a v e t o a s s u m e t h a t J i s a l s o a n N - g r o u p t o make (cf.

1.33,

1.34).

J/Im e a n i n c r f u l

01

18

THE ELEMENTARY THEORY OF NEAR-RINGS

1 . 3 1 DEFINITION A s u b n e a r - r i n g M o f N i s c a l l e d i n v a r i a n t i f MNZM and NMEM .

I n v a r i a n t s u b n e a r - r i n q s and i d e a l s c o i n c i d e i n r i n g s , b u t n o t i n n e a r - r i ngs: 1 . 3 2 PROPOSITION

( a ) N o 4, N , b u t n o t g e n e r a l l y

N o It N .

( b ) N, i s a n i n v a r i a n t s u b n e a r - r i n g o f N , b u t i n g e n e r a l n e i t h e r a r i g h t nor a l e f t i d e a l . and Proof. ( a ) No i s a l e f t ideal: f o r a l l n , n ' E N we h a v e ( n + n o - n ) O = nOtnoO-n0 = 0 , so noEN 0 = n(n'O+noO)-nn'0 = n+no-ncNo , a n d [ ( n ( n ' + n o ) - n n ' ) ] O 0 , h e n c e n ( n ' + n J - n n ' ~ N 0' N o i s n o t n e c e s s a r i l y an i d e a l : N:= M(1R) , i d I R c N0 = l = l & M o ( IR) = Mo(IR) , 1: IR+ IR E M ( I R ) , b u t i d ox + l ( b ) N, i s a n i n v a r i a n t s u b n e a r - r i n g :

-

.

tj

ncEN

= ncn

C

,

( n n c ) O = n n C and ( n c n ) O = n c O = w h i c h i m p l i e s t h a t nnccNc a n d n c n E N c . :

nEN

N, i s n o t a l e f t o r r i g h t i d e a l i n g e n e r a l , s i n c e N, i s n o t a l w a y s a normal s u b g r o u p o f ( N , t ) : Take a n o n - a b e l i a n group r a n d y , 6 ~ r w i t h y+6 6+y r + r E M C ( r ) . Now ( i d t f Y - i d ) ( o ) = y,

+

-

.

( i d t f y - i d ) ( 6 ) = 6 + y - 6 $; y i m p l y i n g t h a t id+fy-id&Mc(r) So Mc(r) i s n o r m a l i f f r i s a b e l i a n .

but

.

1.33 R E M A R K I n g e n e r a l t h e r e i s no d i r e c t c o n n e c t i o n b e t w e e n N - s u b g r o u p s a n d l e f t i d e a l s , a s we h a v e seen a b o v e . T h i s i s t h e r e a s o n f o r a v o i d i n g t h e terms " n e a r - m o d u l e s " a n d " s u b m o d u l e s " : s u b m o d u l e s would n o t be n e a r - m o d u l e s i l l g e n e r a l , f o r i d e a l s o f N-groups a r e n o t n e c e s s a r i l y N-subgroups. So i n g e n e r a l N-groups a r e not"R-groups" ("groups w i t h m u l t i p l e o p e r a t o r s " ) i n the sense o f (Kurosh) o r ( H i g g i n s ) . T h i s does n o t happen f o r z e r o symmetric n e a r - r i n g s ( s e e a l s o Prehn ( 1 ) - ( 3 ) ) :

19

l a Fundamental definitions and properties

1.34 PROPOSITION

(a) L 9, N

NoL

->

E

L

(b) N = No <=+ each left ideal of N is an N-subgroup o f N. (c) N = No => (A dN r Proof.(a)

N

L 9,

(b) ->: <=:

=+

->

A sN

~ L E L

by (a) {O} 9, L

r ) for

all

r E Nij+

w n o ~ N o : not = no(OtL)-noOEL.

10) SN N => NO = {o) ->

->

N = No

.

(c) is settled similarly. 1.35 PROPOSITION

(a)

NY SN

yENr:

(b)wAsNI':

r.

N o = NC o F A .

is the smallest under all N-subgroups o f F,r. Throughout t h i s monograph w e will w r i t e

So No

NO Of

course, YET:

N =

R = NCy

.

=

NCo = : R

No implies R = { o l . Also, N, .

. By l.l9(d),

1.36 DEFINITION

(a) N(Nr) (b)

i s simple: <->

Nr is called N-simple: e x c e p t R and r

N(NT) <-=>

has no non-trivial ideals.

Nr has no N-subgroups

(cf. 1.35).

is simple then all (N-) homomorphic 1.37 PROPOSITION If N(Nr) images a r e (N-) isomorphic either to (0) o r t o N({oI o r r ) . Proof: by 1.29.

8 1 THE ELEMENTARY THEORY OF NEAR-RINGS

20

1 . 3 8 EXAMPLES ( a ) I n 5 7 we w i l l s e e t h a t s i m p l e nr,'s

(7.30,

M(r) ( I r l

> 2) and

Mo(r) are

7.33).

( b ) S e e B l a c k e t t ( 4 ) f o r some m o r e e x a m p l e s o f s i m p l e n r ! s o f real functions. (c) If

No

N =

then N - s i m p l i c i t y implies (by 1.34(c))

s i m p l i c i t y f o r each

NI"Nq.

S i n c e {O) i s a l w a y s m i n i m a l i n t h e s e t o f a l l i d e a l s o f N ,

we d e f i n e m o r e i n t e r e s t i n g o n e s t o b e m i n i m a l : 1.39 DEFINITION A m i n i m a l i d e a l o f N i s a n i d e a l w h i c h i s minimal i n t h e s e t o f a l l non-zero ideals. one d e f i n e s minimal r i c l h t i d e a l s , (minimal under a l l N-subgroups Dually,

l e f t i d e a l s , N-subgroups

.

$. D), e t c .

one g e t s t h e concepts o f maximal i d e a l s e t c .

%r

1 . 4 0 PROPOSITION I d N i s m a x i m a l i n N ( A i f f N / I (I'/A) i s simple. Proof:

Similarly,

.

i s maximal i n

1.30.

Near-rings i n which every (one- o r two-sided) maximal a r e s t u d i e d i n F e r r e r o - C o t t i

ideal # (01 i s

- Rinaldi (1),(2).

5 . ) ANNIHILATORS

We w i l l n e e d t h e " n o e t h e r i a n q u o t i e n t s " q u i t e f r e q u e n t l y : 1 . 4 1 DEFINITION L e t (A1

: A2):

= {nENlnA2

Abbreviations: (A:&),

(o:A)

be subsets of

A1,~*

({6)

e

All

.

Nr

: A2) = : (6 : A2)%

E

Ng .

similarly for

(&:A).

i s c a l l e d t h e a n n i h i l a t o r o f A.

I f n e c e s s a r y , we i n d i c a t e t h e n r . N i n v o l v e d b y w r i t i n g ( A 1 : A2)N.

r)

21

l a Fundamental definitions and properties

1 . 4 2 PROPOSITION N o t a t i o n a s a b o v e . I f Al i s a subgroup (normal subgroup, N-subgroup, ( A , : A 2 ) i n ". o f N r , t h e same a p p l i e s t o

ideal)

The p r o o f i s e a s y and t h e r e f o r e o m i t t e d . 1.43 COROLLARY (a)

(b)

b b

:

YEr

A

( 0 : ~ A, )

sNr : ( o : A ) A N

1 . 4 4 PROPOSITION Let

(Ai:A) =

(.(I

n

hi

(

E

A,Ai

1EI

iE1

<==> n

N

: A)

Ai

: A)

(isI) and

be s u b s e t s of Nr. Then (Ai:A) 5 ( A i :A). iE1 is1

u

u

and s i m i l a r l y for the u n i o n .

iEI

1 . 4 5 P R O P O S I I X Let A b e a s u b s e t o f

(a)

( 0 : ~ = )

n

Nr

E

N9.

(o:a)

~ E A

(b)

Nr

'2,

N r b -2

( 0 : r ) = (0:r')

Proof: straightforward. Consider t h e h y ' s o f 1.26. 1 . 4 6 PROPOSITION Ker h y = ( o : y )

,

so

Ny

c

N/(o:y).

P r o o f : homomorphism t h e o r e m .

Nr i

1 . 4 7 DEFINITION 1 . 4 8 PROPOSITION

Nr

s called faithful i f

f a i t h f u l =>.

N

4

phism

h: N

n

with

+

M(r)

+

fn

= {O?.

M(r).

P r o o f : C o n s i d e r f o r e a c h neN t h e map Then

(o:r)

f,,:r+r

.

fncM(r). r*ny t u r n s o u t t o b e a n e a r - r i n g homomot-

Ker h = { n E N J f n = 6 ) = { n E N ( n E ( o : r ) ) = {Ol.

So h i s a n e m b e d d i n g map.

31

22

THE ELEMENTARY THEORY OF NEAR-RINGS

Nr

1 . 4 9 PROPOSITION Let

(a) If

r

be f a i t h f u l .

i s a b e l i a n then s o i s N .

v

then

.

(b) If

tj

Proof:

( a ) by 1 . 4 8 a n d ( b ) b y a s t r a i g h t f o r w a r d c a l c u l a t i o n .

nEN

: n ( y t 6 ) = ny+n6

y,6Er

nENd

More g e n e r a l l y o n e c a n p r o v e t h a t , i f N r i s f a i t h f u l , e v e r y " i d e n t i t y which h o l d s i n r " ( c f . ( G r a t z e r ) ) a l s o " h o l d s i n N " .

1 . 5 0 PROPOSITION Let (by 1.48). ( a ) R = Io) <->

(b) R =

r

f a i t h f u l . We a s s u m e t h a t

= (0) <->

N, N,

<->

Nr be

% -N

= M c ( r ) <->

Then So

t a k e some

3nccNc : n c ( o ) =

N,

nc(y) =

.

meMc(r).

m ( o ) =:yo.

yo.

: m(y) = m ( o ) = y o = n c ( o ) = n c ( y ) , a n d N, = Mc(r).

vyEr

m = nc

If

r,

=

M(r)

N c'

.

R =

F

NEn,,;

P r o o f : ( a ) I f R = { o l then V YEr V nceNc: = n c ( o ) = o = a ( y ) , s o n c = 6. I f N, = {Ol t h e n N = No€%, I f NEBO t h e n R = ( 0 1 by 1 . 1 9 ( c ) (b) If

N

Mc(r) t h e n t h e map

h:

i s an

Nc+T

nc+nc

hence

(0)

N-isomorphism.

zN

r N, Finally,if an a r b i t r a r y yEr

.

by some N - i s o m o r p h i s m h , t a k e h ( y ) =: ncEN,. Then h ( y ) =

= nC = ncnc = nch(y) = h ( n c ( y ) ) = h ( n c ( o ) ) .

So

y = nc(o)

E

n and

R =

r.

See a l s o F e r r e r o - C o t t i ( 7 ) a n d S c o t t ( 1 5 ) .

23

l a Fundamental definitions and properties

6.) GENERATED OBJECTS

1 . 5 1 PROPOSITION ( a ) The s e t s o f a l l i d e a l s ( r i g h t i d e a l s ,

N-subgroups,

l e f t ideals,

invariant subnear-rings)

form inductive

M o o r e - s y s t e m s o n N.

( b ) The s e t s o f a l l i d e a l s (N-subgroups) f o r m i n d u c t i v e Moore-systems on r .

of

an N-group

(...)

Hence i t makes s e n s e t o speak a b o u t t h e " i d e a l

Nr

generated

by a s u b s e t " . 1 . 5 2 PROPOSITION ( S c o t t ( 6 ) ) L e t

RcN

with

RNER

.

Then t h e

l e f t i d e a l LR generated b y R i s an i d e a l . P r o o f : RN e R c L R b y 1.42,

that

,

so

R

LR c (LR:N)

E

.

(LR:N).

Since

Therefore

( L R : N ) qQ N

LRN c L R

showing

L R A N.

See a l s o 2 . 1 6

and 9 . 1 7 4 .

1 . 5 3 THEOREM

(Beidleman (1))

( a ) If N i s f g . ( 0 . 3 ) a s a n i d e a l ( c f . Van d e r W a l t ( 2 ) , ( 3 ) ) ( e . 9 . i f N ~ 3 , t)h e n e a c h i d e a l ( r i g h t i d e a l , M - s u b g r o u p ) different

f r o m Pi i s c o n t a i n e d i n a m a x i m a l o n e .

(b) I f NT i s f.g.

a s i d e a l w i t h N €71, t h e n e v e r y p r o p e r i d e a l N r i s c o n t a i n e d i n a maximal o n e .

ideal ( N - s u b g r o u p ) o f

Proof ( f o r i d e a l s I o f N ) . L e t N b e g e n e r a t e d b y ( s a y ) z:= IIIIaN). T a k e some c h a i n 1 1 C 1 2 E ... xl, ...,x k . o f e l e m e n t s o f $. I : = IJ In9 N by 1 . 5 1 ( a ) . nc IN If

I=N

then a l l o f SEIN w i t h x l ,

some

xl, ...,x k c I . ...,x k € 15 ' B u t

Hence t h e r e i s then

I s = N,

a contradiction. So (r,Z)f u l f i l l s t h e h y p o t h e s i s o f Z o r n ' s Lemma (unless N = (01, a t r i v i a l c a s e ) a n d c o n s e q u e n t l y c o n t a i n s a maximal element.

If

N

E??~,

one proceeds as i n r i n g t h e o r y .

24

$ 1 T H E E L E M E N T A R Y THEORY OF N E A R - R I N G S

b ) CONSTRUCTIONS

1.) PRODUCTS, DIRECT SUMS A N D S U B D I R E C T P R O D U C T S

-

F o r 1.54

1.62 c f . each book on g r o u p s ,

a l g e b r a . We c i t e e . g . 1.54

DEFINITION l e t

X

iE1 and

(Ni)icI

be a f a m i l y o f n e a r - r i n g s .

w i t h t h e component-wise d e f i n e d o p e r a t i o n s

Ni "

.

'I

(iEI).

e l e m e n t s w i t h a l l components

8

Ni

equal t o zero,

'I+"

of t h e

iE1

1.55 D E F I N I T I O N The s u b n e a r - r i n g o f

INo-

ll Ni

i s called the d i r e c t product

n e a r - r i n g s Ni

E

rings o r universal

from (Gratzer).

-

ll Ni

ieI

c o n s i s t i n g of

those

e x c e p t a f i n i t e number

i s c a l l e d t h e ( e x t e r n a l ) d i r e c t sum

o f t h e Nils.

ic1

ll N.i

More g e n e r a l l y ,

every subnear-ring N o f

p r o j e c t i o n maps

xi ( ~ E a r e Is) u r j e c t i v e ( i n o t h e r words, : ni i s t h e i - t h c o m p o n e n t o f some e l e m e n t

~ E I \ niENi

where a l l

ieI

o f N) i s c a l l e d a s u b d i r e c t p r o d u c t o f t h e Ni's. The d e f i n i t i o n s o f p r o d u c t s , d i r e c t sums a n d s u b d i r e c t p r o d u c t s o f N - g r o u p s s h o u l d be c l e a r now ( f o r d i r e c t s u m s YOU need N = N o ) . A g a i n we r e f e r t o P r e h n ' s p a p e r s . 1.56

N O T A T I O N I f t h e Ni ( i E I ) a r e a s a b o v e , l e t fii A, : = { ( . ,0,ni ,o, ) I nic:Ni I .

..

...

b e g i v e n by

25

1 b Constructions

1 . 5 7 P R O P O S I TION (a)

b

- N. 3 -

i c I : N~

A N i G

8

ic1

J

4 Ji Ni

Ni

N.1

$ N . A N i 9

jcI (b)

A

1

8

j EJ

1 . 5 8 REMARKS ( c f .

N~ 4

A

n N j ; jcI

8

N~ A

i€ 1

(ieI)

n N~

4

jcJ

(Gratzer)).

near-rings Ni

n j cn I N J.

8 N A Ti jc1 j

;

ic1

( c ) J ~ I->

51

n

N~

.

ic1

I f N i s a subdirect product o f

then the

Nils

a r e homomorphic

i m a g e s o f N ( u n d e r t h e p r o j e c t i o n maps n i ) . If K e r vi =:Ki o f ideals o f N with zero i n t e r we g e t a f a m i l y (Ki)icI section. Conversely,

[I

with

if a f a m i l y of

o f som? n r . N i s given then N i s isomorphic t o a

= {Ol

Ki

icI

ideals

subdirect product o f the near-rings Of

course,

1.56

(Ki)icI

= N/Ki.

Ni:

- 1.58 can be t r a n s f e r r e d t o N-groups i n t h e

o b v i o u s way. 1 . 5 9 D E F I N I T I O N A s u b d i r e c t p r o d u c t N o f n e a r - r i n g s Ni ( i c 1 ) i s called t r i v i a l i f 3 i c I : ni i s an isomorphism. NEW i s c a l l e d s u b d i r e c t l y i r r e d u c i b l e i f N i s n o t isomorphic t o a n o n - t r i v i a l subdirect product o f nearrings. T h e same i s d e f i n e d f o r N - g r o u p s . 1 . 6 0 T H E O R E M ( ( G r a t z e r ) , F a i n ( 1 ) ) . The f o l l o w i n g c o n d i t i o n s

f o r a nr. (a)

N

{O)

N

( b ) If ( I a ) a c * I a = (01

n

acA

(c

1

are equivalent:

i s subdirectly irreducible;

n

{01+14N

I

+

i s a f a m i l y o f i d e a l s of N w i t h then

3

aEA

: I a = CO);

{UI;

( d ) N contains a unique minimal ideal, o t h e r non-zero i d e a l s .

contained i n a l l

81

26

THE ELEMENTARY THEORY OF NEAR-RINGS

Replacinq " N " N-groups.

by 'INTI' y i e l d s a n a n a l o q o u s t h e o r e m f o r

1 . 6 1 C O R O L L A R Y Each s i m p l e n r . ( N - g r o u p ) i r r e d u c i bl e .

i s subdirectly

1 . 6 2 THEOI?EM ( ( G r a t z e r ) , p.124). ( a ) Each n e a r - r i n g i s i s o m o r p h i c t o a s u b d i r e c t p r o d u c t of subdirectly irreducible near-rinqs.

( b ) Each N-group i s N - i s o m o r p h i c t o a s u b d i r e c t p r o d u c t o f s u b d i r e c t l y i r r e d u c i b l e N-groups. The i n t e r s e c t i o n o f a l l n o n - t r i v i a l in ilartney ( 3 ) .

ideals i s a l s o considered

2 , ) N E A R - R I N G S OF Q U O T I E N T S 1 . 6 3 DEFINITION L e t N be a n r . a n d S a s u b s e m i g r o u p o f ( N , . ) . A n e a r - r i n g N S i s c a l l e d a n e a r - r i n g of riclht ( l e f t ) q u o t i e n t s of N w.r.t. S i f ( a ) NS

E n ,

( b ) N a N S ( b y h, s a y )

( c ) v SES: h ( s ) (d)

qENS

3

is invertible in

SES

3

nEN

(NS,v)

: q = h(n)h(s)-'

(q = h ( s ) - l h ( n ) ) .

Of c o u r s e t h e r e a r i s e t h e q u e s t i o n s a b o u t e x i s t e n c e a n d u n i q u e ness o f s u c h n e a r - r i n g s o f q u o t i e n t s . We w i l l s e t t l e t h e s e questions a f t e r the following 1 . 6 4 DEFINITION N i s s a i d t o f u l f i l l t h e r i q h t ( l e f t ) @ r e c o n d i t i o n (Ore ( 1 ) ) w . r . t . a g i v e n s u b s e m i g r o u p S o f (N,.) if

V

(s,n)

E

SxN

3

(sl,nl)

E

SxN

:

nsl = s n l

(sln = nls)

1b Constructions

1 . 6 5 THEnREM ( G r a v e s - M a l o n e

<->

N

L e t S be a subsemiaroup of

(1)).

I

(N,.).

27

has a n r . o f r i g h t q u o t i e n t s w . r . t .

(a)

s

(b)

v

S

+0

<->

SES : s i s c a n c e l l a b l e (on b o t h s i d e s )

( c ) N s a t i s f i e s t h e l e f t Ore c o n d i t i o n w . r . t . A s s u m e t h a t N h a s a n r . NS o f

P r o o f . ->:

w.r.t. (a):

S,

le f t

S.

quotients

and l e t h b e as i n 1.63.

By 1 . 6 3 ( d )

(b):b

SES

m,nEN

h ( m ) = h ( n ) ->

->

Take

= h(n)h(s)->

m = n.

q: = b ( s )

scS.

nEN,

h(m)h(s)

.

m = n

sm = s n ->

Similarly, (c):

: ms = n s =>

-1 h(n)

,

N,

E

so b y

1.63(d) 3nlcN

: h(s)-'h(n) h(nsl) = h(snl)

]sics

Therefore

,

(n,s)%(n',s'):

3nlEN

nsl = n'nl). H.J.Weinert

n 'n ~

SCl

%

on NxS

slES

by

: ( s s l = s ' n l and

b e t h e equivalence c l a s s o f (11,s) and we m i g h t f o l l o w a s u g g e s t i o n o f n'

I f s,7~NS,

_

p. 262):

Jacobson),

t o clet "common d e n o m i n a t o r s " :

5s

=

3 S S I '

37

w i t h ( n , , ~ , ) E N ~ Sf u l f i l l i n n s ' n ,

_

=

ssl E S

ss 1

!Je t h e n a r e a b l e t o d e f i n e w i t h t h e s e n o t a t i o n s :

.-

n n' + 7.-

s

ns t n ' n 1 1

s

ss 1

w h e r e (n,,~,) t

and

E

and

n n' '"2 s-7:= I,

s2 llxS f u l f i l l s n s 2 = n ' s 2 E S .

a r e shown t o b e w e l l - d e f i n e d

turns o u t t o be a nr. with i d e n t i t y element o f 5 ) .

If

t E S ,

t h e map

i s a mononorphism a n d e v e r y inverse

st'

=

I

n'n, 1_-

n

n n '

NS.

IlxS/% =:

-

Let

3

.

n s l = snl

whence

(N.

<==: S i m i l a r t o r i n q t h e o r y ( c f . Define an equivalence r e l a t i o n

1 .,

= h(nl)h(sl)-

Every

11 S

h(s)=-i-

e = h:N

s t

(Ns,t,*)

and

+

S S

( s any nt Ns: n + t

ch(S)

c a n be w r i t t e n a s

has t h e h(n)h(s)-'

81

28

THE ELEMENTARY THEORY OF NEAR-RINGS

1 . 6 6 THEOREM If N h a s n r ! s o f l e f t then

N,

- NA

quotients

N,,N;

-

left

S

.

%

(We may t h e r e f o r e s p e a k a b o u t " t h e n r . o f

w.r.t.

w.r.t.

quotients

S".)

P r o o f . N f u l f i l l s t h e r i g h t Ore c o n d i t i o n b y 1.65. by

h , h ' b e a s i n 1 . 6 3 ( b ) . D e f i n e 6 : N S + N; h(n)h(s)-' * h'(n)h'(s)-l. Ti i s w e l l d e f i n e d :

if

h(n)h(s)-'

Let

=

stl

and n t l

= h(m)h(t)-l = mn,.

So

then

h'(n)

=

3(tl,n1)ESxN

:tnl=

h'(m)h'(nl)h'(t)-'

and

h'(n ) = h'(t)-lh'(s)h'(t,). T h e r e f o r e we g e t 1 h ' ( n ) = h ' ( m ) h ' ( t ) - ' h ' ( s ) , w h i c h a l l o w s us t o c o n c l u d e

that

h'(n)h'(s)-l

=

h'(m)h'(t)-l.

C l e a r l y , Ti i s an isomorphism.

1.67 REMARKS As Maxson (1) a n d G r a v e s - M a l o n e

(1) p o i n t e d o u t , 1.65 does n o t h o l d f o r n e a r - r i n g s o f l e f t q u o t i e n t s , because a d d i t i o n i n NS (as i n 1.65) i s n o t n e c e s s a r i l y we1 1 - d e f i n e d , T e w a r i (1) e v e n showed t h a t t h e r e e x i s t n e a r - r i n q s h a v i n g n r l s o f l e f t ( r i g h t ) q u o t i e n t s b u t no n r ! s of r i g h t ( l e f t ) quotients.

1.68

D E F I N I T I O N If S = ( S E N I S c a n c e l l a b l e } exists) i s called the

of

t h e n NS ( i fi t

r i g h t ( l e f t ) quotient near-ring

N.

I n t h e s e c t i o n o n n e a r - i n t e g r a l d o m a i n s (59b2)) we w i l l c o n s i d e r t h e c a s e t h a t Ns i s a n e a r - f i e l d . 1 . 6 9 R E M A R K I n Chan-Chew

(1) a characterization o f r i g h t

q u o t i e n t n e a r - r i n a s b y means o f " s e m i - N - h o m o m o r p h i s m " i s given.

See a l s o H o l c o m b e ( 6 ) , O s w a l d ( 9 ) , S e t h ( l ) , (21, Shafi (1).

Seth-Tewari

29

1b Constructions

3 . ) F R E E N E A R - R I N G S A N D N-GROUPS

For t h i s number, we a g a i n u s e G r a t z e r ' s t e r m i n o l o g y and r e s u l t s . Let P b e any v a r i e t y of n e a r - r i n g s ( e . g . a l l n e a r - r i n g s , a l l abelian near-rings, a l l near-rings w i t h unity o r a l l d i s t r i b u t i v e near-rings). Let X b e any non-empty s e t .

F Y c V i s c a l l e d a f r e e - n e a r - r i n q i n 'v f:X+FX''v NEV b g:X+N 3 hEHom(FX,N):hof=g

1 . 7 0 DEFINITION A n r .

over X i f

3

( i n diagram n o t a t i o n :

f

X \

ff4

?Fx 4

-

From ( G r a t z e r ) we deduce 1 . 7 1 T H E O R E M I n t h i s c a s e , f i s i n j e c t i v e w h i l e X can be

r e g a r d e d as a s u b s e t of F x a n d g e n e r a t e s F X . F x i s ( u p t o isomorphism ) u n i q u e l y d e t e r m i n e d by V a n d 1x1 a n d has t h e form i n d i c a t e d i n ( G r a t z e r ) , p . 1 6 3 . One t h e r e f o r e i s a b l e t o speak a b o u t "the f r e e n r . i n V d e t e r m i n e d by some c a r d i n a l a". A f t e r s e v e r a l g l a s s e s o f wine one would d e s c r i b e . F x l o o s e l y a s t h e " s e t of a l l sums a n d p r o d u c t s o f e l e m e n t s o f Xu{Ol ( a n d p o s s i b l y more 0 - a r y symbols such a s 1 ) where one can c a l c u l a t e a c c o r d i n g t o t h e laws which h o l d in V " . 1 . 7 2 DEFINITION If

If =??,

we s i m p l y speak a b o u t t h e f r e e n r . on X . A n e a r - r i n g i s c a l l e d free i f i t i s f r e e o v e r s o m e s e t X.

1 . 7 3 R E M A R K In t h e same way we d e f i n e f r e e N-qroups

i n some v a r i e t y o f N-groups, f r e e N-qroups o v e r some c a r d i n a l n u m b e r and f r e e N-qroups. 1 . 7 1 can be t r a n s f e r r e d t o N-groups by making t h e o b v i o u s a l t e r a t i o n s .

30

Q 1 THE ELEMENTARY THEORY OF NEAR-RINGS

1.74 EXAMPLES If

N E ?7,

then

i s f r e e over { l l .

NN

I t i s h a r d e r t o be a f r e e nr. (N-group) t h a n a f r e e r i n g ( m o d u l e ) . T h i s f a c t i s rewarded by 1.75 THEOREM If

then

F,F1

F 2 F'

P r o o f . <=>:

E?? %

( N @ , N @ ' & Na( rae) f r e e o v e r s e t s X , X '

( @ a N a ' ) c->

1x1

=

IX'I

.

i s s e t t l e d by 1 . 7 1 .

n

( N ~ c)o n t a i n f i n i t e s t r u c t u r e s w i t h more t h a n

one e l e m e n t ( f o r n r l s e . g . t h e f i e l d Z+, N-groups e . g . (Z,,+) w i t h nO = n l = 0 n & N ) . Now a p p l y ( G r a t z e r ) , p . 1 9 7 .

for for all

1.76 R E M A R K Note t h a t t h e theorem above a l s o h o l d s e . g . f o r a f r e e n r . (N-group) i n t h e v a r i e t y of a b e l i a n nr!s ( N groups) ( " f r e e a b e l i a n nr!s (N-groups)").

1 . 7 7 R E M A R K Let be f r e e o v e r X . The usual c h a r a c t e r i z a t i o n ( i n t h e c a s e o f u n i t a r y ( r i n g - ) modules) o f X a s a b a s e ( ' l i n e a r l y i n d e p e n d e n t g e n e r a t i n g s e t " ) does not c a r r y o v e r t o t h e c a s e of N-groups d i r e c t l y : N-groups d o n o t have t o be u n i t a r y , t h e l a c k o f c o m m u t a t i v i t y i n N@ c a u s e s " l i n e a r c o m b i n a t i o n s " ( d e f i n e d as u s u a l ) t o be i n f l u e n c e d by t h e o r d e r o f t h e summands ( a s Maxson ( 1 ) p o i n t e d o u t , one has t o d e f i n e l i n e a r c o m b i n a t i o n s i n t e r m s o f o r d e r e d a n d - most t r o u b l e s o m e of a l l s e t s of elements of N @ u s u a l l y c o n s i s t s o f more then t h e s e t o f a l l l i n e a r combinations, s i n c e i n qeneral n ( n l y l + . . + n k y k ) i s no l i n e a r combination any more. Anyhow, g e n e r a l i z i n g t h e c o n c e p t o f l i n e a r independence g i v e s something l i k e a b a s e : l e t W n ( n & I N o ) be t h e s e t o f a l l n - a r y words o v e r some s e t X i n a v a r i e t y V o f N-groups a n d ( f o r , r E V ) w r t h e induced f u n c t i o n r n + r . D e f i n e i n W n W % ~ W ' : <=> b A E 0 :wA = w;

-

.

.

31

1 b Constructions

1 . 7 8 DEFINITION A s u b s e t B o f

NrcV

i s c a l l e d independent i f

WrcR: r M = 1 . 7 9 REMARK L e t RM be a u n i t a r y r i n g - m o d u l e w i t h = (01 -> r = 0 ( o t h e r w i s e RF1 w o u l d h a v e n o l i n e a r l y independent subset a t a l l ) . Then each s u b s e t o f RM i s l i n e a r l y i n d e p e n d e n t i f f i t i s i n d e p e n d e n t i n t h e s e n s e o f 1.78.

Nr

1.80 DEFINITION B E ( a ) B generates

i s called a base f o r

Nr i f

Nr

B i s independent.

(b)

As u s u a l , t h e f o l l o w i n g q u e s t i o n s a r i s e : ( a ) Which N-groups have a base ? (b) Are d i f f e r e n t bases e q u i p o t e n t ? 1.81 THEOREM B t:

B

+

r

Nr

E

Nr i f f

i s a base f o r

t h e i n c l u s i o n map

can be e x t e n d e d t o an N - i s o m o r p h i s m

O+r,

where 0 i s t h e f r e e N - g r o u p on B.

-P r o o f .

->:

L e t B be a b a s e f o r

S i n c e 0 i s f r e e o n 6,

with hof = 1 h i s an N - i s o m o r p h i s m : (a) Bth(@) A B generates I#JE@

w(f(Bl),

Consider the diagram

t h e r e i s e x a c t l y one

hEHomN(O,r)

(b) L e t

Nr.

We h a v e t o show t h a t

.

r

=>

h(4)

=

r

b e i n K e r h . R e p r e s e n t I$ b y some w o r d

...,f ( B 6 ) )

over

o = h(O) = h(W(f(B1) = wr(h(f(a1) = w r ( 6 1,...,3,).

f(B)

,..., f(B,)))

,..., h ( f ( B , ) ) )

( B ~ + R ~f o r i + j ) . f . ( o w =

= w,(~(B1)s***s~(B,)) 20 6 = 5 .

B i s inCeq2nient.

=

32

$ 1 THE ELEMENTARY THEORY OF NEAR-RINGS

e x t e n d s t o a n N-isomorphism N @ , f ( B ) i s independent a n d g e n e r a t e s N@. So f ( B ) i s a b a s e f o r N @ . Hence B = I(B) = h ( f ( B ) ) i s a b a s e f o r h ( 0 ) = r . Suppose t h a t

<=:

I

h:@+r. By t h e c o n s t r u c t i o n of

From t h i s theorem we immediately deduce 1.82 T H E O R E M

(a) (b)

Nr Nr

has a base

<-o

Nr i s f r e e .

has a b a s e B <-> each m a p f from B t o some N-group N A can u n i q u e l y be extended t o a n N-homomorphism

Nr

.+

N~

.

And from 1 . 7 5 we g e t 1 . 8 3 T H E O R E M Let NI'be a non-zero N - g r o u p p o s s e s s i n g a b a s e .

Then a l l b a s e s a r e e q u i p o t e n t . I f N i s i n 3, t h e n NN has a b a s e (namely 1 1 ) ) a n d a l l o t h e r b a s e s c o n s i s t of one s i n g l e e l e m e n t .

1.84 EXAMPLE

1 . 8 5 R E M A R K See more on f r e e p r o d u c t s e t c . i n F r o h l i c h ' s

paper (4), i n Meldrum ( 2 ) , ( 3 ) a n d i n Rao ( 1 ) . Cf. a l s o F r o h l i c h ( 4 ) a n d Maxson ( 1 ) f o r a c h a r a c t e r i z a t i o n o f a b a s e i n terms o f f r e e p r o d u c t s . See Zeamer ( 1 ) f o r a n " a r i t h m e t i c " i n f r e e n e a r - r i n g s . Free sums ( p r o d u c t s ) a r e s t u d i e d i n Prehn ( 3 ) and Rao ( I ) . See a l s o Meldrum (13), Banaschewski-Nelson ( 1 ) a n d J o h n ( 1 ) .

33

Ic Embeddings

c ) EMBEDDINGS

1 . ) EMBEDDINGS

I N M(r)

The r e a d e r m i g h t be w o n d e r i n g i f a l l n e a r - r i n g s a r e n e a r - r i n g s o f f u n c t i o n s o n some g r o u p

r.

This i s true,

although near-rings

a r e a l s o c o n s i d e r e d under t o t a l l y d i f f e r e n t aspects. The m a i n r e s u l t i s 1.86 THEOREM

w NEW

3

rev : N 4 M(r). (1)). Let

Proof.(Heatherly-Malone

r

be any g r o u p p r o p e r l y

containing (N,+).

v

e a s i l y see, A f n o f n ,= fnnc

*

n,n'EN

: fn+fnl h :

T h u s t h e map

= fn+,,'

N n

+ +

M(r)

A

is

f,

a homomorphi sm.

I f h(n) = h ( n ' )

.

fn = f n ' In particular, V ycT\N : n = f n ( y ) = f n l ( y ) = n ' . T h i s i m p l i e s t h a t h i s i n f a c t a monomorph srn a n d a n e m b e d d i n g map, 1.87

then

as desired.

R E M A R K T h e r e a r e s e v e r a l p r o o f s f o r 1.86,

See e 9 . B e r m a n - S i l v e r m a n ( 3 ) , Nobauer ( 8 ) , H e a t h e r l y - M a one ( 1 ) .

W h i l e N o b a u e r embeds i n M ( r ) w i t h Heatherly-Malone suggest 1.86

r:

=

r:

= (M((N,+)),+),

(N,+) 8 ( Z , , + ) .

a n d i t s p r o o f h a v e many i n t e r e s t i n g c o r o l l a r i e s .

are i n

Some

31

34

THE ELEMENTARY THEORY OF NEAR-RINGS

1 . 8 8 COROLLARIES (a) I f N i s a b e l i a n t h e r e i s an a b e l i a n group

.

Nq,M(r)

r

( b ) I f N i s f i n i t e t h e r e i s some f i n i t e

VNE'H~

(c)

r

with

with N+M(r).

NCPMO(r).

(d) ~ N E ? ? ~l r c q : N 4 M c ( r ) .

N€Xl

(e) If

embeds i n M ( r )

then +(n)

b y 0 and

neN

i s invertible

i s bijective.

(f ) Every n e a r - f i e l d i s isomorphic t o a n f .

on a group

r,

where a l l

fEF'

F of

functions

are bijective.

1 . 8 9 THEOREF! E a c h N - q r o u p c a n b e e m b e d d e d i n t o some f a i t h f u l N-group. Proof.

So e a c h n r .

If

NrENq,

r'

Evidently,

N h a s some f a i t h f u l N - c l r o u p .

t a k e some q r o u p

r'3

with

NcpM(r').

i s a n N - g r o u p i n t h e ~ a t u r a lw a y a n d

moreover a f a i t h f u l one. 1.90 REMARK H e a t h e r l y (1) showed t h a t 3r~?:Nc,E(r)

->

(1.15).

N = Nd ==>

See M e l d r u m ( 1 3 ) f o r a n e x a m p l e

o f a f a i t h f u l , s i m p l e N=No-group

r

(with NEW,)

which is

not unitary.

I t i s sometimes d e s i r a b l e t o l o o k f o r an embeddinq o f N i n t o

M(r)

with a "smaller"

e.g.

M(T)

as above.

R e c a l l t h a t i n 1.86 and 1.87

i s e m b e d d e d i n t o t h e much b i g g e r M ( M ( r ) @ ? $ ) !

For doing t h i s ,

we g e n e r a l i z e a c o n c e p t d u e t o M e n g e r :

1 . 9 1 DEFINITION 6

v

r

n,n'cN:

e

N

i s c a l l e d a base ( o f e q u a l i t y ) i f

( b b c B : n b = n ' b ) .->

1 . 9 2 REMARK C l e a r l y 6 f o r m s a b a s e i f f

n = n'.

(0:B) = I O I , s o i t

w o u l d n o t b e n e c e s s a r y t o u s e a s p e c i a l name. B u t we d o it, because i t i s a v e r y s u g g e s t i v e one.

1 . 9 3 E X A M P L E S I n M(r) t h e s e t Mc(r) ( a g r o u p i s o m o r p h i c t o r f o r m s a b a s e . I n Mcont(r) ( 1 . 4 ( a ) ) i t s u f f i c e s t o t a k e a dense s u b s e t o f

r.

)

l c Embeddings

35

This motivates the i n t e r e s t in the case t h a t t h e constants Nc f o r m a b a s e . T h i s csn b e a c h i e v e d by f o r c e : 1 . 9 4 PROPOSITION Let TI be t h e n a t u r a l e p i m o r p h i s m Then n ( N c ) f o r m s a b a s e f o r n ( N )

.

6 = n(n)

N+N/

(O:Nc)

'

E l = n(nl) are ET(N) then ( V 7 i c ~ ~ ( ~ :c n) n c = Ti 1n c ) - > ( b n c ~ N c : n n c - n 1n c€ ( O : N c ) ) = > =-z ( b n c e N C : 0 = (nnc-nl!c)nc = n n c - n l n c = ( n - n ) n ) => 1 c -> ( n - n l ) E ( O : N c ) -> n = n 1 '

Proof. If

and

--

1 . 9 5 EXAMPLES ( a ) In

M(T),

t h e c o n s t a n t s form a b a s e .

t h e c o n s t a n t s I p do n o t f o r m a b a s e . xp-x p o ( z e r o p o l y n o m i a l ) , b u t Va€Zp : ( x P - x ) ( a ) = ap-a = 0. Therefore xp-x~(O:iZp). (0:Z ) c o n s i s t s o f a l l p o l y n o m i a l s w h o s e c o r r e s p o n d i n q P p o l y n o m i a l f u n c t i o n i s t h e z e r o map.

(b) In

Zp[x], In f a c t ,

The f o l l o w i n g s o l v e s t h e p r o b l e m s t a t e d a f t e r 1 . 9 0 . 1 . 9 6 T H E O R E M I f B sNN

,

the following conditions a r e equivalent:

( a ) B i s a base (of equality);

( b ) B i s a f a i t h f u l N-group; ( c ) N4M(B). P r o o f . 1 . 4 8 and 1 . 9 2 . 1 . 9 7 C O R O L L A R Y I f N, i s a b a s e t h e n N c a n b e c o n s i d e r e d a s a n e a r - r i n g o f f u n c t i o n s on N c . I n v i e w o f 1 . 9 5 ( a ) , t h i s I s "the natural representation o f N". 1 . 9 8 DEFINITION L e t r,A b e g r o u p s . fEM(T) i s c a l l e d k e r n e l if free if y c r : ( f ( y ) = ' o -> y = 0 ) . P u t M ( T ) q k M(A) t h e r e i s some h : M ( r ) w M ( A ) such t h a t h s e n d s k e r n e l f r e e elements of M ( r ) i n t o k e r n e l - f r e e ones o f M(A), a n d M ( r ) Zk M(A) i f M(r) qk M ( A ) by an i s o m o r p h i s m h .

v

36 1.99

THE ELEMENTARY THEORY OF NEAR-RINGS

01

r,A

THEOREM ( H e a t h e r l y - M a l e n e ( 1 ) ) . L e t

r+A

Then Proof.

M o ( r ) 4 M o ( A ) <-> k

<->

rCA

(a) Let

b y h.

r

If

'I

arbitrary,

ycr, y

If

,

fEMo(r) f y : A

define

(01 , the r e s u l t i s

=

o b v i o u s . Assume t h a t but fixed

(01

a n d t a k e some

Y

&+C

If f i s kernel-free,

embeds Mo(r) i n t o M o ( A )

( A = (01

i s again t r i v i a

+ 0,

a

f

h i s moreover i n j e c t i v e : h ( f a ) = 6 ==> h ( f a ) ( 6 ) = 0 ,

a = o

diction) or

r

Mz(r)aMz(A)

,

SO

and A

so

a

i s kernel-free, (a contra-

6 = 0

= 6. a (as groups). But

whence

a r e isomorphic groups,

M:(A)

1.

( N o t a t i o n as i n 1 4 ( a ) ) i s a

M:(A)

fa + fs(f,)(6) g r o u p homomorphism. I f

Hence

and

Y

b y ( s a y ) 9 t h e n t a k e some

k

+

64Im h

t h e same a p p l i e s t o f

( b f If Mo(r) 4 M o t h )

h : M:(r)

6cIm h

1

h(f(Y)

~ E A *

by

f €MO(A)

h(f(h-'(d)))

fixed

.

o

A

+

t h e map f + f y

be g r o u p s .

M(r)c,M(A).

.

M:(r)

and

a n d t h e same a p p l i e s t o

r4A.

r+A

(c) If

t h e n p r o c e e d i n o as i n ( a ) one sees

M(r)GM(A)

that

.

(d) I f M ( r ) q M ( A ) t h e n MC(r)+Mc(b) by r e s t r i c t i o n . % Mc(r) r and Mc(d) % A i m p l i e s t h a t r - h

-

1 . 1 0 0 COROLLARY ( B e i d l e m a n ( 5 ) )

<=+

td(r) 2:

Since each group

r

.

I ,

=

A



Mo(r)

M(A).

r

c a n b e embedded i n t o some

-k M o ( A ) ?,

M(S) =

s u i t a b l e s e t ) b y C a y l e y ' s t h e o r e m , we g e t f r o m 1 . 9 9

S

S

<->

(S a

37

I c Embeddings

1.101

COROLLARY ( N o b a u e r ( 8 ) ) .

NEW t h e r e i s some s e t

For a l l

Nc>M(M(S)). (More p r e c i s e l y , f o r e v e r y n r . N

S with

t h e r e i s a s e t S and a n e a r - r i n g N ' i n

N

that

2.)

- N'.

M(M(S))

such

I ,

MORE BEDS.

S i n c e M ( r ) c o n t a i n s an i d e n t i t y ( i d r ) 1.102

COROLLARY E v e r y ( f i n i t e ,

we g e t f r o m 1 . 8 6

abelian,

zerosymmetric) near-

r i n g c a n b e embedded i n t o a ( f i n i t e ,

abelian,

zerosymmetric)

near-ring with i d e n t i t y . 1.103

REMARK Despite t h i s analogy t o r i n g theory the embedding i s t o t a l l y d i f f e r e n t f r o m t h e one i n r i n g theory.

1.104

M o r e o v e r , N i s n o t a l w a y s embedded as a n i d e a l .

PROPOSITION Each r i n g ( r i n g w i t h u n i t y ,

f i e l d ) c a n be

embedded i n t o a n o n - r i n g ( n o n - r i n g w i t h i d e n t i t y ,

non-

field). Proof.

(a) For rinqs (rings w i t h i d e n t i t y ) i t follows f r o m

1.86.

See C l a y ( 3 ) f o r a n o t h e r p r o o f .

( b ) L e t F be a f i e l d (Maxson ( 7 ) ) .

,

F(x)

fl

f = f ~ F ( ~ ) d ( f ) : = deqfl-degf2 2 a .9. 0 p u t e : F ( x ) + F ( x ) . C l e a r l y 0 i s y + ay+b

define f o r a,bcF

Take

Define f o r

f,gEF(x)

f*eg:

and

.

For

1-1. f - 0

=

Cl)ed(f)Og) .f f P 0 Fe(x): = (F(X),+,*~) i s a n e a r f i e l d and a f i e l d i f f 0 = i d

Then i t i s easy t o see t h a t

.

i:F 1.105

+

FO(x)

i s t h e d e s i r e d e m b e d d i n g map.

COROLLARY ( M a x s o n ( 7 ) ) . zero divisors

+0

Each c o m m u t a t i v e r i n g R w i t h o u t

c a n b e embedded i n t o a n o n - f i e l d .

38

01 Proof.

THE ELEMENTARY THEORY OF NEAR-RINGS

R c a n b e embedded i n t o a n i n t e g r a l d o m a i n ,

w h i c h c a n b e embedded i n t o some f i e l d .

Now a p p l y 1 . 1 0 4 . 1 . 1 0 6 R E M A R K See m o r e o n e m b e d d i n g s i n t h e c h a p t e r o n n f . ' s near i n t e g r a l domains.

and

See a l s o B e i d l e m a n ( l o ) , P l o t k i n ( I ) ,

( 2 ) and Prehn ( 1 ) .

d ) SOME A X I O M A T I C C O N S I D E R A T I O N S

I n t h i s s e c t i o n we c o m p i l e some r e s u l t s o n t h e a x i o m a t i c s o f c o n d i t i o n s f o r N t o be a r i n g o r t o be a b e l i a n ,

near-rings:

c a n c e l l a b l e a n d i n v e r t i b l e e l e m e n t s and a b r i e f s u r v e y o f s t r u c t u r e s which a r e c l o s e l y r e l a t e d t o near-rings.

1.) MISCELLANEOUS RESULTS

1.107 P R O P O S I T I O N L e t N be a n r . ( a ) N a b e l i a n A N c o m m u t a t i v e <==> N i s a c o m m u t a t i v e r i n g ; (b) N abelian A

N distributive

<=>

( c ) N2 = N A N d i s t r i b u t i v e -z Proof.

N i s a ring

(Taussky ( 1 ) ) .

( a ) and (b) a r e obvious.

(c):b

n,n'EN

Computing

3

therefore

n = ab

a,brc,dEN:

(a+c)(d+b) n+n' = n'+n.

so

= cd. yields

cd+ab = ab+cd,

Now a p p l y ( b ) .

and t h e f a c t t h a t t h e r e e x i s t

d i s t r i b u t i v e near-rings

A n'

i n t w o d i f f e r e n t ways

ad+cd+ab+cb = ad+ab+cd+cb,

From 1 . 1 0 7 ( c )

N i s a ring;

non-abelian

( 1 . 1 5 ) we d e d u c e

1 . 1 0 8 COROLLARY N o t e v e r y d i s t r i b u t i v e n r . c a n b e embedded i n t o a distributive nr. with identity.

39

I d Some axiomatic considerations

1 . 1 0 9 PROPOSITION Each o f t h e f o l l o w i n g c o n d i t i o n s i m p l y a n e a r - r i n g N w i t h i d e n t i t y t o be a b e l i a n : (a)

nEN

: n(-1) = -n;

( b ) (Ligh ( 6 ) ) N f i n i t e A

( c ) (B.H.

WnEN

: n(-1)

=

n ->

Neumann ( 1 ) ) ( W n E N 3 h E N : n : n ( - 1 ) = n -> n = 0 ) .

n = 0.

h+h)

A

(b ncN

A

n,n'cN : n t n ' = ( - n ) ( - l ) + ( - n u ) ( - 1 ) = = (-n-n')(-1) = -(-n-n') = nl+n.

Proof.

(a)

(.b ). D e f i n e

a: N

.

N C l e a r l y a c A u t ( N , + ) and n(-1) a 2 = i d . a ( n ) = n i m p l i e s n = 0 . S o by a t h e o r e m o f g r o u p t h e o r y ( e . q . ( W . R . S c o t t ) , ~ . 357), N i s abel i an.

n

-+ -+

( c ) a ( a s above) i s again a f i x e d - p o i n t - f r e e autom o r p h i s m o f o r d e r 2 . From g r o u p t h e o r y ( B . H . Netlmann ( l ) , p . 2 0 6 ) we know t h a t N i s a b e l i a n . 1 . 1 1 0 R E M A R K M c Q u a r r i e ( 2 ) showed t h a t 1 . 1 0 9 ( b ) d o e s n o t h o l d i n the i n f i n i t e case. We now c o n s i d e r c a n c e l l a b l e e l e m e n t s .

1 . 1 1 1 PROPOSITION L e t N b e a n r . ( a ) (Maxson ( 1 ) ) neN i s r i g h t c a n c e l l a b l e a right zero divisor; ( b ) (Maxson ( 1 ) )

nENo

l e f t zero divisor;

is l e f t cancellable

<=an

i s not

-> n i s not a <+

( c ) ( T i m m ( 3 ) ) I f N E ~ , t h e n t h e l e f t c a n c e l l a t i o n law implies t h e r i g h t one. P r o o f . ( a ) i s shown a s i t i s d o n e f o r r i n g s . ( b ) I f n i s l e f t c a n c e l l a b l e a n d i f n n ' = 0 = nO t h e n n ' = 0 . To s e e t h e " < + " - p a r t , c o n s i d e r n e a r r i n g s o f t h e t y p e M:(T) (as introduced i n 1 . 4 ( a ) ) .

§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS

40

+

( c ) I f n'n = n"n, n 0. then (n'-n")n = 0 = = (n'-n")O. so t h e l e f t c a n c e l l a t i o n l a w i m p l i e s n ' = n"

.

1.112

REMARKS H e a t h e r l y (1) p r o v e d t h a t a f i n i t e n r .

N has

e i t h e r only r i g h t zero divisors o r a r i g h t i d e n t i t y , L i g h ( 1 ) showed t h a t i f t h e r i g h t i d e n t i t y i s u n i q u e then N i s a n f . Moreover,

L i g h (1) p r o v e d t h a t i n a f i n i t e n o n - a b e l i a n

near-ring without non-trivial

zero d i v i s o r s each element

has a u n i q u e s q u a r e - r o o t . R a m a k o t a i a h - R e d d y ( 1 ) showed t h a t i f N i s g e n e r a t e d by a l e f t zero d i v i s o r then N

I$ 2 , .

An a p p l i c a t i o n o f t h e e m b e d d i n g t h e o r e m 1 . 8 6 i s 1 . 1 1 3 P R O P O S I T I O N ( H e a t h e r l y ( 1 ) ) L e t NcWl

be f i n i t e .

I f nEN has a o n e - s i d e d i n v e r s e t h e n t h i s i n v e r s e i s two-sided. P r o o f . By 1 . 8 8 ( b ) t h e r e i s a f i n i t e g r o u p r w i t h h N 4 M ( r ) . I f ncN h a s a o n e - s i d e d i n v e r s e t h e n h ( n ) h a s t h e same.

Since

r

i s finite,

h(n)

is

a 1-1-map a n d t h e r e f o r e i n v e r t i b l e . 1.114 R E M A R K I f

a r e n r ' s and h i s a homomorphism ( a n

N,N'

automorphism) from N t o N ' t h e n

h/No

and

a l s o nr.-homomorphisms ( n r . - a u t o m o r p h i s m s ) . C o n v e r s e l y one m i g h t a s k w h e t h e r each p a i r and

h,:

N,

-+

N;

h/Nc h o : No

o f homomorphisms ( a u t o m o r p h i s m s )

are +

Mi

can

be "mated" t o g e t h e r t o g i v e a nr.-homomorphism ( a u t o m o r p h i s m ) f r o m N t o N ' . As M a l o n e ( I ) , ( 4 ) p o i n t e d o u t , t h i s i s n o t t h e case i n g e n e r a l : h : NN' no+nC * h o ( n o o ) + h c ( " c )

w n0ENo vnsll

Wnc"Nc

i s a n e a r - r i n g homomorphism i f f

: hc(nc)+ho(no) = ho(no)+hc(nc)

~ m c H o : h(nm) = h ( n ) h o ( m ) .

and

1 d Some axiomatic considerations

41

2 . ) RELATE0 STRUCTURES -

1.115 SEMINEAR-RINGS A s e t S t o g e t h e r w i t h two b i n a r y o p e r a t i o n s

i s c a l l e d a seminear-ring i f (S,+) and semigroups and W s , s ' , s " ~ S : ( s + s ' ) s " =

It+''

(S,.) SS"+S'S"

and are

'I*''

.

EXAMPLES: The s e t s o f a l l m a p p i n g s o n a n ( a d d i t i v e l y w r i t t e n ) semigroup w i t h p o i n t w i s e a d d i t i o n and c o m p o s i t i o n , e.g. M(IN)

.

P i l z ( 5 ) , Van t l o o r n , V a n Hoorn-Van P o o t s e l a a r , Van R o o t s e l a r , N e i n e r t ( 1 4 ) .

REFERENCES:

1.116 NEAR-ALGEBRAS

A v e c t o r s p a c e A o v e r a (skew-) f i e l d F t o g e t h e r w i t h a n additional binary operation "*" is called a near-algebra o v e r F i f (A,+,*) i s a n e a r - r i n g and b a , b & A b XEF : ( X a ) . b = X ( a . b ) .

EXAMPLES:Take t h e s e t s o f a l l m a p p i n g s o f a v e c t o r s p a c e FV i n t o i t s e l f w i t h p o i n t w i s e d e f i n e d a d d i t i o n , c o m p o s i t i o n and forming A-folds. REFERENCES: H . D . B r o w n , M a r i n , Timm (8), W i l l i a m s ( 2 ) , Yamamuro ( 1 ) - ( 4 ) . Holcombe ( 5 ) f o r an e s s e n t i a l l v d i f f e r e n t definition o f a near-algebra.

ATTENTION

Cf.

1.117 COMPOSITION RINGS (TRI-OPERATIONAL ALGEBRAS)

A s e t R t o g e t h e r w i t h 3 b i n a r y o p e r a t i o n s " t " , " * " ' " 0. " i s c a l l e d a c o m p o s i t i o n rinq ( t r i - o p e r a t i o n a l a l c r e b r a , i f (R,+,;) i s a ring, ( R , + , o ) a n e a r - r i n g and if b r , r ' , r ' ' E R : ( r . r ' ) O r ' ' = ( r o r " ) * ( r ' o r ' ' ) .

x)

EXAMPLES: The s e t s o f a l l m a p p i n g s o f a r i n g i n t o i t s e l f w i t h pointwise a d d i t i o n , m u l t i p l i c a t i o n and composition.

81

42

PEFERENCES:

THE ELEMENTARY THEORY OF NEAR-RINGS

A d l e r , B u r k e , Clav-Doi

(2), H e l l e r ,

Yannos, Yenger, P i l g r a m , N6bauer ( 1 ) - ( 9 ) , Penner ( l ) , P i l z ( 1 ) , ( 3 ) , R i e d l , S t e i n e q g e r , Stueben, S u v a k .

1.118 A G E N E R A L P R O C E D U R E Take a u n i v e r s a l a l g e b r a

A = (A,R),

form the s e t

M(A)

o f a l l self-maps o f A and d e f i n e t h e o p e r a t i o n s o f R pointwise on M(A). Adding t h e b i n a r y operation "0" o f c o m p o s i t i o n y i e l d s a new a l g e b r a EXAMPLES:

A

semi g r o u p

semi g r o u p

semi n e a r - r i ng n e a r - r in g n e a r - a 1g e b r a

ring

composition r i n g

near-ri ng

( - 1

n-group

Rdo)).

( - 1

v e c t o r space

l i n e a r algebra

-

= (M(A),

M(A)

naked s e t group module

(

I

M(A)

( - ) R c o m p o s it io n - q r o u p 'I

'I

-

) : t h e r e e x i s t s n o s p e c i a l name.

R E F E P E N C E S : Berman-Silverman ( 3 ) , H u l e , Lausch ( 2 ) , La us ch - Pliibaue r , Y 1 i t z , PJKbauer ( S) ,( 1 0 ) ,( 11 ) , Pi 1 z ( 4 ) , P o l i n , Stefanescu ( 1 ) .

A l g e b r a s of f u n c t i o n s i n more t h a n one v a r i a b l e a r e s t u d i e d e . g . i n Menger ( 3 ) , Nobauer ( 2 ) a n d S t u e b e n ( 1 ) . Such a l o t o f i n t e r e s t i n g s t r u c t u r e s ! S i n c e one might be a t t e m p t e d t o s t a r t l o o k i n g a t them more t h o r o u g h l y we s w i t c h back t o n e a r - r i n g s very q u i c k l y .

43

$ 2 IDEAL THEORY

I n t h i s p a r a g r a p h we d e v e l n p a n i d e a l t h e o r y s i m i l a r t o t h a t one f o r r i n g s . A f t e r d e f i n i n g sums a n d ( i n t e r n a l ) d i r e c t sums of i d e a l s ( o f N a n d N r ) we n o t e t h a t , u n l i k e t h e r i n g c a s e , i n t e r n a l and e x t e r n a l d i r e c t sums o f i d e a l s a r e n o t n e c e s s a r i l y i s o morphic. We c a l l a n i n t e r n a l d i r e c t sum w i t h t h i s p r o p e r t y a “ d i s t r i b u t i v e s u m ” , a n d prove t h a t f o r N = N o each d i r e c t sum i s d i s t r i b u t i v e . A l s o , we c o n s i d e r t h e l a t t i c e s o f i d e a l s ( a n d l e f t i d e a l s i n N o ) . Of c o u r s e , t h e s e l a t t i c e s a r e c o m p l e t e modular o n e s . I f N = N o a n d no n o n - z e r o homomorphic image o f N i s a r i n g t h e n t h e l a t t i c e o f l e f t i d e a l s i s even d i s t r i b u t i v e . Chain c o n d i t i o n s p l a y a n i m p o r t a n t r 6 l e t h r o u g h o u t t h i s m o n o g r a p h . We prove f o r example t h a t N has some c h a i n c o n d i t i o n i f f a d i r e c t summand I of N a n d N / I have t h e same one. I f N h a s t h e D C C o n i d e a l s t h e n i t i s a f i n i t e d i r e c t sum of indecomposable n e a r - r i n g s . N i s c a l l e d c o m p l e t e l y r e d u c i b l e i f i t i s t h e d i r e c t sum of s i m p l e i d e a l s . T h i s i s t h e c a s e i f f e v e r y i d e a l i s a d i r e c t summand, a n d t h e n each i d e a l o f N has t h e same p r o p e r t i e s . N i s a f i n i t e d i r e c t sum o f s i m p l e i d e a l s i f f N i s c o m p l e t e l y r e d u c i b l e and has D C C a n d A C C o n i d e a l s or ( e q u i v a l e n t l y ) i f f N i s c o m p l e t e l y r e d u c i b l e a n d has one o f t h e c h a i n c o n d i t i o n s o n i d e a l s or ( a g a i n e q u i v a l e n t l y ) i f f N i s c o m p l e t e l y r e d u c i b l e a n d f i n i t e l y g e n e r a t e d which { s i n term e q u i v a l e n t t o t h e e x i s t e n c e o f f i n i t e l y many maximal i d e a l s w i t h z e r o i n t e r s e c t i o n . Any t w o s u c h decompos i t i o n s a r e isomorphic. F i n a l l y , we d e v e l o p t h e t h e o r y o f ( s e m i - ) prime a n d n i l ( p o t e n t ) i d e a l s which r u n s f a i r l y p a r a l l e l t o r i n g t h e o r y : e v e r y n e a r r i n g has minimal prime i d e a l s ; t h e i n t e r s e c t i o n of prime i d e a l s i s semiprime; i f I 4 N t h e n N i s n i l ( p o t e n t ) i f f I a n d N/I a r e n i l ( p o t e n t ) ; i f N = N o h a s D C C on l e f t i d e a l s t h e n N i s

5 2 IDEAL THEORY

44

a prime n e a r - r i n g i f f N has a u n i q u e minimal i d e a l which i s not nilpotent. Many r e s u l t s c a r r y o v e r t o N - g r o u p s w i t h N E ? ~ .

a )

S U M S

1 . 1 S U M S A N D DIRECT SUMS

2 . 1 - 2.11 a r e f o r m u l a t e d f o r i d e a l s o f n e a r - r i n g s ; b u t a l l ( e x c e p t 2 . 6 ( b ) ) c a n be t r a n s f e r r e d t o N - g r o u p s w i t h NE%, by m a k i n g t h e u s u a l c h a n g e s . The p r o o f s i n t h e s e c o n s i d e r a t i o n s r u n p a r a l l e l t o g r o u p o r r i n g t h e o r y and a r e t h e r e f o r e o m i t t e d . A l s o , i t i s p o i n t e d o u t t h a t t h e s e r e s u l t s f o l l o w from t h e general theory o f "Q-groups" ( s e e (Kurosh) o r ( H i g g i n s ) ) ( n o t e t h a t nr!s a r e R - g r o u p s , a n d N - g r o u p s a r e R - g r o u p s i f NEIY)O). See a l s o P r e h n ( 1 ) .

2.1

( I k ) k c K be a f a m i l y o f i d e a l s o f a nr. N . Then t h e f o l l o w i n g s e t s a r e e q u a l : ( a ) The s e t o f a l l f i n i t e sums o f e l e m e n t s o f t h e I k ' s ; ( b ) The s e t o f a l l f i n i t e sums o f e l e m e n t s o f d i f f e r e n t Ik's; ( c ) The sum o f t h e normal s u b g r o u p s ( I k $ + ) ; ( d ) The s u b g r o u p o f ( N , + ) g e n e r a t e d by U !k ;

THEOREM Let

kEK

( e ) The normal s u b g r o u p o f

(N,+)

( f ) The i d e a l o f N g e n e r a t e d by

g e n e r a t e d by IJ I k

.

U

Ik ;

kEK

kEK

2.2

-

DEFINITION The s e t ( a ) ( f ) a b o v e i s c a l l e d t h e sum o f t h e i d e a l s I k (kEK) a n d d e n o t e d by 1 Ik kEK

(for K = {1:2,.,.} From 2 . 1 ( d )

-

a l s o by

11+12+.,.).

( f ) we r e a d i l y d e d u c e

45

2a Sums

2.3 --

COROLLARY ( a ) T h e sum o f i d e a l s o f N i s a g a i n a n i d e a l o f N .

( b ) F o r m i n g sums o f i d e a l s i s a n a s s o c i a t i v e a n d c o m m u t a t i v e operation. C e r t a i n sums a r e o f p a r t i c u l a r i m p o r t a n c e :

2.4

DEFINITION A g a i n l e t

1

kEK

of

( I k ) kbe e iK d e a l s o f N . T h e i r sum

Ik i s c a l l e d a n ( i n t e r n a l ) d i r e c t sum i f e a c h e l e m e n t

1 Ik h a s

a u n i q u e r e p r e s e n t a t i o n a s a f i n i t e sum o f

keK

Ik's.

elements o f d i f f e r e n t

I*

I n t h i s c a s e we w r i t e f o r t h e sum as i n 2.2).

2.5

PROPOSITION F o r e a c h f a m i l y

Ik

(or

I~~I~:...

keI

(Ik)kEK

o f ideals o f

N the

following conditions are equivalent: ( a ) T h e sum o f t h e I k ' s i s d i r e c t . ( b ) T h e sum o f

2.6 -

(Ik,+)

t h e normal subgroups

l*Ik

PROPOSITION L e t

be d i r e c t , a , a '

E

i s direct.

Ii, b , b '

E

kEK

I .

J'

i # j . Then (a) a + b = b + a (b) a ' ( a + b) = a'a

( c ) ab

=

a0

( d ) If N = N ,

2.7

then

ab = 0.

EXAMPLE I n t h e n o t a t i o n o f 1 . 5 6 ,

IONi = ieI

2.8

THEOREM ( " F i r s t i s o m o r p h i s m t h e o r e m " )

If I , J 9 N 2.9

then

InJ A J

A

ItJ/I

(cf.

%

.

8 Ni i€ 1

Prehn ( I ) ) .

J/Ir\J

.

R E M A R K I f t h e r e a d e r s h o u l d h a v e t h e same d i f f i c u l t i e s

as t h e a u t h o r i n remembering t h i s f o r m u l a he m i g h t n o t e t h e a l t e r n a t i n g appearance o f I and J i n t h e isomorphism statement.

5 2 IDEAL THEORY

46

2.10

i s c a l l e d a d i r e c t summand ( o f N ) i f

DEFINITION I 9 N

3 JdN:

N = IiJ.

J i s t h e n c a l l e d a d i r e c t complement o f I i n 2 . 1 1 PROPOSITION I A N

i s a d i r e c t summand <=>

: a c a n be e x t e n d e d t o an e p i m o r p h i s m

N

+

N.

VaEAut I :

I.

The f o l l o w i n g r e s u l t w i l l b e used f r e q u e n t l y . I4 N

2 . 1 2 THEOREM I f of

i s a d i r e c t summand t h e n e a c h i d e a l

I i s a n i d e a l o f N. Ni

Ji

and

4 Ni

then

i E I

Ti 4 N . 2.14

0

N =

2 . 1 3 COROLLARY I f , a s i n 1.56,

REMARK I n g e n e r a l ,

the (group-theoretic)

sum o f t w o N - s u b -

groups i s n o t an N-subnroup any more. B u t : 2 . 1 5 PROPOSITION ( F a i n ( 1 ) ) .

If

sN r

A

-P-r o o f .

v

and

E dN r

then

A t E s . ~r

.

WnEN : n ( 6 t n ) = n ( 6 t n ) - n 6 t n 6

~ E A VQEE

E

EtA =

= AtE.

2 . 1 6 COROLLARY ( M l i t z ( 2 ) ) . of

Nr g e n e r a t e d

If

a, r

A

b y A i s g i v e n by

t h e n t h e N-subgroup AtR.

P r o o f : 2.15 and 1.35(b). W i t h t h i s e q u i p m e n t we c a n c o n s i d e r t h e r e l a t i o n o f i d e a l s o f N a n d o f some h o m o m o r p h i c i m a g e N ' m o r e c l o s e l y t h a n i n 1 . 3 0 . 2 . 1 7 PROPOSITION L e t h : K e r h =:K. of

N,N'

L e t A,A'

,

N+

N'

be an e p i m o r p h i s m w i t h

be i d e a l s ( l e f t

i d e a l s , N-subgroups)

r e s p e c t i v e l y . Then

(a) h(h-l(A')) (b) h-'(h(A))

= A'. = A t K 2 A.

T h e same a p p l i e s t o i d e a l s o r N - s u b g r o u p s o f N - g r o u p s .

47

2a Sums

Proof. (for ideals A,A' of near-rings Let n E N , n'EN' . (a) n'eh(h-'(A')) (b) nch-'(h(A)) = h(n)-h(a)

2.18 PROPOSITION

<->

h(n)Eh(A)

<->

R 4, N ->

]n~h-l(A'):

= 0

<->3aeA:

N,N'). n' = h(n)

n'EA'.

<-->

3 a ~ A :h(n-a) n-aEK <=> neA+K

=

<->

.

R = R n ( N o + N c ) = R n N 0 + R n N c = Ro+Rc.

Proof. kl PER 3fiO€NO 3nCENc : r = n 0+n c R 9, N -> nc = n C 0 = (no+n C )O = rOER , s o T h e rest is trivial.

nOER, too.

2.19 REMARKS 2 1 8 does not hold f o r left ideals L of N. All E E L h a v e the form S! = n +n o c with noEN 0 and nc€Nc , but in general no$L and nc$L: Consider N = P [ x ] and L:= I C a i xi 1 C a i ~ 2 =Z I0,+2,+4, 1 1 . L i s a left ideal o f N (even a maximal one - see S o ( l ) ) , but E:= x + ~ E L decomposes a s S! = no+n C with no = x&L .

...

2.20 THEOREM Under forming sums and intersections, the ideals of N (Nr with NEno ) form a complete modular lattice.

Proof. follows from (Kurosh), p . 143. 2.21 REMARK T h e s e lattices are not necessarily distributive. But cf. t h e following considerations and 2.18 (and also Scott ( 3 ) ) . 2.22 PROPOSITION (Scott (4)) If A,B qN r and A , B sN r then \I nEN b a cA b B E B : n(a+B) z na+nB(mod A r r B ) .

Proof. n(a+B)-nB-na E A+A = A So n(a+B) 5 na+nb(mod A). Similarly, n(a+B) z nB+na(mod and t h e result f o l l'ows.

B ) : na+nB(mod

O n e c a n suspect that 2.22 will be particularly important for A n B ( 0 1 : s e e 2.29.

B),

82

48

IDEAL THEORY

2.23 PROPOSITION ( W i e l a n d t ( 2 ) ) . I f

NE??,

and

A,B,A

%r

then rl:

( A + A )0 ( B + A ) / ( A

i s commutative a n d

ncN

B)+A

y1,y2€r'

: n(y1+y2) = nyl+ny2

.

.

( B e t s c h ( 5 ) ) . E : = ( A n B)+A ; H: = ( A + A ) ~ ( B + A ) Let nl,r12€H a n d ncN Then 1 a E A 3 B E B : rll a(mod E ) A n 2 : B(mod E ) . Now a+B I B+a(mod A A B ) a n d n ( a + B ) E na+nB(mod AnB) by 2 . 2 2 . S i n c e A O B t E we g e t and q 1 + q 2 E a+B z B+a I n2+ql(mod E ) and the n(r11+n2) I n ( u + B ) : n a + n @ E nrll+nq2(mod E ) , p r o p o s i t i o n i s proved.

Proof.

.

2 . 2 4 C O R O L L A R Y ( B e t s c h ( 6 ) ) . With t h e a s s u m p t i o n s a n d n o t z t i o n s of 2.23, = N/(o:r,) is a ring.

m:

P r o o f . r' c a n be c o n s i d e r e d a s a f a i t h f u l TI-group i n t h e o b v i o u s way. Now t h e r e s u l t f o l l o w s f r o m 1 . 4 9 .

NET

a n d i f no 2 . 2 5 C O R O L L A R Y ( B e t s c h ( 6 ) ) . I f N€VO a n d n o n - z e r o homomorphic i m a g e o f N i s a r i n a then t h e l a t t i c e of l e f t i d e a l s of N i s d i s t r i b u t i v e . Proof. Let L 1 , L 2 , L 3 be l e f t i d e a l s o f N . C o n s i d e r t h e N - g r o u p r : = ( L 1 + L 3 ) a ( L ~ + L ~ ) / ( LL 2~ ) ~+ ~ 3

If r 9 { o j t h e n (o:r) N, f o r NET$. From 2 . 2 4 we k n o w t h a t N / ( o : r ) i s a r i n g (9N ) , a c o n t r a d i c t i o n . S o r = I01 a n d t h e l a t t i c e o f l e f t ideals i s distributive. F i n a l l y , l a t t i c e t h e o r y p r o v i d e s us w i t h two more l a w s f o r t h e i d e a l l a t t i c e o f a n r . N o r ,,,T ( N E % ) . L e t I , J , K b e i d e a l s . Modular l a w : I f K e I t h e n I n ( J + K) = ( I n J ) + K . C a n c e l l a t i o n law: I f I E J t h e n I n K = J n K , I u K = J u K i m p l i e s I = J ,

49

2a Sums

2.)

2.26

D I S T R I B U T I V E SUMS

DEFINITION

I'

( a ) A d i r e c t sum

Ia =:I

o f ideals

Ia o f

N (aeA)

ib) =

.EAiaih .

aEA

i s called distributive:

1i

W



,

aeA a

( b ) A d i r e c t sum

Nr

BEA

i;

I*Aa aEA

<->

: (

EI

1 ia)( I

aEA

BEA

o f ideals

=:A

i s called distributive:

Aa

of

(aoA)

<->

( N o t e t h a t t h e sums i n v o l v e d a r e a c t u a l l y f i n i t e o n e s ; a l l summands s h o u l d come f r o m d i f f e r e n t i d e a l s . )

2.27

@ Na

N =

EXAMPLES I f

o f the ideals

ma

then N i s t h e d i s t r i b u t i v e sun

aEA

(1.56).

The same a p p l i e s t o N - g r o u p s . 2.28

PROPOSITION Let sum i s d i r e c t .

(Ia)aEb Then aEA

Moreover:

N whose i s distri-

be a f a m i l y o f i d e a l s o f

Ia

8

?,

I*

Ia<=>

Ia

aEA

aEA

b u t i ve.

N

The a n a l o g o u s r e s u l t h o l d s f o r N - g r o u p s w i t h

=

No

.

Proof. obvious. 2.29 P R O P O S I T I O N ( H e a t h e r l y ( 2 ) ) . o f ideals o f

Nl' w i t h

1 Aa

Let

=

aeA

v Conversely,

ncNo

b

C6a~A:

be a f a m i l y

(Aa)aEA I*Aa aEA

n ( E a a ) = En&,

.

i f Nl' i s f a i t h f u l a n d i f f o r

~ C & , E A : n(Z6,)

= CnCa

then

nENo

Then

=:A.

nEN

.

P r o o f . The f i r s t a s s e r t i o n f o l l o w s f r o m 2.22

and b y

i n d u c t i o n . See a l s o 2 . 6 ( b ) .

I f f o r n c N a n d a l l Z 6 a ~ A n(CSa) = n(oto) = notno, h e n c e no = 0. So

tf

yEI' : ( n 0 ) y = n ( 0 y ) = no = o = O y nO = 0.

then and c o n s e q u e n t l y

50

9 2 IDEAL THEORY

From 2 . 2 9 we g e t t h e f o l l o w i n g s a t i s f a c t o r y r e s u l t ( r e c a l l that for

I*Aa

?r uN

N

p

8

ha

No

,

a EA

acA

t h e r e i s no chance a t a l l t h a t a l w a y s f o r the

2 . 3 0 THEOREM ( B e t s c h ( 3 ) ) . if N

a r e n o t n e c e s s a r i l y N-groups).

bats

Each d i r e c t sum o f i d e a l s i n N ( a n d ,

a l s o in N r ) i s d i s t r i b u t i v e .

P r o o f . The -

If

statement f o r

l*Ia =:I

aeA

and

Nr i s c l e a r f r o m 2 . 2 9 .

1 la,1 ii € 1

then

BEA

aaA

b) CHAIN CONDITIONS

2 . 3 1 REMARKS By 1.51, t h e i d e a l s f o r m a n i n d u c t i v e M o o r e system.

I t makes s e n s e t o s p e a k a b o u t t h i n g s l i k e

"the i d e a l s f u l f i l l the DCC" e t c . By 0.10,

i s f.g.

.

i f t h e i d e a l s f u l f i l l t h e ACC t h e n each i d e a l

2 . 3 2 CONVENTION I f t h e s e t o f i d e a l s f u l f i l l s t h e D C C we s a y t h a t " N f u l f i l l s t h e DCC f o r i d e a l s " o r more b r i e f l y t h a t

" N has t h e D C C I " . To s i m p l i f y s t a t e m e n t s , t h e p h r a s e " L e t N have t h e D C C I " w i l l be a b b r e v i a t e d b y " D C C I " . S i m i l a r c o n v e n t i o n s a p p l y t o r i g h t i d e a l s (DCCR),

left

i d e a l s (DCCL) a n d N - s u b o r o u p s (DCCN). O f course,

t h e same i s d o n e f o r t h e A C C .

2.33 REMARK C l e a r l y t h e DCCN i m p l i e s t h e D C C I i f D C C R o r DCCL i m p l y t h e D C C I .

i m p l i e s t h e DCCL. The same h o l d s f o r t h e A C C .

If

N = No

N = N

-

0'

i n N,

t h e n t h e DCCN

51

2b Chain conditions

2.34 E X A M P L E S

( a ) (Beidleman ( 1 ) ) . L e t a g r o u p r c o n t a i n o n l y f i n i t e l y many normal subgroups b u t an i n f i n i t e c h a i n r = = A 1 = A 2 =... o f s u b g r o u p s ( s u c h g r o u p s a r e known t o e x i s t ) . N: = { f e M ( r ) l t l i e l N : f ( A i ) E A i l . has t h e D C C I b u t n o t Then i t i s immediate t h a t 'I,, t h e D C C N ( s i n c e a l l A i sN r ) . (b)

Each r i n g s a t i s f y i n g t h e A C C I b u t n o t t h e D C C I (Z, f o r i n s t a n c e ) o r c o n v e r s e l y i s o f c o u r s e a n example of a nr. w i t h t h e same p r o p e r t i e s .

2.35 T H E O R E M

( a ) I f I A N and N has t h e D C C I ( D C C N , D C C L ) t h e n t h e same a p p l i e s t o J / I .

I 4N

a n d I i s a d i r e c t summand t h e n N has t h e D C C I ( D C C N , D C C L ) <=> I a n d N/I have t h e D C C I ( D C C N ,

(b) If

DCCL), (c) If

A

AN r (NEW,)

i s a d i r e c t sumniand t h e n

DCCI (DCCN) i f f A a n d

r/A

r has t h e

have t h i s p r o p e r t y .

Proof. ( f o r i d e a l s o f N and the DCCI)

...

( a ) Let J l d 2 ? be a d e s c e n d i n g c h a i n of i d e a l s o f N / I . I f J i : = T -1 ( J i ) ( i E I N ) t h e n J 1 3 J 2 3 by 1 . 3 0 . S o 3 neIN V k r n : J k = J n . S i n c e V icIN : n ( J i ) = = l r ( T - 1 ( 3 i ) ) = Ji by 2 . 1 7 ( a ) , Jk = Jn f o r a l l k z n .

...

I t r e m a i n s t o show t h a t I h a s a l s o t h e D C C . B u t t h i s f o l l o w s from t h e f a c t t h a t e a c h i d e a l o f I i s an i d e a l o f N. ( b ) ->:

L e t I a n d N/I have t h e D C C and l e t J 1 3 J 2 ? , . . be a c h a i n of i d e a l s o f N. The c h a i n s J l o 1 3 J 2 n 1 2 ... and a(J1+I)?a(J2tI)?. g e t c o n s t a n t a f t e r some nEIN. Therefore b k 2 n : JknI = J n n I A n ( J k + I ) = = n(Jn+I)

<-:

..

.

82

52

IDEAL THEORY

Since n - ' ( a ( J i t I ) ) = J i t I t I = J i + I , J k t I = Jn+I f o r a l l k 2 n . Now XEJ, : xeJn+I = J k + I , so 3 YEJ, 3 iE1 : x = y + i . Therefore X - Y E J k I=~J,," I c J n a n d s o X E J , T h i s shows t h a t b k r n : J k = J n

v

.

.

( c ) The p r o o f i s s i m i l a r t o t h e one o f ( b ) .

has t h e O C C N 2.36 R E M A R K Lausch ( 4 ) showed t h a t i f N E T $ and ecN h a s some e'EN w i t h e ' e = 1 t h e n e e l = 1. The " J o r d a n - H o l d e r - t h e o r y " c a r r i e s o v e r t o n e a r - r i n a s a n d N-groups w i t h N E n o ( b u t we o n l y f o r m u l a t e i t f o r n e a r - r i n g s ) . The p r o o f s a r e n e a r l y word f o r w o r d t h e same a s i n g r o u p o r r i n g t h e o r y a n d hence o m i t t e d . This o m i s s i o n i s a g a i n j u s t i f i e d by t h e f a c t t h a t a l l o f 2.37 - 2.41 i s a s p e c i a l c a s e of t h e Jordan-Holder-theory of Q-groups ( s e e e . g . (Kurosh), I V B 5 2 ) . 2.37 D E F I N I T I O N A f i n i t e sequence

N = N,=N1=N

*=...= N n = {ol

t*)

o f s u b n e a r - v i n g s N i of N i s c a l l e d a normal s e q u e n c e o f N <-> b i c { l . n l : Ni A Ni-l I n t h e s p e c i a l c a s e t h a t a l l N i A N we c a l l t h e normal sequence ( * ) a n i n v a r i a n t s e q u e n c e . n i s c a l l e d t h e l e n g t h o f t h e sequence ( " ) a n d t h e n e a r rings Ni-YNi ( i E f 1 , n l ) are called the factors

.

...,

...,

of ( * ) . Another normal ( i n v a r i a n t ) sequence N

M o a M 1=M 2=.. . = M m

= {O

1

i s c a l l e d a refinement o f (*) i f iE{O B n l 3 j E { O ,... B m l : N i = M j

.

( * ) and ( * * ) a r e c a l l e d i s o m o r p h i c i f

n = m

....

and t h e f a c t o r s o f (*) a n d (**) a r e ( a f t e r a possibly necessary r e - o r d e r i n g ) isomorphic.

53

2c Decomposition theorems

(++) i s c a l l e d a p r o p e r r e f i n e m e n t of

(*) i f (*) i s n o t

a refinement o f (**).

A normal ( i n v a r i a n t ) sequence ( * ) i s c a l l e d a c o m p o s i t i o n s e q u e n c e ( p r i n c i p a l s e q u e n c e ) i f ( * ) h a s no p r o p e r refinement

.

2 . 3 8 P R O P O S I T I O N ( * ) i s a c o m p o s i t i o n ( p r i n c i p a l ) s e q u e n c e <=> <->

a l l f a c t o r s are simple.

2 . 3 9 COROLLARY A s e q u e n c e i s o m o r p h i c t o a c o m p o s i t i o n ( p r i n c i p a l ) sequence i s i t s e l f a c o m p o s i t i o n ( p r i n c i p a l ) sequence We now s t a t e t h e f a m o u s J o r d a n - H o l d e r - t h e o r e m :

2 . 4 0 COROLLARY L e t

N have a c o m p o s i t i o n ( p r i n c i p a l ) sequence.

Then e a c h n o r m a l ( i n v a r i a n t ) s e q u e n c e c a n b e r e f i n e d t o a c o m p o s i t i o n ( p r i n c i p a l ) s e q u e n c e and a l l t h e s e sequences aye i s o m o r p h i c .

2 . 4 1 THEOREM

N has a p r i n c i p a l sequence

the ideals o f N

<->

f u l f i l l both chain conditions.

See a l s o K a a r l i ( 2 ) , ( 4 ) , ( 6 )

a n d Oswald ( 8 ) , ( 1 0 ) .

c ) DECOMPOSITION THEOREMS

2 . 4 2 DEFINITION N

(Nr)

i s c a l l e d decomposable i f i t i s t h e d i r e c t sum o f n o n - t r i v i a l i d e a l s ( o r , e q u i v a l e n t l y , i f

i t h a s a n o n - t r i v i a l d i r e c t summand),

otherwise

indecomposable.

2 . 4 3 EXAMPLES C l e a r l y e a c h s i m p l e n r . The r i n g

(N-group)

i s indecomposable.

Z i s indecomposable, b u t n o t simple.

Between t h e c o n c e p t s o f s i m p l i c i t y ,

indecomposability

a n d m i n i m a l i t y o f a n i d e a l ( w h i c h i s a t t h e same t i m e s u p p o s e d t o b e an N - s u b q r o u p i n t h e c a s e o f N - g r o u p s ) there are the following relations:

54

9 2 IDEAL THEORY m i n i m a l <-

s i m p l e ->

indecomposable

I f t h e i d e a l i n q u e s t i o n i s e v e n a d i r e c t summand, we g e t s i m p l e =>

m i n i m a l <->

2.44

indecomposable.

REMARK T h e n e x t c o n s i d e r a t i o n s c o n c e r n m e r e l y N - g r o u p s with

NE'I,.

The r e a s o n i s o b v i o u s :

i n qeneral t h e i d e a l s

o f N-groups a r e n o t n e c e s s a r i l y N-groups again.

.

we h a v e t o s p e a k a b o u t " s i m p l e i d e a l s " e t c .

But

also

Cf.

R o t h (1) a n d t h e r e m a r k s p r e c e d i n g 2 . 1 .

2 . 4 5 THEOREM L e t N

(Nr)

Proof

( N rw i t h NEW,)

N

have t h e D C C I . Then

i s t h e f i n i t e d i r e c t sum o f i n d e c o m p o s a b l e i d e a l s . (for near-rings).

I f N i s n o t indecomposable then

there are non-trivial

.

N = Il;12 If

11,12

I1

or

11,12

with

a r e i n d e c o m p o s a b l e , we a r e t h r o u g h .

I2

t h e DCCI,

ideals

decompose a g a i n p r o p e r l y ,

If n o t ,

e t cetera.

By

these decompositions stop a f t e r f i n i t e l y

m a n y s t e p s t h e r e b y p r o v i n g t h a t N i s t h e d i r e c t sum

o f f i n i t e l y many i n d e c o m p o s a b l e i d e a l s . The c o r r e s p o n d i n g a s s e r t i o n i n n u c l e a r p h y s i c s i s m u c h h a r d e r t o prove! 2.46 DEFINITION

N ( N rw i t h

NEWo)

i s called completely

reducible i f

N (NT)

I$N

i s completely reducible i f

(ASr,r)

i s t h e d i r e c t sum o f s i m p l e i d e a l s . I (A)

i s com-

p l e t e l y r e d u c i b l e when c o n s i d e r e d a s a n e a r - r i n g (PI-group). 2.47

R E M A R K A n o t h e r u s u a l name i s " s e m i s i m p l e " .

However,

"semi-

s i m p l e " w i l l h a v e a n o t h e r m e a n i n g i n 95. M o r e on t h a t c a n be f o u n d i n O s w a l d ( 2 ) , ( 3 ) a n d ( 5 ) .

55

2c Decomposition theorems

2.48

T H E O R E M ( R o t h ( l ) , Beidleman ( 1 ) ) . I f conditions are equivalent:

NE??,

the following

( a ) Every i d e a l of N i s t h e sum o f s i m p l e i d e a l s . ( b ) N i s t h e sum o f simple i d e a l s , ( c ) N i s t h e d i r e c t sum o f

(d) (e) (f) (9)

simple i d e a l s . N i s completely r e d u c i b l e . Each i d e a l of N i s a d i r e c t s u m m a n d . I 9 N : I and N / I a r e c o m p l e t e l y r e d u c i b l e , N i s t h e s u m o f minimal i d e a l s .

The a n a l o g o u s t h e o r e m h o l d s f o r N-groups w i t h

NENo.

Proof.(for near-rings). ( a ) =.> ( b ) : t r i v i a l .

( b ) => ( c ) : I f

N =

1 Ia ,

define

aEA

4 :=

J c o n t a i n s a maximal e l e m e n t ( w . r . t . 5 ) B . acA: ( I a n N ' = I a v I a n N ' = {O}) I n n N 1 = {Ol

-

of

B.

( c ) ->

So

'd

i s aEA

.

a c o n t r a d i c t i o n t o the maximality : 1,cN'

a n d hence

N = N1 =

I*IB

€3 E B

*

( d ) : by d e f i n i t i o n .

( c ) =.> ( e ) : I f I 9 PI, c o n s i d e r a n i d e a l J maximal ( Z o r n ! ) w i t h t h e p r o p e r t y t h a t J n I = {Ol. N ' : = I i J . I f N =f N ' , 3 J 0 d : J o s i m p l e A J o ~ N i A J o ~ I O Then l.

JonN' = ( 0 1 ,

so

J+Jo=J.

( J t J o ) n I = (01,

Also,

since x = j t j o c ( J + J o ) n I implies t h a t j o = = x - j c ( I t J ) n J o = N 1 n J o = {Ol. T h i s c o n t r a d i c t s t h e m a x i m a l i t y of J . T h e r e f o r e IGJ = N a n d I i s a d i r e c t summand.

( e ) -> ( a ) : I f I 9 N, d e n o t e by T t h e sum o f a l l s i m p l e i d e a l s o f I. Assume t h a t I I. T 9 I A N A I i s d i r e c t summand => T 9 N Hence T i s i t s e l f a d i r e c t summand a n d t h e r e i s some

-

.

J A N with

TGJ = N.

3 2 IDEAL THEORY

56

C o n s e q u e n t l y each s i m p l e i d e a l of T i s a simple i d e a l o f N . T G ( J n 1 ) = I , s i n c e e a c h iE1 h a s t h e form i = T + j w i t h TET a n d j E J ; because of TSI we know t h a t j c 1 . We n o w show t h a t J n I c o n t a i n s a s i m p l e n o n - z e r o i d e a l of N and a r r i v e a t a c o n t r a d i c t i o n . By a s s u m p t i o n , J n I $. {O). I f J n I i s f.g. then t h e r e e x i s t s a maximal i d e a ? I* i n J n I , a n d e a c h d i r e c t complement ( e x i s t e n c e a s b e f o r e ) o f I* i n J n I i s a s i m p l e n o n - z e r o i d e a l of JnI and o f N. I f J n I i s n o t f . g . , t a k e any f g . i d e a l F I01 o f J n I . T h e n F 9 J n I . F c o n t a i n s a maximal i d e a l M I f ) : S i n c e ( c ) => ( a ) , e v e r y I N i s t h e sum (and by ( b ) => ( c ) t h e d i r e c t sum) o f s i m p l e i d e a l s , i m p l y i n g t h a t I i s completely r e d u c i b l e . I f I d N, t a k e some J g N ( a g a i n , J i s c o m p l e t e l y 2r J by 2 . 8 r e d u c i b l e ) w i t h 1;J = N . B u t t h e n N/I and N/I i s c o m p l e t e l y r e d u c i b l e .

-

( f ) => ( d ) :

trivial (take

( a ) ->

(9): trivial.

( 9 ) ->

( e ) : a s i n ( c ) =>

I = N).

(e).

2 . 4 9 C O R O L L A R Y The d i r e c t s u m o f c o m p l e t e l y r e d u c i b l e n e a r - r i n g s i s again completely r e d u c i b l e . (N-groups w i t h NE??,)

Near-r.ings ( N - g r o u p s ) which decompose i n t o f i n i t e l y many simple i d e a l s a r e e s p e c i a l l y important. The f o l l o w i n g theorem w i l l b e used f r e q u e n t l y t h r o u g h o u t t h i s book. M u c h more o n t h i s s u b j e c t can be found i n B l a c k e t t ( l ) , C h a o ( I ) , Hartney ( Z ) , Oswald ( 3 ) , ( 4 ) , ( 5 ) , ( l o ) , N a t a r a j a n ( 1 ) , Ramakotaiah ( 3 ) and S c o t t ( 7 ) .

57

2c Decomposition theorems

.

2 . 5 0 THEOREM (Beidleman ( l ) , Betsch ( 3 ) ) . Let N be a n r . E q u i Val e n t a r e : ( a ) N i s t h e sum o f f i n i t e l y many s i m p l e i d e a l s . ( b ) N i s t h e d i r e c t sum of f i n i t e l y many s i m p l e i d e a l s . ( c ) N i s c o m p l e t e l y r e d u c i b l e and has t h e D C C I a n d t h e ACCI. ( d ) N i s c o m p l e t e l y r e d u c i b l e and has t h e ACCI. ( e ) N i s c o m p l e t e l y r e d u c i b l e a n d has t h e D C C I . ( f ) N i s completely reducible a n d every ideal o f N i s f ( 9 ) There e x i s t maximal i d e a l s 11, I n of N with zero i n t e r s e c t i o n , b u t a l l J r : = Ik (0). n k+r ( i n t h i s c a s e , N = T'Jr and Jl J n are simple). r =1

...,

n

+

,...,

( h ) There e x i s t maximal i d e a l s

n

n

Il,...,In

with

I,. = {OI.

r =1

The u s u a l changes y i e l d a n a l o g o u s r e s u l t s f o r N-groups w i t h Ncno (remark a l s o t h e a d d i t i o n a l r e s u l t s i n Oswald ( 2 ) ) . P r o o f . ( a ) <->

( b ) : as in 2.48.

n

1 I k ( a l l I k simple) then k=l N i s c l e a r l y c o m p l e t e l y r e d u c i b l e . Moreover, N = I ~ / /I, a I ~ / . . . / I , - ~ a a I~ (01 i s a p r i n c i p a l s e r i e s , so N f u l f i l l s both c h a i n c o n d i t i o n s by 2 . 4 1 . ( b ) =>

(c): If

N =

...

( c ) ->

( d ) <->

( d ) a n d ( c ) ->

...

(e) are trivial.

( f ) : by 0.10.

A C C ( D C C ) f o r c e s A t o be f i n i t e .

$ 2 IDEAL THEORY

58

( b ) -> (9): I f d e f i n e Ik: =

F

...;Jn

N = J1;

r kJ r

I k a r e maximal i d e a l s . I f

3

and i f

(jiEJi)

(Ji

. Because x

E

...,n l :

kE{l,

simple ideals).

of

N/Ik

fi

k=l

p

a1 1

...+j n

x = jl+

Ik,

jk

- Jk, 2r

0

xBIk

then

n

a contradiction.

n

Since

(h):

=+

= (0).

k=1

+ IO),

I k = J,

kSr (9)

fl I k

So

we a r e t h r o u g h .

trivial.

...,

( h ) -> ( b ) : L e t 1 1 , Inb e m i n i m a l w . r . t . the p r o p e r t y t h a t t h e i r i n t e r s e c t i o n = {Ol, Then e a c h Jr: = Ik (01. Since rsil, n l : J, I,,

+

n

v

+

...,

k4-r b u t J r n I r = f O ? , we h a r e N = Jr;Ir. Hence -9 and J , i s s i h p l e . Jr N/Ir K r i = I1n nIr L e t f o r r E f 1 , ...,n )

-

N

We c l a i m t h a t

= J1;

.

...

...tJr;Kr

and p r o v e t h i s by

i n d u c t i o n on r.

If

r

= 1

then

K,

= I1

and

Assume t h a t i t i s shown f o r the assertion for Since

IrtltKr

Since

YIrt1

= N

and

Jrtl

Also Hence

5

Krtl

Kr

Krtl~Jrtl

9

r (< n).

We show

(by m a x i m a l i t y ) ,

t h e same a p p l i e s t o

i s a maximal i d e a l i n

J r + l $ Kr+l a n d N = J1;

but

= K,

. . . +.J r +.J r t 1 ~ K r t 1

= J1+.

= N.

rtl.

i s simple,

KdKrt1

Jl;If

.

.

K,

...t.J r +.K r

=

,

2c Decomposition theorems

2 . 5 1 R E M P R K S T h e p r o o f o f ( h ) ->

59

( b ) i n 2.50 c o u l d a l s o be done

b y u s i n g s u b d i r e c t p r o d u c t s and "words g e n e r a t i n g p r i m e i d e a l s " s i m i l a r tofMcCoy),

p. 59.

Cf.

also (Higqins),

A t a f i r s t g l a n c e o n e m i g h t assume t h a t " f . g : '

$9.

implies

a1ready"completely reducible'.' T h i s i s n o t t h e case: t h e zero-nr.

N o n t h e d i h e d r a l g r o u p D8 o n 8 e l e m e n t s .

D8 i s k n o w n

Then normal subgroups and i d e a l s c o i n c i d e . B u t G 9

t o have

take

D8

H 4 G,

and

but

H $I

D8

.

By 2 . 4 B ( e )

and 2.12 N c a n n o t be c o m p l e t e l y r e d u c i b l e . 2.52

COROLLARIES ( a ) I f N f u l f i l l s one (and hence a l l ) o f t h e c o n d i t i o n s i n 2.50

I 9 N

and i f

2.48(f),

2.48(e)

t h e n t h e same a p p l i e s t o I ( u s e

and 2 . 3 5 ( b ) ) .

( b ) I f N h a s t h e DCCI a n d i s a s u b d i r e c t p r o d u c t o f s i m p l e n e a r - r i n g s Ni

N

Aqain,

=

8 N jEJ j

(ieI)

then

(apply 2.50(h)

IJcI, J finite: and 1.58).

corresponding statements h o l d f o r N-qroups w i t h

2 . 5 3 D E F I N I T I O N Two d e c o m p o s i t i o n s o f N : N = a r e c a l l e d isomorphic if and

JB's

are

-

/A1 = IBI

-

up t o o r d e r

The K r u l l - S c h m i d t - T h e o r e m

1'1,

acA and t h e

=

N

=

No

I'JB

B€B

Is's

isomorphic.

r e a d s as

2 . 5 4 THEOREM ( R o t h ( 1 ) ) . I f N (,,,r w i t h Ncr),) f u l f i l l s one (and hence a l l ) o f t h e ' c o n d i t i o n s o f 2.50 t h e n any two

(Nr) i n t o s i m p l e i d e a l s a r e

decompositions of N

is o m o r p h ic Proof

.

( f o r rrr.ls)

If

(Iky JL simple)

N =

then

~ ~ t . n, =. tJ ~~; . . . ~ J , N=I1;...;In-l=...=I1={o]

and

N = J ~ ; , . . + J ~ _ ~ = . . . = J ~ ~ ~a or e? t w o i n v a r i a n t s e q u e n c e s with simple factors

60

8 2 IDEAL THEORY

By 2.40

t h e s e sequences and t h e r e f o r e t h e s e de-

c o m p o s i t i o n s a r e isomorphi,c. Compare t h e f o l l o w i n g r e s u l t w i t h 59 o f ( H i g g i n s ) .

2 . 5 5 THEOREM n

( a ) If N =

!'Ir

a n d if I A N

( a l l Ir simple)

r=l

there i s a subset S o f

...,n l

{l,

then

with

(b) I f I,J 9 N a r e such t h a t N / I a n d N/J a r e c o m p l e t e l y r e d u c i b l e t h e n N/In i s completely reducible, too,

a l l o f whose s i m p l e summands b e i n g i s o m o r p h i c t o o n e o f t h e s i m p l e components o f Again,

N/I

or

N/J.

t h e c o r r e s p o n d i n g theorem h o l d s f o r N-groups w i t h

N = No. Proof.

(a) Let

Then

K,

Kr:

...+ ( l s r s n ) , ...,n - 1 1 : Kr 9 N

= ItI1+ I r

= N.

rE{l,

K r n'r+1g1r+1* T h u s we h a v e e i t h e r

or

A

Kr+l

= K,

= Kr'Ir+1.

Kr+l

Hence

KO: = I .

3

T5{1,

...,n l :

N = 1;

l'It,

a n d s o by 2.8

tET

S: = {l, ...,n l \ T

.

( b ) L e t K: = I + J . Then K/I 9 N / I a n d 3 MsN: N / I = = (K/I)YM/l), whence K+M = N a n d K n M = I . So M n J = M n K n J = I n J a n d M+J?I+J = K, M + J 3 M t K = N, h e n c e MtJ = N. Consequently

61

2d Prime ideals

M I I n J = M/MnJ

%

"In

J

N/I nJ

%

MtJIJ = N/J

and

are completely reducible, s o

K / I 9 N/I

is completely reducible by 2 . 4 9 .

The rest follows from (a) and the first line on this page. 2

Ferrero-Cotti showed that N = I , I ? , N # N,, I , * #to}# I 2 implies that all ideals o f N are given by { O l , I , , I 2 and N.

d ) PRIME IDEALS

1 . ) PRODUCTS OF SUBSETS. 2.56

NOTATION If S , T c N then ST: = (StlsESAtET?. For nsIN, the definition o f Sn is then clear.

2.57 PROPOSITION (Maxson ( 1 ) ) . (a)

1 R,S,T

E

N : (RS)T = R(ST).

m

(b) If h: N * and tl SITc

(c)

I 4N

then tl S,T E N : h(ST) = h(S)h(T) W: h-'(YT) ? h-'(?)h-'(T).

VS,T

E

N:

( S t I ) ( T t I ) = STtI.

Proof. (a) and (b) are immediate. (c) follows from (b) for 2 . 5 8 REMARK Note that

T ~ : N* N / I .

S T has no particular structure fn general. Even if S I T are ideals, S T i s not even except i n some very special a subsemigroup o f ( N , t ) cases.

62

$ 2 IDEAL THEORY

2 . ) PRIME IDEALS 2.59 DEFINITION P A N -> I e P v J c P .

i s c a l l e d prime i f

2.60 N O T A T I O N F o r ScN,

let

(Cn}) =: ( n )

I,J4N:

IJCP ->

be t h e i d e a l g e n e r a t e d by S .

(S)

.

2 . 6 1 P R O P O S I T I O N ( V a n d e r Walt ( 1 ) ) . Let P be a n i d e a l o f N . Equivalent a r e

( a ) P i s a prime i d e a l . (b)

v

(c)

v (e) v (d)

I,J A N:

(IJ)

i,jEN: i

4

P

A j

4

1,J A N:

I

a

P

J

I , J A N: i

P

F

I\

P A J

P r o o f , ( a ) <=> ( b ) <-> ( a ) ==> ( c ) : I f s o iEP v j E P .

( c ) -> Then

I s P v J

=+

P.

( i ) ( j ) $ P.

P ->

4

5

4

P ->

IJ

P

IJ $ P.

-7

P.

(e) i s trivial. (i)(j)cP

then

(d): If IZPAJaP, t a k e ( i ) ( j ) $ P , so IJqP.

(i)cP

or

iEI\P

and

(j)cP,

jEJ\P.

( d ) => ( e ) : I f I+PAJ$P, t a k e i E I \ P and jcJ\P. Then ( i ) + P = P a n d ( j ) + P = P . Then ( ( i ) + P ) ( ( j ) + P ) $ P . So 3 i ' E ( i ) 3 j ' E ( j ) 3 p , p ' ~ P : (i'+p)(j'+p')&P. T h e r e f o r e i'(j'+p')-i'j'+i'j'+p(j'+p')&P. But since i ' ( j ' + p ' ) - i ' j k P and p ( j ' t p ' ) E P , i ' j 'BP, hence I J$P

.

2.62 PROPOSITION L e t ( P a ) a E A be a f a m i l y o f prime i d e a l s , t o t a l l y o r d e r e d by i n c l u s i o n . Then P =:P i s a pr ime aEA a i d e a l , too.

n

P r o o f . We may a s s u m e t h a t A is o r d e r e d such t h a t f o r a i B -> Pa c P B . a,B€A

63

2d Prime ideals

Of

P i s an i d e a l . L e t I , J b e i d e a l s o f N . P a -> b a E A : I J 5 P a . I f 3 a c A : I 4 P a '.

course,

fl

IJ 5

acA

then then SO

tl

J E Pa. B l a : J 5 P B , I f 3 y < a : J % Py I c P Y' s o I t P a' a c o n t r a d i c t i o n . acA: J c pa and J c pa.

n

acA

2 . 6 3 PROPOSITION ( M a x s o n ( 1 ) ) . I f I A N i s a d i r e c t summand and P A N i s prime then P n I i s a prime i d e a l i n I. I f JlJ2 F PI\ I (J1,J2 4 I ) then J1J2 c P and J1,J.p N , so J1 c P o r J 2 e P and t h e r e f o r e J1 c P n I o r J 2 c P n I .

Proof.

2.64 PROPIISITION I f I 3 N a n d I c P A N a n d i f a : N -+ N / I = : N i s t h e c a n o n i c a l e p i m o r p h i s m a s u s u a l t h e n : P i s p r i m e c-> n ( P ) i s p r i m e . J ~ J ~ ~ ~ (~ P l) ;Aj IT), ~ l e t J ~ : =a - l ( ~ ~ ) -1 -1 ( i ~ { 1 , 2 1 ) . By 2 . 5 7 , J 1 J 2 = TI ( J l ) r ( J p )c If

P r o o f . ->:

ra-1(JlJ2)En-l(Tr(P)) = P t I = P. So or

JlcP v J2cP, hence J2cn(P).

<==: I f

So

J1J2cP

n(Jl)ea(P)

then or

J1

= a(a

-1

(J1)) = a ( J 1 ) ' T ( P )

n ( J 1 ) r ( J 2 ) = T(J1J2)'a(P). n(J2)tn(P).

T h i s shows t h a t

J I C J 1 + I = n - l ( ~ ~ ( J ~ ) ) c ~ r - ~ (= n P( tPI ) =) P J2eP.

either or

2 . 6 5 DEFINITION C a l l N a p r i m e n e a r - r i n q i f

{01

i s a prime

ideal.

7 i n a l d i (1) s t u d i e d n r . ' s whose p r o p e r i d e a l s a r e a l l prime I f N €'h0 i s n o t s i m p l e a n d h a s t h i s p r o p e r t y t h e n N h a s a t n o s t two minimal a n d two maximal ideals.

64

2.66 --

9 2 IDEAL THEORY

EXAMPLES ( a ) Every i n t e g r a l n e a r - r i n g i s o f c o u r s e a prime n e a r r i n g ( f o r I - J 5 { O l a n d I $- { O ) , J $. CO) would

g u a r a n t e e t h e e x i s t e n c e o f some with i j = 0 ) .

(p)

N i s a prime i d e a l o f N , s o More g e n e r a l l y :

2 . 6 7 PROPOSITION I f

I A N,

f0)

iEI*,

jEJ*

( s e e p . 1)

i s a prime r i n q .

I i s a prime i d e a l i f f

N/I

is

a prime r i n g . P r o o f . Take

P

=

I

i n 2.64.

2 . 6 8 E X A M P L E I f NEVc t h e n e a c h normal subgroup o f i s a prime i d e a l .

(N,+)

2 . 6 9 C O R O L L A R Y Each c o n s t a n t n e a r - r i n g i s a prime n e a r - r i n g . 2 . 7 0 PROPOSITION N s i m p l e =+ N i s prime or N i s a z e r o - n e a r r i n g , The p r o o f i s t r i v i a l . More g e n e r a l l y : 2 . 7 1 PROPOSITION I f I 4 N prime o r N 2 c I .

i s a maximal i d e a l t h e n I i s e i t h e r

P r o o f . N/I i s s i m p l e . By 2 . 7 0 , N/I i s e i t h e r prime ( i m p l y i n g t h a t I i s a prime < d e a l ) o r N / I is a 2 z e r o - n r . which c a u s e s N 5 I . 2.72 COROLLARY If

I

4

NET

i s maximal t h e n I i s p r i m e .

2 . 7 3 R E M A R K I f I i s p r i m e , I i s n o t n e c e s s a r i l y maximal ( n o t

even f o r f i n i t e d g n r ' s . : and Laxton-Machin ( 1 ) ) .

s e e Beidleman ( 8 ) , Laxton ( 4 )

2 . 7 4 DEFINITION A n i d e a l minimal i n t h e s e t of a l l prime i d e a l s c o n t a i n i n g s o m e g i v e n i d e a l I i s c a l l e d a minimal prime ideal of I .

2d

Applying 2.66(b) w e get

Prime ideals

and Zorn's lemma on

65

(EPsNIP3IAP prime),?)

2.75 PROPOSITION F o r each ideal I there exists a minimal prime ideal o f I . 2.76 DEFINITION A minimal prime ideal o f minimal prime ideal (in N).

{O}

is called a

2 . 7 7 COROLLARY

(a) Each prime ideal contains a minimal prime ideal. (b) N has a minimal prime ideal. Proof. Take

I = {Ol

in 2.75.

As i n ring theory (cf. e.g.(McCoy)), ideals deserve some interest.

the complements o f prime

2.78 DEFINITION M C N is called an m-system if b a , b E M 3 alc(a) 3 blE(b) : alblcM. 2.79 EXAMPLES (a) b and N are trivial examples of m-systems. (b)

2.61(c)

v

nEN: {n,n2,n3

,... 1

is an m-system.

gives us

2.80 COROLLARY If an m-system.

P

A

N,

P is a prime ideal iff

2.81 PROPOSITION (Van der Walt ( I ) ,

N\P

is

Ramakotaiah ( 3 ) ) . Let M c N be a non-void m-system in N and I an ideal o f N with I A M = 0. Then I is contained in a prime ideal P N with P n M = 0.

$ 2 IDEAL THEORY

66

P r o o f . z: = IJgN: J 1 I A J n M = p ) . I E Z . B y Z o r n ' s L e m m a , Z c o n t a i n s a m a x i m a l e l e m e n t P . P i s a n i d e a l p N. P is in fact a prime ideal: I f Jl=P A J2==P t h e n t a k e s o m e j l E J l n M a n d j2EJ2n M.

and

( j 1 ) ( j 2c ) J 1 J 2 ,

(J1J2)n M

So

f 0,

3

jjc(j,)

4

(J1J2)

3

j;€(j2)

and

P

JlJ2

: jijhEM.

4

P.

3 . ) SEMIPRIME IDEALS 2.82 -

i s semiprime:

DEFINITION S A N

<->u

I sl N :

I 2 eS => I E S .

Evidently, each prime ideal is semiprime. S i m i l a r to 2.61 w e g e t 2 . 8 3 PROPOSITION F o r a n i d e a l S o f N t h e f o l l o w i n g c o n d i t i o n s

are equivalent: (a) S is semiprime. 2 (b) ISN: ( I )ES ->

v

IES.

2 . 8 4 PROPOSITION If

then

I] S a

(Sa)aEA is a family o f semiprime ideals is again semiprime.

acA

Proof. If

I A N

and

I2 c

n

Sa

aEA

SO

V

aEA:

I E S ~ , hence

I

E

then

n

V

aEA:

I 2ES

a'

Sa.

aEA

As in 2 . 6 3 and 2.64 w e g e t

2.85 PROPOSITION L e t

be semiprime then

I 9 N

be a direct summand and Sn I i s semiprime in I .

S 4N

67

2d Prime ideals

2 . 8 6 PROPOSITION I4N

w(S)ENN/I

ICSSN.

A

Then S is semiprime iff

i s semiprime.

2 . 8 7 DEFINITION N is called a semiprime near-rinq if

is

{O]

a semiprime ideal. 2 . 6 7 , and 2 . 7 4

-

2 . 7 7 c a n again be transferred t o semiprime

near-rings (Ideals). 2 . 8 8 DEFINITION (Maxson ( 1 ) ) .

if

SES

3

S e N sl,s2€(s) : sls2cS.

is called an sp-system

2 . 8 9 PROPOSITION (Maxson ( 1 ) ) .

(a) Each m-system is an sp-system. (b)

S A

N:

S

is semiprime <->

N\S

is an sp-system.

T h e proof is trivial.

2.90

PROPOSITION (cf. 2 . 8 1 ) . L e t S be a non-void sp-system in N. L e t I be an ideal o f N with I n S = 0. Then I is contained in a semiprime ideal N.

Now w e study s o m e relations between prime and semiprime ideals. Since each prime ideal is semiprime we get a t o n c e from 2 . 8 4 2 . 9 1 PROPOSITION Any intersection of prime ideals is a semiprime

ideal, 2 . 9 2 PROPOSITION (Maxson (1)).

Let S be an sp-system and T h e n there i s s o m e m-system M w i t h S E M E S .

SES.

Continuing this process, o n e gets a sequence

with

v

kEIN

:

slk)sjk)~S

and

( s ) ? ( s 1 s 2 )a - ( s 1' s 2' ) ? . .

. .

68

5 2 IDEAL THEORY

Take M : = I s , s 1 s 2 , a d e s i r e d m-system.

If

s ( ~ ) s ( ~ s) ( ' ) s ( ' ) E M 1 2 ' 1 2

I A N,

2 . 9 3 DEFINITION I f

prime r a d i c a l o f

p(1)

O f course,

sisi,

2 . 9 4 PROPOSITION

call

...1 .

We show t h a t M i s

Lsk) t h e n

( w . 1 .o.Q.

?(I):=

np

the

P prime id. P 3 I 1 . Gojan ( 1 ) c a l l s i t t h e B a e r - r a d i c a l .

i s a semiprime ideal nEfJ(1)

->

3

(by 2 . 9 1 ) c o n t a i n i n g I .

k

kEIN : n € 1 .

i s an m-system ( 2 . 7 9 ( b ) ) . Proof. M:= ( n , n 2 , n 3 , . . . ) If I n M = ff t h e n b y 2 . 8 1 t h e r e i s some p r i m e i d e a l P I I w i t h P n M = fl, a c o n t r a d i c t i o n t o n E r ( 1 ) . k Hence I o M .f $ a n d 3 k t z l N : n €1.

G r o e n e w a l d ( I ) , V . S . R a o ( 1 ) ) L e t I y-N . ( a ) I i s semiprime i f f ?(I) = I . ( b ) I f I i s semiprime then I i s t h e i n t e r s e c t i o n of a l l prime i d e a l s c o n t a i n i n g I . ( c ) ? ( I ) i s t h e i n t e r s e c t i o n of a l l semiprime i d e a l s containing I. ( d ) A semiprime n e a r - r i n g i s a s u b d i r e c t p r o d u c t of prime n r ' s .

2.95 T H E O R E M (Gojan ( I ) ,

S i n c e ( a ) - > ( b ) = > ( c ) a n d by 1 . 5 8 a n d 2 . 9 1 , we o n l y h a v e t o show => i n ( a ) . S u p p o s e p s T ( I ) \ I . S i n c e N\I i s a n s p - s y s t e m , 2 . 9 2 p r o v i d e s u s w i t h a n m-system M s u c h t h a t

___ Proof.

aEMsN\I.

But

M A I C ( N \ I ) ~ I= @ c o n t r a d i c t s

asrfI).

F o r more a n d some o t h e r r e l a t e d m a t e r i a l s e e B e i d l e m a n ( 7 ) . F e r r e r o - C o t t i ( 7 ) , G o j a n ( l ) , Oswald ( 5 ) , ( 8 ) and R a m a k o t a i a h Rao ( 5 ) , S a n t h a k u m a r i ( 2 ) . S e m i p r i m a r y n e a r - r i n g s w e r e c o n s i d e r e d in a s e r i e s o f papers by K a a r l i . See in p a r t i c u l a r K a a r l i ( 7 ) and 9.260.

2e Nil and nilpotent

69

e) NIL A N D NILPOTENT

2.96

DEFINITION

(b)

S

N

C

is called nilpotent if is called nil i f all

(c) S e N 2.97

3

i s c a l l e d n i l p o t e n t if

(a) nEN

k E I N : n k = 0.

3

kcIN: S k = CO)

SES

are nilpotent.

REMARKS ( a ) S C N n i l p o t e n t - > S nil. ( I n 3 . 4 0 w e w i l l s e e t h a t i f NEW has the DCCN t h e n " n i 1 " a n d " n i l p o t e n t " c o i n c i d e f o r N-subgroups.)

T

(b) S 5

2.98

2.99

S

N

A

nil(poter,t) => S nil(potent).

T

EXAMPLES ( a ) In

n4[xJ,

( b ) If

ncNc

COROLLARY If

2x

i s nilpotent.

is n i l p o t e n t t h e n

I A N

n = 0.

i s nil t h e n

I

c

No.

P r o o f . B y 2.18, I = IotI C' so by 2.97(b) Ic = NcnI i s n i l , h e n c e by 2.98(b) I c = ( 0 1 a n d I = 1, c N 0' 2.100 T H E O R E M ( R a m a k o t a i a h ( 3 ) ) . I 9 N . <=> I a n d N/I a r e nil(potent).

N i s n i l ( p o t e n t ) <=>

Proof (for nilpotence) ==>: by 2.97(b), I is n i l p o t e n t . I f 3 k c l N : N k = CO} t h e n (N/I)k kl'Nk: k2€IN : I

<=-'

3

Therefore

(N/1lk'

= {I},

so

N k/ I

N k l eI

=

(11.

and

= (01.-

k k ( N ')

*=

CO)

a n d N is n i l p o t e n t .

T h e proof f o r "nil" is similar.

70

52

IDEAL THEORY

2.101PROPOSITION (Ramakotaiah (1)). Let

Ie

(CIEA)

be ideals

of N. (a) (

baEA:

Ia nilpotent

A

A finite) =>

1 Ia

i s nilpotent.

aEA

Proof. Let

1L

I,J

be nil(potent) ideals. I+J/I J/I n J ’ By 2.100 JJIn and by assumption I are nil(potent). Harnessing 2.100 again, I+J is nil(Potent1. By induction we get (a) and (b) for a finite A . I n (b), let ic 1 Ia : i = i +...ti (say). asA al “k Then iEI +...+I and i is again nilpotent. al ak Scott ( 1 1 ) even s h o w e d t h a t the s u m o f a nilpotent N-subgroup and a nilpotent i d e a l i s nilpotent if N = N o . 2.102 PROPOSITION (Polin (2)). Let N be isomorphic to a sub-

direct product Qf near-rings N u ( ~ E A ) without non-zero nil(potent) N-subgroups, left ideals or ideals. Then N has the same property. Proof (for N-subgroups of N). Let M sN N be nil(potent). Let x u : N -r) Na be the usual epimorphisms ( 1 . 5 8 ) . Then all aa(M) are nil(potent) i n N u , hence = {Ol, s o M c fl Ker aa = {Ol and therefore M = {Ol. acA

(If N+Vo then 3 a c A : N a ~ V o . S o neither i n N nor in Na there are nil(potent) N-subgroups and the proposition is meaningless in this case). The following proposition will be useful later on. I,J A N A I is nil(potent) in N/J.

2.103 PROPOSITION (Polin (2)).

Then

ItJ/J

Proof: by 2.8 a n d 2.100.

nil(potent).

2e Nil and nilpotent

71

There a r e s e v e r a l c o n n e c t i o n s between n i l ( p o t e n t ) a n d ( s e m i ) p r i m e i deal s : 2 . 1 0 4 P R O P O S I T I O N (Maxson ( 1 ) ) . I f I 9 N . Then n i l p o t e n t i d e a l s i f f I i s semiprime.

N/I

h a s no

P r o o f . ->: Assume t h a t N / I has n o n i l p o t e n t i d e a l s a n d 2 2 t h a t J A N , J e I . Then ~ ( 5 =) a ~( J ) = 111 ( z e r o i d e a l o f N / I ) . So n ( J ) = {I} and J c I . P i s prime a n d J O N w i t h J k 5 P t h e n ( J k - ’ ) J t ( Jk ) 5 P b y r i g h t d i s t r i b u t i v i t y . So

<-.

. If

J c P b y i n d u c t i o n . By 2 . 9 5 ( b ) , t h i s a l s o h o l d s i f I i s o n l y s e m i p r i m e . By 2 . 1 0 3 , N/I has n o nilpotent ideals. 2.105 T H E O R E M ( a ) ( P o l i n ( 2 ) ) . I f I -4 N n i l p o t e n t i d e a l s of N .

then

p(1)

contains a l l

(b) p ( { O l ) i s n i l . P r o o f . ( a ) ?(I) i s s e m i p r i m e . So N / P(I) has no n i l p o t e n t i d e a l s . Assume t h a t J i s a n i l p o t e n t i d e a l o f N. By 2 . 1 0 3 , Jtp(I)/p(I) i s n i l p o t e n t i n N / F ( I ) and t h e r e f o r e z e r o . Hence J c P(1). ( b ) f o l l o w s from 2 . 9 4 .

2 . 1 0 6 T H E O R E M ( L a x t o n ( 4 ) ) . I f N E n 0 has D C C I t h e n N i s a prime near-rinq c==> N has a s m a l l e s t i d e a l I u n d e r a l l n o n - z e r o i d e a l s and I i s n o t n i l p o t e n t . P r o o f . =>: I f N i s p r i m e , E O I i s a prime i d e a l . L e t I be a minimal i d e a l ( e x i s t e n c e g u a r a n t e e d by t h e DCCI). I i s n o t n i l p o t e n t by 2 . 1 0 4 . I f J i s a n o t h e r minimal i d e a l t h e n CO) IJ E I , so ( I J ) = I . S i m i l a r l y , ( I J ) = J , s o I = J. If K i s a n o t h e r n o n - z e r o i d e a l , K c o n t a i n s a minimal i d e a l of N, s o K c o n t a i n s I . Hence I i s t h e s m a l l e s t of a l l non-zero i d e a l s .

§ 2 IDEAL THEORY

72

Conversely, let I be the unique minimal ideal and suppose that I is not nilpotent. If J 1 , J 2 A N , J l J 2 = ( 0 1 , but J 1 4 I01 and J2 (01 then J 1 , J 2 contain a minimal ideal o f N by the DCCI and this minimal ideal = I . Hence I c J 1 A I c J 2 , so I 2 c J I J p = ( 0 ) and I is nilpotent, a contradiction. <-:

From 1.60 we get the following is prime and has the DCCI then N is subdirectly irreducible.

2.107 C O R O L L A R Y I f

NE%

2.108 R E N A R K S See Oswald (2) for a discussion o f "strictly (semi-)prime" near-rinqs and Holcombe (1) for "0-, 1-

and 2-(semi-)prime ideals" and their connection to vprimitive ideals (4.2 (c), cf. 4,341. Thereby I 4 N is called 0-(1-,2-) prime if for all ideals (left ideals, N-subgroups) A , B of N: A B c I =>A 5 I v B c I , and similar for 0-(1-,2-) semiprime ideals. So 0-(semi-)prime ideals are just our (semi-)prime ideals. See also Ramakotaiah-Rao (2). The correspondence between " n i l " and "nil potent" is further discussed i n 3.40 and 5.48 below, a s well as in Kaarli ( 4 1 , Oswald ( 6 ) and Scott (16). Local nilpotency is studied i n Gringlaz ( I ) . See also Beidleman ( 7 ) and Gojan ( 1 ) .

PART II

STRUCTURE THEORY

$ 3 ELEMENTS OF THE STRUCTURE THEORY 5 4 PRIMITIVE NEAR-RINGS

$ 5 RADICAL THEORY

74

33 ELEMENTS OF THE STRUCTURE THEORY

Irreducible (ring-)

m o d u l e s RM ( i . e .

s i m p l e ones w i t h RM

p l a y a n i m p o r t a n t r 6 1 e i n r i n g t h e o r y . They have e.g. property that N-groups

b

mcM:

Nr w i t h

Rm =

{ o l v Rm = M.

Nr f {ol

However,

p

{O))

the

simple

do n o t e n j o y t h i s p r o p e r t y .

It

m i g h t a l s o come t o m i n d t o u s e N - s i m p l i c i t y o r N o - s i m p l i c i t y (both equivalent t o s i m p l i c i t y i n the r i n g case).

So we d e f i n e

3 types ( t y p e 0 , l and 2 ) o f N-groups, a l l c o i n c i d i n g w i t h i r r e d u c i b i l i t y i n t h e case o f modules, w i t h t y p e 2 i m p l y i n q t y p e 1 a n d t h i s i n t u r n t y p e 0 . M o n o g e n i c N - g r o u p s (3 Y E N : Ny = r ) a r e p a r t i c u l a r l y i m p o r t a n t . F o r i n s t a n c e , we p r o v e t h a t e v e r y monogenic N-subgroup o f N c o n t a i n s a r i g h t i d e n t i t y has t h e DCCN.

If N=N,

We t h e n s t u d y t h e e f f e c t o n t h e t y p e o f

Nr o f c h a n q i n g N i n t o

N f l , No o r N., I n c),

m o d u l a r l e f t i d e a l s a r e i n t r o d u c e d i n t h e same w a y a s

f o r r i n g s . Many t h e o r e m s o f r i n q t h e o r y c a r r y o v e r t o n e a r - r i n g s : each modular l e f t i d e a l i s c o n t a i n e d i n a maximal one,

modular

l e f t ideals are e x a c t l y t h e a n n i h i l a t o r s of generators o f monogenic N-groups,

t h e i n t e r s e c t i o n o f two maximal modular

l e f t ideals i s modular, etc. I t i s advisable t o c a l l a modular l e f t i d e a l L u - m o d u l a r if N/L i s an N-group o f t y p e v. We p r o v e e . g . t h a t i f I i s a d i r e c t summand o f N = N o t h e n e v e r y v-modular l e f t i d e a l i n I i s t h e " t r a c e " o f one i n N . We a l s o i n t r o d u c e q u a s i r e g u l a r i t y f o r a b u s i n g i t t o s h o w t h a t

" n l l " and " n i l p o t e n t " c o i n c i d e f o r

M

sN

N

if

N i s a zero-

s y m m e t r i c n e a r - r i n g w i t h DCCN.

If

NEW,

has a r i g h t i d e n t i t y e and i f N i s t h e f i n i t e d i r e c t

sum o f l e f t i d e a l s t h e n d e c o m p o s i n g e i n t o gonal" idempotents

ei.

yields "orthoCei A n o t h e r m e t h o d ( d u e t o S.D. S c o t t ) t o

g e t o r t h o g o n a l idempotents i s p r e s e n t e d and c e n t r a l i d e m p o t e n t s

a r e discussed.

3a Types of N-groups

75

F i n a l l y , we c o n s i d e r z e r o - s y m m e t r i c n e a r - r i n q s N w i t h minimum c o n d i t i o n on N - s u b g r o u p s and show e . q . t h a t e v e r y " m i n i m a l n o n - n i l p o t e n t " N-subgroup ( l e f t i d e a l ) c o n t a i n s a r i q h t i d e n t i t y ( a non-zero i d e m p o t e n t , r e s p e c t i v e l y ) , and t h a t e v e r y m i n i m a l i d e a l i s a f i n i t e d i r e c t sum o f N - i s o m o r p h i c m i n i m a l l e f t ideals.

a \ T Y P E S OF N - G R O U P S 3 . 1 DEFINITION

(a)

N r i s monogenic:

<->

3

YEr:

( I n t h i s c a s e we s a y t h a t

Ny

r.

=

N r is

" m o n o g e n i c by

y"

and y i s called a generator f o r Nr.)

(b)

Nr i s tf YETI:

s t r o n g l y m o n o q e n i c : <-> (Ny = {o) v Ny = r ) .

Nr

i s monogenic a n d

3 . 2 R E M A R K O b s e r v e t h a t a s t r o n g l y m o n o q e n i c N-group has R = {ol o r n = r.

Nr

3 . 3 EXAMPLES

( a ) Each (b)

M(r)r

Nr and

with

MJr)

R =

r

r

i s s t r o n g l y monogenic.

are s t r o n g l y monogenic.

See a l s o the examples 3 . 8 , 3.9 below. Now we l i s t some p r o p e r t i e s o f m o n o g e n i c N - g r o u p s w h i c h a r e useful for the sequel.

53 ELEMENTS OF THE STRUCTURE THEORY

76

3.4

Nr b e m o n o g e n i c ( b y

PROPOSITION L e t

( a ) L 9%N =-> L y o AN

r. b

(b) I f e i s a l e f t i d e n t i t y o f N then (c)

Then

yo).

I f e i s a l e f t i d e n t i t y o f N and i f

e y = y.

YETI:

i s faithful

Nl'

then e i s a two-sided i d e n t i t y .

r i s No-simple t h e n e i t h e r NI' =

(r

(d) If

c a n be c o n s i d e r e d as an N o - g r o u p ! ) Nr i s s t r o n g l y m o n o g e n i c .

(01 o r

%

Nr -N (f)

(9)

"{o:y0)

Nr i s s i m p l e

<->

or

= N.

Nr

i s N - s i m p l e <=+

between

r

t h e r e i s no N - s u b g r o u p s t r i c t l y

L 1 ,L 2 4L N

but

-Proof. V

i s a maximal

(O:yo)+Nc

(o:yo)

i s a maximal No-subgroup

N.

Nr

( i )( B e t s c h ( 6 ) ) I f

3

<->

= N.

i s N o - s i m p l e <->

or

N

and

(o:yo)

N-subgroup o r

(h)

i s a maximal l e f t i d e a l

(o:yo)

Lln (a)

b

L2

with

3

yEr

y = nyyo.

nyEN:

b

I l y o ~ l y ob nEN

= n ( n y tIly ) - n n y

Y O

0

ycr:

b'

b

yEr

(d)

b

y e r : Ny 5

ncN: NO

r.

Ly,.

E

i s a normal

Lyo

e y = e n y y o = n y Y O = Y.

(c)

or R =

,

so

= (n(nytL)-nny)yo

r.

( b ) As i n ( a ) ,

= N

n(ytlyo)-ny =

I n t h e same way one shows t h a t

subgroup o f

and i f

NE??,,

= L2t(o:yo)

then N i s a ring.

(o:yo),

E

Y O

i s faithful, L1t(o:yo)

o = ny-ney = (n-ne)y,

r

So e a c h

implies Ny

(y = 0)

h:

a p p l y t h e homomorphism t h e o r e m .

N n

= Io)

R t o be

e q u a l s e i t h e r I01

( e ) Consider t h e N-epimorphism

n = ne.

so

-+

-+

r nY0

or and

r.

77

3a Types of N-groups

-

(f)

(h) follow from (e),

2.16

and t h e "second

is o m o r p h i sm t h e o r e m " . 2,

(i)

-N

By

=

/ (L1

= (L1+(O:Yo))n (L2+(o:yO)) (0:r')

3.5

= (o:r) = io).

So b y 2 . 2 4 ,

r

DEFINITION A monogenic N-group

=:I-'.

L2)+(o:yo)

with

N i s a ring.

r

Co)

i s said

t o be o f t y p e 0 : <=+

Nr

t y p e 1: <=>

N r i s s i m p l e and r i s No-simple.

t y p e 2 : <=>

i s simple s t r o n g l y monogenic

The d e f i n i t i o n o f " t y p e 2 " c r i e s f o r 3.6

R E M A R K O f c o u r s e i t seems m o r e n a t u r a l t o d e f i n e " t y p e 2 " by "N-simple"

F a i n (1)). B u t N - s i m p l i c i t y says

(see e.g.

v e r y l i t t l e a b o u t r . F o r i n s t a n c e , e v e r y n o n - z e r o subn e a r - r i n g N o f M(r) w i t h N ? M c ( r ) h a s ,4F o f " t y p e 2 " t h e n . So one c a n g e t n e a r l y e v e r y t h i n g e x c e p t n i c e s t r u c t u r e

theorems. 3.7

M o r e o v e r , we w o u l d n o t g e t

PROPOSITION

(a)

Nr

Nr o f t y p e 1

o f t y p e 2 ->

(b) If

Nr

i s o f type 1 o r 2 then

(c) If

NT

i s a unitary

r N

t y p e 1 c->

b

NY = ?!

(a):by

3.4(d)

Proof.

Nr

o f t y p e 0.

R = {oj

N = No-group

i s o f type 2.

ycr* :

->

Nr

then

r.

R =

or

is of

I n t h i s case,

(see a l s o 3.19(a)!). and 1.34.

( b ) f o l l o w s f r o m ( a ) and 3 . 2 . ( c ) ->:

Let

Nr

~ E A : N6 = { o )

s i n c e each

<-:

by (a).

b e o f t y p e 1. I f or

6~N6.

=

r.

A sN 0

Hence

A = {o)

r

then or

A =

r,

5 3 ELEMENTS OF THE STRUCTURE THEORY

78

3.8

r

EXAMPLES I f

Z4,

=

define

(Betsch ( 3 ) )

No:= ~ f ~ M o ( ~ ) ~ f ( 2 ) ~ { 0 , 2 ~ l N1:=

{ f c M o ( r ) l f ( 2 ) = 01

N2:= { f E M o ( r ) l f ( 3 )

r

Then

i s o f t y p e 0,

N2 (where

but not o f type 1

i s o f t y p e 1, b u t n o t o f t y p e 2

Or

%

= 0).

r

i s o f type 2

ny

i s d e f i n e d as i n l . l 8 ( c ) ) .

T h i s c a n be seen b y s i m p l e c a l c u l a t i o n s . 3.9

N = Nc

EXAMPLES I f

Nr

then

i s o f t y p e 0 <=>

n

group w i t h

=

Fir

i s o f t y p e 1 <=+

r

i s a simple

r,

N r i s o f t y p e 2 <=> r i s a c y c l i c g r o u p o f p r i m e o r d e r . T h i s h o l d s s i n c e N - k e r n e l s (No = { O l - s u b g r o u p s ) i n r c o i n c i d e w i t h normal subgroups (subgroups, r e s p e c t i v e l y ) o f r ; 3 y0€r: = r r e s u l t s i n R = N~ = %o = r . 3.10 P R O P O S I T I O N (Betsch ( 3 ) ) .

L sN N

g e n e r a t o r y ) and l e t with

L

( 0 : ~ ) . Then

P r o o f . By 3 . 4 ( a ) ,

h: L

I , AN

-c

""Lv

r

E

be a minimal l e f t i d e a l

-N r . ?I

L

r.

Nr i s s i m p l e ,

Since

Nr b e o f t y p e 0 ( w i t h

Let

HomN(L,r)

By

L, and

L$(O:y),

=

Ly

{ol.

r. Ker h = L n ( o : y )

=

(01

(since L i s minimal). 3 . 1 1 COROLLARY L e t

N =

1 Li€:.n0,

w h e r e I i s some i n d e x s e t and

iE1

all

Li

a r e m i n i m a l l e f t i d e a l s o f N. L e t v be

~{0,1,21.

( a ) E a c h N - g r o u p o f t y p e v i s N - i s o m o r p h i c t o some Li. ( b ) I f i n i t e -> t h e r e a r e o n l y f i n i t e l y many c l a s s e s o f non-N-isomorphic N-groups o f t y p e v.

79

3a Types of N-groups

P r o o f . ( a ) L e t N r be o f t y p e v a n d g e n e r a t e d by y . S i n c e r = NY {ol, 3 ieI: Li 9 ( o : ~ ) . Now a p p l y 3 . 1 0 .

+

( b ) Follows from ( a ) . 3 . 1 2 L E M M A ( b c o t t ( 4 ) ) Let N h a v e t h e D C C N and l e t M 5 N N be m o n o g e n i c ( b y m o ) a n d 3 m l € M : (O:ml) = {Ol. T h e n M c o n t a i n s a r i g h t i d e n t i t y a n d (O:mo)M = {Ol. P r o o f . Let a l l a n n i h i l a t o r s be t a k e n i n M . ( a ) S i n c e (O:m,) = (01, t h e map h : M + Mml is an N-isomorphism; moreover, Mml E M . I f Mml = M and so o n , t h e n a p p l y i n g h we g e t Mm12 = M m l , c o n t r a d i c t i n g t h e D C C N . So M m l = M and 3 e E M : eml = m l . B u t t h e n 'd m E M : meml = m m l , s o b m E M : me-rnE(0:m ) = { o l e 1 T h e r e f o r e e i s a r i g h t i d e n t i t y i n !,I. ( b ) Mmo = M -> 3m2sM: m2mo = e . I f mm2 = 0 t h e n a l s o me = rnm2mo = Omo = 0, so (O:m2) t ( 0 : e ) = ( 0 ) . as i n the beginning of ( a ) So (0:m2) = I 0 1 and M = Mm2. S u p p o s e t h a t r n 3 ~ ( O : m o ) . 3 rn4cM: m 3 = m 4 m 2 . So m4 = m4e = m m m = 0 a n d m3 = m4m2 = 0 . 4 2 0 = m3mo T h i s shows t h a t (0:mJ = CO).

-

3 . 1 3 T H E O R E M ( S c o t t ( 5 ) ) . I f N = No has t h e D C C N and M sN N i s m o n o g e n i c ( b y m o ) t h e n M c o n t a i n s a r i g h t i d e n t i t y and ( O : m o ) M = { O l . P r o o f . Again, a l l a n n i h i l a t o r s a r e t o be t a k e n i n M . In view o f lemma 3 . 1 2 we " o n l y " h a v e t o show t h a t 3 m l E M : ( O : m l ) = {Ol. S u p p o s e t h a t 3 M ' 3N N 3 mAcM': M' m o n o g e n i c by m i a n d ( 0 : m A ) =/= {Ol. W.1.o.g. we may a s s u m e t h a t M ' i s m i n i m a l f o r c o n t a i n i n g s u c h an m ' . Let m i E M ' b e s u c h t h a t ( 0 : m i ) i s m i n i m a l i n {(O:m')IM'm' = M ' ) (so a l s o = MI). Therefore

M'mi

80

4 3 ELEMENTS OF THE STRUCTURE THEORY

3 If

mjEM': mimi = mi and (0:mi) 5 (0:mi) a s in 3.12(b). m i g e n e r a t e s M i t h e n rninimality o f (0:mi) f o r c e s (0:mi) = (0:mi). m'eM':m'mimi = m ' m i , s o V m'EM': m'mi-m'E(0:mi) = (0:m;) a n d hence m'EM': (m'mi)mi = m'm; s h o w i n g t h a t m i is a r i g h t identity i n M'mi = M' and s o (O:mi)vl = { o } . By 3.12(b), ( O : m o ) = {Ol.

If

d o e s n o t g e n e r a t e M' t h e n M'm; < M I . C o n s i d e r . the s e q u e n c e M' > M'm; t M'(mi) 2 t 3 kEIN : M'(mi) k = M'(m;) ktl = . Thus (M'(mi)k)(mi)kt' = M'(m;)k and s i n c e m i : = = ( m i ) kt 1 E M ' ( m i ) k , m i g e n e r a t e s M'(mi) k . mi

...

By t h e minimality o f

M',

(0:m;)n

...

=

{Ol.

A g a i n using the minimality o f M I w e see that e a c h h a s ( 0 : m i ) n M'mj = g e n e r a t o r m i o f M'(mi)k=W'm; = (0).

We shall s h o w t h a t

mi: = mimi violates this statement. (a) m i m j generates M ' m j , for mi = m j m i and M ' m i = M ' imply t h a t (M'mj)(mimi) = = M'mimj = M ' m i . f C O I , f o r otherwise M ' -N M ' m j 5 M'm'2 < MI. T a k e s o m e n o n - z e r o mgc(0:mj). 3 mieM': mkmi m i , since m i generates M ' . 6 3 1 3 ' Now 0 = m;mj = m k m i m j = m'm'm'm' H e n c e mkmje(0:rnimj)n M ' m j , b u t m k m i $. 0 s i n c e mimimi = m k m i = m k 0.

( b ) Observe that

(0:mi)

%

+

S o we a r r i v e a t

a c o n t r a d i c t i o n and the p r o o f is

complete. N-aroups o f type 0 o v e r a semiprimary ( s e e 9 . 2 6 0 ) near-ring N are studied in Kaarli ( 2 ) , ( 4 ) and (6). Holcombe-Walker ( 1 ) study N-aroups Nr o f type 3 (i.e. ,.,r is o f type 2 with (f/ n E N : ny = ny')=> y = y ' . The s u m of all left ideals L o f N = N o , where NL i s o f type 1 , i s called socle o f N (see e.a. Ramakotaiah (3)).

81

3b Change of the near-ring

b ) C H A N G E OF T H E N E A R - R I N G

Up t o now we h a d a n u n j u s t s i t u a t i o n : a n e a r - r i n g

keeps an harem o f N - g r o u p s , b u t n o t c o n v e r s e l y . Now we l e t an N - g r o u p N r c h a n g e i n t o N / I r ( f o r some I A N), N o r , N cr . T h e s e changes w i l l be an i m p o r t a n t t o o l i n l a t e r c o n s i d e r a t i o n s . 3 . 1 4 PROPOSITION ( B e t s c h ( 3 ) ) . L e t I b e a n i d e a l o f N , group and u E I 0 , 1 , 2 1 .

(a) If

r

i s a n N-group w i t h

(o:r)

I c

r a

then

( n t 1 ) y : = ny

r

makes

i n t o an

N/I-group

N/Ir.

I f Nr i s o f t y p e w , s o i s If (b) If

Nr

N/Ir* i s f a i t h f u l , t h e same a p p l i e s t o

N/Ir'

r i s a n N/I-group t h e n ny: = ( n t I ) y

makes If If

r i n t o an N - g r o u p

N/Ir i s NIIr i s

Nr

with

o f t p y e v , so i s f a i t h f u l then

I

C

(o:r)"

Nr.

I = (o:r)N.

The p r o o f i s a c o l l e c t i o n o f s t r a i g h t f o r w a r d a r g u m e n t s a n d therefore omitted. Observe t h a t ( N / I ) o = { n o t I l n o c N o ) . and a s an Each N - g r o u p r c a n b e v i e w e d a s an N o - g r o u p N,-group r i n a n o b v i o u s way ( b y r e s t r i c t i o n ) . I n 3 . 4 ( d ) NC

we a l r e a d y m e n t i o n e d t h i s f a c t . We now s t u d y t h e r e l a t i o n between r, r a n d Nr: NO

NC

8 3 ELEMENTSOF THE STRUCTURE THEORY

82

r be a n N - g r o u p and A a s u b s e t o f r . ( a ) Nr is faithful i f f r and r a r e faithful.

3.15 P R O P O S I T I O N Let

NC

NO

( b ) A AN

r



A AN,

( c ) A sN r <=> A

r r

5

A R c A.

NO

Proof. ( a ) I f Nr i s f a i t h f u l , t h e s a m e t r i v i a l l y a p p l i e s r and r . C o n v e r s e l y , l e t n r b e = {03. to NC

NO

T h e n ( w i t h n = no+n C a s in 1.13) \I YEr: noy+nco = = n o y + n c y = ny = 0 . T a k i n g y = o y i e l d s nco = 0 . So y c r : noy = o and no = 0. But nco = o g i v e s y E r : n,y = 0, h e n c e n c = 0. T h e r e f o r e n = n o +n c = 0. ( b ) => is t r i v i a l . If

A AN

r

then

0

i

6cA yEr ncN: n(6ty)-ny = no(6+y)tnc(6+y)- n c o - n 0y = n 0 (6ty)+nco-n,o-noyEA. ( c ) is even m o r e trivial.

Nr a n d

The r e l a t i o n b e t w e e n 3.16 COROLLARY

Let

Nr E

NO

r

i s particularly important.

N(ZI. r

( a ) Nr is s i m p l e <-> NO

is simple.

(b) Nor is m o n o g e n i c by y ->

Nr is m o n o g e n i c by

( c ) Nor is s t r o n q l y m o n o g e n i c -> genic o r (d)

iol

r is N,-sirnple

!,r

y.

is s t r o n n l y m o n o -

+ R + r. ->

r is N-simple.

3.17 EXAMPLES I f N = Mc(Z4) t h e n Z4 is N - s i m p l e but not No-simple ( s i n c e IO,23 i s a n No = IO3-subgroup). So N - s i m p l i c i t y d o e s n o t i m p l y N,-simplicity.

P l u g g i n g all t o g e t h e r y i e l d s

83

3b Change of the near-ring

Nr

3.18 THEOREM Let

be an N - g r o u p and

( a ) Nr i s of t y p e w

r

-?

wc{O,l,21.

Nor = C o ) .

i s of type w o r

NO

( b ) Nor i s o f t y p e w ( f o r

Nr n

or

= Co)

Nr

P r o o f . ( a ) Anyhow,

r)

=

Q.

w = 1

assume t h a t i n

Nr i s o f t y p e

->

w.

r

i s simple, t h e r e f o r e a l s o NO

by 3 . 1 6 ( a ) . L e t Nr be monogenic by y . Then Hence Noy = ( 0 1 o r Noy = r . If

Noy =

r,

r

Noy

sk

r

by 3 . 4 ( a ) .

i s monogenic, t o o .

NO

If

Noy =

(01

then

r

= Ny = NoytR = R

implies

t h a t b YEr: Ny = r . Again by 3 . 4 ( a ) , b y e r : Noy = C o ) o r = r . So e i t h e r r i s monogenic o r N o r = C o ) . If

Nr If

NO

i s of t y p e 1 t h e n

{ol

or

R =

r.

t h e n b y c r : Ny = NoytR = Noy i s a g a i n o f t y p e 1.

C2 =

r

=

Col

and

l\10

If = r t h e n e a c h y e r g e n e r a t e s Nr s o ( a g a i n by 3 . 4 ( a ) ) t/ Y E r : Noy = I01 o r Noy = So r i s e i t h e r o f t y p e 1 o r Nor = Io).

r.

NO

The a s s e r t i o n f o r

w = 2

i s trivial.

( b ) By 3.16.

3.19 R E M A R K S ( a ) 3 . 1 8 ( a ) and ( b ) show t h a t 3 . 7 ( c ) h o l d s f o r a r b i t r a r y near-rings! ( b ) Information a b o u t t h e behaviour of

can be found i n M l i t z ( 3 ) .

Mr

with

M sN N

$ 3 ELEMENTSOF THE STRUCTURE THEORY

84

c ) MODULAR1 TY

3.20

L A,

DEFINITION

3

<->

b

eEN

N

i s c a l l e d m o d u l a r : <->

n&N : n - n e c L .

I n t h i s c a s e we a l s o s a y t h a t L i s " m o d u l a r b y e " and t h a t e i s a " r i g h t i d e n t i t y modulo L " ( s i n c e

b

ncN:

ne

n (mod L ) ) .

f

3 . 2 1 REMARKS

(a) If

A,

L1,L2

L1

with

N

t h e n L 2 i s m o d u l a r b y e, ( b ) I01

L2

5

i s modular by e

a n d L1

too.

i s modular i f f N contains a r i g h t i d e n t i t y .

( c ) E v e r y normal subgroup o f (Nc,+) i s a modular l e f t i d e a l o f Nc {by a n y e l e m e n t o f N c ) ,

(d) I f L i s modular by e i n

Nc'YlO

then

ecL

iff

3 . 2 2 PROPOSITION ( B e t s c h ( 3 ) ) . E a c h m o d u l a r l e f t i d e a l i s c o n t a i n e d i n a maximal one ( w h i c h i s modular, Proof.

L =

N.

L)NEV, too).

L e t L b e m o d u l a r b y e . A p p l y Z o r n ' s Lemma t o t h e

I

set o f a l l l e f t ideals

3

L

with

e

4

I

and

use 3.21(a). P r o p o s i t i o n 3.22 3.23

i s n o t a l w a y s t r u e if N

PROPOSITION ( B e t s c h ( 3 ) ) . <==>

3

re 9 N N

P r o o f . ->:

3

ycr:

Nr

L 3, N

f

= {n+LlncN)

y:

i s m o d u l a r <=>

<->

n ( e t L ) = L <->

<-:

Let

But then

NT

tl

since

= e+L,

= N/L =

see 3.21(c).

monogenic b y y A L = ( 0 : ~ ) .

L e t L b e m o d u l a r b y e . Then

m o n o g e n i c by

No:

r.

N(e+L)

Moreover,

n e c L <->

Nr: = N/L

i s

= {ne+LlnsN)

ne(o:y)

ncL.

b e m o n o g e n i c by y . T h e n 1 e c N : e y = y. s o b ncN: n - n e c ( o : y ) = L

ncN:ney = ny,

a n d L i s shown t o b e m o d u l a r b y e .

=

= (L:e+L)<=>

85

3c Modularity

Applying 3.4(e)

we g e t

3 . 2 4 COROLLARY

L dt N

i s m o d u l a r =>

L

2

(L:N).

r

P r o o f , T a k e some ( b y y ) m o n o q e n i c N - g r o u p L = (o:y)

Then 3.22

-

(o:r)

3

= (o:N/L)

with

3.24 a r e s i m i l a r t o t h e r i n g case ( ( J a c o b s o n ) ,

Looking a t

/L:N)

L =

( 0 : ~ ) .

= (L:N).

more c l o s e l y g i v e s f o r f u t u r e use

pp.

5-6).

(cf.

Ramakotaiah ( 1 ) ) :

3.25

L e t L b e m o d u l a r b y e. Then

PROPOSITION

(L:N)

= (L:Ne)

and t h i s i s t h e g r e a t e s t i d e a l o f N c o n t a i n e d i n L . Proof.

b b

c (L:Ne)

(L:N)

i s clear.

n'EN : nn'eEL. n'EN:

nn'EL.

But

If

n&(L:N)

So

nE(L:Ne)

nn'-nn'eEL, and

then

hence (L:N)

= (L:Ne).

By 1 . 4 2 , (L:N) i s a l e f t i d e a l and i t i s e a s y t o s e e t h a t i t i s e v e n a n i d e a l o f N , (L:N) 5 L h o l d s t y 3.24.

If

I 9N

I

with

5

L

then t r i v i a l l y

1

C

(L:N).

3 . 2 6 THEOREM ( c f . ( K e r t e s z ) , p , 1 2 2 ) . I f N = Ll+L2, where Ll,L2 a r e modular l e f t i d e a l s , then L 1 n L 2 i s again ~

modular. Proof. Let

L1,L2

Decompose

be modular b y

el,e2,

el = %+%2 e 2 = "1+%2

where

We c l a l m t h a t

L1n L2

If

n-ne = n-n(k21+t12)

-n( But

ntzN

then

kll,f.21~Ll,

n-nelEL1,

(51++)

Therefore Similarly,

b

k12'k22EL2'

i s modular by

%21+k12) = n-nel+nel-n

nk12-n

respectively.

el,e2:

't21+k112 = : e .

= n-na12+nt12-

( - E l l+el ) + n t 1 2 - n ( R Z 1 + L l 2 ) .

nel-n(-kll+el)EL1

and

'L1ntzN:

n-necL1.

nEN: n - n e E L 2 ,

a n d we a r e t h r o u g h .

86

$ 3 ELEMENTS OF THE STRUCTURE THEORY

3.27 C O R O L L A R Y

( a ) I f L i s a m o d u l a r a n d M a maximal m o d u l a r l e f t i d e a l then L n M i s modular. ( b ) A f i n i t e i n t e r s e c t i o n o f maximal m o d u l a r l e f t i d e a l s

i s modular.

( t ) I f N i s a d i r e c t sum o f two m o d u l a r l e f t i d e a l s t h e n N contains a riqht identity. ( d ) ( B e t s c h ( 3 ) ) . I f N c o n t a i n s a f i n i t e f a m i l y o f maximal modular l e f t i d e a l s w i t h z e r o i n t e r s e c t i o n t h e n N contains a right identity.

D-E F I N I T I O N L e t v be ~ ~ 0 ~ 1A l ~e f t2 i d~e a .l L o f N i s 3.28 c a l l e d v-modular i f L i s modular a n d N / L i s an N-group ( v i a n ( n ' t L ) : = n n ' t L ) of type v. L e t gv(N) be t h e s e t o f a l l u - m o d u l a r l e f t i d e a l s o f N . 3.29 R E M A R K So a 0-modular l e f t i d e a l i s j u s t a modular m a x i m a l o n e and a 2 - m o d u l a r l e f t i d e a l L i s a m o d u l a r maximal l e f t i d e a l w i t h no N o - s u b g r o u p s t r i c t l y b e t w e e n L a n d N . (Beidleman c a l l s t h e s e l e f t i d e a l s " s t r i c t l y m a x i r a l " . ) v - m o d u l a r l e f t i d e a l s t u r n o u t t o be v e r y u s e f u l i n d e t e r m i n i n g r a d i c a l s of r e l a t e d near-rinqs. 3 . 3 0 PROPOSIlION L e t p r o d u c t . Let L i Denote

ll M j jEI

( N i ) i c I be n e a r - r i n g s a n d N t h e i r d i r e c t be a l e f t i d e a l o f N i f o r some i E I .

with

Mj:

Then f o r v ~ C 0 , 1 , 2 ) , L i v-modular i n N .

[

=

i + j Nj Li

by

i = j

i s v-modular i n N i

ITi qk N . iff

Li

is

Proof. ( a ) I f Li i s v-modular i n N i then (Ni/Li) n i ( ( . . . , n: , . . . ) t L i ) : Ni = i s o f type v. By = (... , O , n . n ' , O , , . . ) t r i , becomes a n N i - g r o u D 1 i 2, a n d c l e a r l y N/Ci = Ni/Li. S o N/Ti i s an

N/ri

Ni

3c Modularity

If

o f t y p e v.

Ni-group

the statement),

Eli ( n o t a t i o n

Ji:=

% =

Ni

a7

s o 3.14(b)

N/Ji,

as i n

shows t h a t

i s an N-group o f t y p e v (and t h e m u l t i p l i c a t i o n N/Ei i s t h e same a s i n 3 . 2 8 ) . H e n c e Ei i s v - m o d u l a r i n N. (b)

ri

If

N/Ti

i s v-modular i n N then

N-group o f t y p e v. S i m i l a r t o ( a ) , (...,ni,...)(n;tLi):=

where

of Ni/Li 3.14(a) of 3.28. From 2.28, 3.31

2.30

N

i n Ni/Li

Li

=N Ni/Li

The a n n i h i l a t o r

(as i n (a)),

i s an Ni-group

Therefore

N/SCi

n 1. n !1t L i .

Ji

contains

i s an 'Ir

s o by

of type v i n t h e sense

in

i s u-modular

Ni.

a n d 3 . 3 0 we g e t

I e- N

COROLLARY I f

i s a d i r e c t summand i n N a n d i f L c y v ( I )

t h e n t h e r e i s some L t Y , , ( N ) T h e r e e x i s t e x a m p l e s ( s e e e.g. o f l o w o r d e r ) such t h a t 3.31

L = CnN.

with

N) 1 ) i n t h e a p p e n d i x ( n r .

no.

' S

does n o t n e c e s s a r i l y h o l d f o r v = l i f

I i s n o t a d i r e c t summand. See a l s o E x . 6 . 3 2 i n Y e l d r u m ( 1 3 ) . Is,

i n 3.30,

i s a counterexample 3.32

ci?

e v e r y L c Y V ( N ) g i v e n b y some

L

=

i(x,y)(x:y

mod 2)

i n t h e c o n s t a n t n e a r - r i n g on k x 2 . B u t :

P R O P O S I T I O N L e t N b e t h e d i r e c t sum ( o r p r o d u c t ) o f t h e n r . ' s Ni

EL!.

iI ) ~ and

(

I f QNi$L

vided t h a t Proof.

L 2,N.

For i

E

I

E ~ ~ ( tNh e)n L i c

N = N0 i f

l e t L i : = { l . €1 N . 11 (

z,(Ni)

If L i s m o d u l a r b y ( . . . , e i , . . . ) I f a l l L .1= N .1 t h e n

t3N.G 1

L.

I. ( 0 ) I f v = O , assume t h a t Li i n Ni.

t h e n Li

i s m o d u l a r b y ei.

S o s u p p o s e now t h a t Li

Ni

4

i s n o t a maximal l e f t i d e a l strictly

Now i f L~:={(..,Oy1~,O,..)~l!~L~}

and Ni.

L+C; i s

1

a l e f t ideal i f N properly containing =

N.

T a k e n .1€ N .1\ L ! .1

(..,li,..)t(..,O,l;,O,..)

a l l j+i a n d n i = l i + l ; So Li

i s a l e f t i d e a l o f Ni.

T h e n t h e r e i s some l e f t i d e a l L;

L , w h e n c e L t L-; =

I (pro-

E

b e t w e e n Li then

E

v=I).

I t i s e a s y t o show t h a t e a c h Li

f o r some i

. . . ,O , l i , O , . . . )

f o r some i

E

E

Li+L;=L;,

L+L!. -1

T h e n (..,O,ni

,O,..)

Hence 1 . = O f o r J

a contradiction.

i s a 0 - m o d u l a r l e f t i d e a l i n Ni.

88

8 3 ELEMENTS OF THE STRUCTURE THEORY (2)

I f w=2, Li

we p r o c e e d s i m i l a r l y .

i s a maximal l e f t i d e a l .

L i s 0-modular,

too,

m a x i m a l t h e n t h e r e i s some b i g g e r ( N i ) o - s u b g r o u p Define L;

s t r i c t l y monogenic. L . < N.(n.+Li) 1

We m u s t show t h z t N i / L i

S u p p o s e t h a t ni€Ni

< Ni/Li.

1

If n:=(.

N(ntL)=O+L o r N(n+L)=N/L. =

( . . ,O,Nini,O,..) 1

n"=(.

t a k e some n;ENi

w i t h n'+L=n"n+L.

we g e t n 1! - n '1! n1. ~ L1 ., w h e n c e n 1! + L i contradiction. 3 . 3 3 COROLLARY

t h e n Ti 3.34 -__

If N

H e n c e Li

=ONi

i s in N,

SNSS

=NO

t h e n L i s an i d e a l o f NS and S/L If n E N ,

Nn =

a contradiction. not i n

w i t h n;+Li

T h e r e i s some

I n t h e i - t h component E

Ni(ni+Li),

i s I-modular

( a n d N=No f o r

(as d e f i n e d i n 3.30)

THEOREM ( K a a r l i ( 4 ) ) I f S <

Proof. ____

then

I n t h e f i r s t case,

and l e t n':=(..,O,n;,O,..).

. ,n'!1 ,. . )

fulfills

,O,..)

hence Ni(ni+Li)=O+Li,

1

I n t h e second case, Ni(n.+Li)

. ,O,ni

i s

( s i n c e N=No) and t h i s i s i n L ,

whence N . n . s L i ,

1

L;.

a s i n ( 0 ) and a p p l y 2.15.

( 1 ) Now l e t v = l a n d N = N o .

1

so

I f i t were n o t s t r i c t l y

again a

i n Ni.

v = l ) and L E Z " ( N ) $I,(N), too. and i f

L-

.,f2(S),

L8So.

i s an N-grouo o f t y p e 2 .

1 E L a n d s E S , we h a v e t o s h o v t h a t

n(l+s)-ns E L ; w.1.o.g.

we c a n a s s u m e t h a t n E N

0 '

If N o s & L t h e n L t N o s = S . T a t e s o ~ S o , s ' E S , s ' = l ' + n ' s . Now s s ' = s ( l ' + n ' s ) = s ( l ' + n ' s ) - ; n ' s t n ' s ~ L + S s = 0 0 0 0 0 0 0 0 0 0 Case I : S o s C - L .

= L . Hence S o S c L .

Now s o - s o e ~ L f o r some eES. H e n c e SO_cL,

a contradiction.

Therefore S S C L always i n p l i e s N o s s L .

But f o r SES,

0

s o ( l + s ) ~ L , h e n c e n o ( l t s ) E L , whence

n o ( l t s ) - n o s E L , as d e s i r e d .

_C a s_e 1_1 : ~S o s * L .

Then S o s t L = S . L e t Y ( s ) : = ( L : s )

Y(s)nS=Y(l+s)nS holds f o r a l l b y h:No+S/L,

n+ns+L.

Y ( s ) i s maximal

in

EL. A l s o ,

0

0

.

Then

No/Y(s)%S/L

F r o m t h i s i s o m o r p h i s m we know t h a t

sN 0 '

So i f Y ( s ) i % ( l t s )

Y ( s ) + Y ( l + s ) = N o . T h i s shows t h a t S o S o s Y ( s ) , S S sGL.

NO

we g e t t h a t hence

S i n c e S o s t L = S , we d e r i v e SoSnL a n d f r o m t h i s

t h e c o n t r a d i c t i o n L = S . S i n c e S O k Y ( s ) , we g e t Y ( s ) + S o = N o . Now e a c h nENo c a n b e w r i t t e n as n=Y+So w i t h y E Y ( s ) = Y ( l t s ) and

s o ~ S oH.e n c e

n(l+s)-ns

=

f o r EL we f i n a l l y g,et w h a t we w a n t : y ( l t s ) + s o ( l + s ) - F s-jis E L . 0

89

3d Quasiregularity

d \ OUASIREGULARITY

3.35 NOTATION F o r set

ZEN,

Cn-nzlnEN1

denote t h e l e f t i d e a l generated by t h e

by

(Note t h a t f o r

L,.

z = 0,

L = N,

No h a s s t i l l o n e s i n g l e

meaning.)

3.36 D E F I N I T I O N ( a ) ZEN

i s c a l l e d q u a s i r e g u l a r ( = : F)

( b ) SEN

i s c a l l e d q u a s i r e g u l a r (=:

zcLZ.

if

qr):

<=>

SES:

s i s qr.

3.37 REMARKS (a) I f

NET),,

( b ) L,

i s modular (by

ZEN

is

q r <->

L,

= N.

z).

( c ) B e i d l e m a n (1) c a l l s ( f o r a n e a r - r i n g

NEW,

with iden-

Z E N q u a s i r e g u l a r i f 3 y c N : y ( 1 - z ) = 1. I n t h i s case, z i s a l s o q u a s i r e g u l a r i n t h e sense

tity 1 )

o f 3.36.

3.38 P R O P O S I T I O N ( R a m a k o t a i a h (1)). ( a ) ZEN

n i l p o t e n t ->

L e t N be

cvo.

z i s qr.

( b ) Each n i l s u b s e t o f N i s q r . (c) If

L Qk N

i s modular by e then e i s n o t q r .

( d ) I f e is a n o n - z e r o Proof.

(a) I f

X-XZCLz, Hence

idempotent then e i s n o t q r .

z n = 0, c o n s i d e r a n y xcN. n - 1 - x z n ELZ,. 2 xz-xz ELZ xz

x-xz

Then

,*..,

n

cLZ,

so

X E L ~ and

L,

= N.

(b) Follows from (a).

( c ) tj n c N : n - n e E L .

I f e i s qr. t h e n

( d ) Assume t h a t t h e i d e m p o t e n t e i s t h e N-endomorphism

he:

N * N x * xe

.

L = L e = N.

-1.

0.

Consider

he(e) = e 2 = e 9 0

90

$ 3 ELEMENTS OF THE STRUCTURE THEORY

shows t h a t h e 0. XEN: h e ( x - x e ) = xe-xe2 = 0 , and e c a n n o t be q u a s i r e g u l a r .

V

3 . 3 9 PROPOSITION Each n i l i d e a l quasi requ 1a r .

L e e Ker h e

so

=/=

N

I o f a near-rinq N is

P r o o f . P r o c e e d i n g a s i n 3 . 3 8 ( a ) one s e e s t h a t ! i s 1 , No = { x - x O l x ~ N l c L i . is1

If

t h e n by 2 . 9 9

isNO,

so

iELi

and i i s qr.

3 . 4 0 T H E O R E M ( R a m a k o t a i a h ( 1 ) ) . Ns’yl0, D C C N , M sNN . M i s qr <=> M i s n i l p o t e n t <-> M i s n i l .

Then

l e t M ( k ) be t h e Proof. L e t M be q r . F o r k E I N , N - s u b g r o u p o f N g e n e r a t e d by M k ( 2 . 5 6 ) . Me g e t a c h a i n M 2 M(2) 3 M ( 3 ) 3 . . . By t h e D C C N , 3 k E m : M(k) = M(k+l) = =:P. I f P = I O ) , we a r e t h r o u g h . I f n o t , o b s e r v e t h a t P ( * ) = P st. {Ol, s o

...

.

=+

‘ =+ 0. P E ~ K I Ks NN A KcP A PK ( 0 1 1 = : P . So P T h e DCCN a s s u r e s the e x i s t e n c e o f a minimal e l e m e n t K O i n P - S i n c e P K o $. {O), 2 k o s K o : P k o -f= (0). P k o sN N,

P2

E

P k O s K O ~ P , P ( P k o ) $. l o )

(O:ko),

so

P = P(2)

e

(O:ko),

(since otherwise a contradiction).

.

T h e s e t h r e e a s s e r t i o n s q u a l i f y P k o t o be E P Since P k o c KO, Pko = KO. T h e r e f o r e 3 P E P : p k , = k,. So b n E N : ( n - n p ) k o = n k o - n p k o = 0. Hence tl n E N : n - n p s ( O : k o ) N, so L f N and p i s P n o t q u a s i r e g u l a r . The r e s t f o l l o w s f r o m 3 . 3 8 ( b ) .

+

3 . 4 1 R E M A R K S K a a r l i ( 4 ) showed t h a t i n 3 . 4 0 t h e D C C f o r rnonogen i c N - s u b g r o u p s s u f f i c e s . I f N + N o t h e n M S N N , M q r . imp l i e s M n i l p o t e n t . B u t f o r N = Nc with D C C N , n i l does n o t imply n i l p o t e n t . I f q i s q r , the l e f t i d e a l L g e n e r a t e d by q i s n o t n e c e s s a r i l y qr. ( t a k e e . g . N = Z 3 [ x ] , q = 1 , L = ( 0 : l ) a n d 1 4 L ) . Rarnakotaiah ( 3 ) showed t h a t i f L a R N = N o a n d q E L i s q r . i n L t h e n q i s qr. i n N . S e e a l s o Oswald ( 6 ) a n d R a m a k o t a i a h - S a n t h a k u m a r i ( 1 ) .

91

3e ldernpotents

e ) IDEMPOTENTS

3.42

DEFINITION A s e t e,fcE:

E o f idernpotents i s c a l l e d o r t h o q o n a l

if

f => e - f = 0.

e

The s t a n d a r d method t o g e t o r t h o g o n a l i d e m p o t e n t s i s t o decompose a r i g h t i d e n t i t y : 3.43 THEOREM ( B e i d l e m a n ( 1 ) , ( 6 ) ) . i d e n t i t y e,

if

k c'Li

N =

I f NET?,

(Li

then

...,e k

and i f

1 ei

e =

i = l

are orthogonal idempotents

i s a right identity i n

a n d e a c h ei Li

el,

k

N)

A,

i=l (eicLi),

contains a r i g h t

Li

which generates

Nei.

Proof.

If e = el+

k

... t e k

then

nE.N: n = n e = n

k

1 nei

=

1

ei

=

i = l

If

(by 2.30).

nELi,

the uniqueness o f

i = l

the representation yields right identity for

Li.

idempotent, w h i l e f o r by t a k i n g

n = e

n = ne

i'

so

In particular, i $. j o n e g e t s

ei

i s a

ei

i s

ejei

= 0

above.

j s i n c e each Finally, Li = Nei, w r i t t e n a s Ri = R . e i

Li€Li

can be

i'

I f o n e has a r i g h t i d e n t i t y i n 3 . 4 3 ,

b u t no d i r e c t d e c o m p o s i t i o n

i t i s sometimes s t i l l p o s s i b l e t o g e t o r t h o g o n a l i d e m p o t e n t s :

NE%. D C C N , M SN N , L1,L2 dII N, I f M c o n t a i n s a r i g h t i d e n t i t y e and i f b o t h a r e m i n i m a l f o r t h e p r o p e r t y t h a t L1tL2 = M t h e n

3 . 4 4 THEOREM ( S c o t t ( 5 ) ) . L1,L2 L1,L2

c M.

t h e r e e x i s t o r t h o g o n a l idernpotents ( a ) el+e2 ( b ) (O:el)n

el€L1,

: e (mod L 1 n L 2 ) M = L2

and

(0:e2)nM

= L1'

e2cL2

with

92

9 3 ELEMENTS OF THE STRUCTURE THEORY

Proof. I f e = R1tL2 = 9. ( a , +R ) . But

1

1

(R1€Ll,

L2€L2)

then

2

Rl(R1tR2)-R1L2-Rl~L1,2

R 1 = Lle

=;

SO

L1(k!1+L2)

a n d Rl(RltR2)-R~-Rlk2~L2, E R ~ R ~2 + (mod R ~ L2).

so

R 1 (k! 1 tR 2 ) t

F r o m 2 . 2 2 we c o n c l u d e t h a t

k l E R 1 R 2 + R i (mod L 1 n L 2 ) ,

: k !21 + R l R 2

hence

(mod Ll)

= R 1 II 2 E L 1n L 2 ,

Rl-R!

so

mEIN:

!ZTERl

11TzR2 (mod L 1 n L2). Now l e t i b e ~ ( 1 , 2 l . k k+l A l l L i R l SN N . BY D C C 3 kEJN: LiRi = L i l i

V

Similarly,

LIE!

.

mEIN:

(LiR!)R!tl

= Lie!

Moreover,

k!+l~Li~:.

Therefore

(mod L 1 f i L 2 ) .

and

--

...

generates

We c a n a p p l y 3.13 a n d

get

( 0 : 1 5 r * 1 ) t ) LiL!

= (0)

and L i t ! has a r i g h t i d e n t i t y eit:+l = R~k t l

.

M = Melt(O:el)0

( b ) By 1 . 1 3 , From

elkit'

=

ei

with

M = L1+(O:el)n

M.

we g e t

( O : e l ) n M s (o:e:+l)n

M

E

(

~ ~ ~~: ~ : n+ l ) n M.

By t h e r e m a r k s a t t h e b e g i n n i n g o f t h e p r o o f ,

Thus

(mod Lln L2). ~ k + l f)i M: m = me : m n k + l + m t ktl 2 =m€(L1nL2: 1

z ma!+'

(mod L1" L 2 )

e E R ktl+Ri+l 1

*ut

L:t1EL2,

50

(since

m€L2,

me = m(Rl+R2)l.

hence

(LIn

a n d b y t h e m i n i m a l i t y o f L~ we g e t BY s y m m e t r y , (0:e2)nM = L ~ .

L ~ :k +&l )~s t 2

( o : ~ ~ ) M, ,= L ~ .

3e ldempotents

93

Finally, since elcL1 = (O:e2)n M , ele2 = 0 and by symmetry, e2el = 0. S o el,e2 are orthogonal idempotents. 3.45 REMARK See Lausch (5) for applyinq sets E of orthogonal

idempotents to get a decomposition o f N into "blocks" (similar to (Artin"spanned" by some partitions of E Nesbitt-Thrall)). See also Deskins ( Z ) , Williams ( 1 ) . See Fain (1) (Th. 6.4) and Lyons (3),(4) for more decompositions induced by orthogonal idempotents. 3.46 DEFINITION An idempotent ecN is called central if it is in the center o f (N,.), i.e. if b n&N: en = ne.

3.47 PROPOSITION (Betsch ( 3 ) ) . Let e be a central idempotent

with Ne A N. Then N is the direct sum o f the ideals Ne and (0:Ne) = ( 8 : e ) . Proof. Clearly Ne (by assumption) and (0:Ne) (by 1.43(b)) are ideals. By 1.13, N = Net(0:e) and NeA(0:e) = {Ol. But (0:e) = (O:Ne), since e is central. 3.48 PROPOSITION (Fain ( 1 ) ) . Let E be a set o f orthogonal

central idempotents and

Cei

any sum o f distinct elements

o f E . Then

(a) E c Nd. (b) rei (c)

i s idempotent.

ncN: nCei-Cneie(O:E) A

( d ) (OEE v I E 1 2 2)

-7

N

=

N.

NO +

53 ELEMENTS OF THE STRUCTURE THEORY

94

P r oof. ( a ) i s t r i v i a l . ( b ) (Zei)

(c)

\1

2

= z e J. ( C e i )

CZejei = C e i e i = Z e i .

eEE: ( n X e i - C n e i ) e = 0 .

(1 ( 0 : e ) 9, N ( b y 1 . 4 3 ( a ) ) . Moreover, eEE (0:E)N c ( 0 : E ) s i n c e b n E N b mE(0:E) eEE: (mn)e = men = O n = 0 . (0:E) =

v

(d) If

If nEN,

OEE then c l e a r l y N = No. / E l 2 2, let e f be i n E . Then f o r a l l nO = nef = e f n = On = 0.

+

3.49 R E M A R K A r i n g i s c a l l e d b i r e g u l a r i f each p r i n c i p a l i d e a l i s g e n e r a t e d by a n i d e m p o t e n t . I n ( 3 ) , B e t s c h d e f i n e d a n e a r - r i n g t o be b i r e o u l a r i f t h e r e e x i s t s some s e t E of c e n t r a l idempotents with (a) (b)

V eEE: rie 9 N . b n s N 3 eEE: Ne

=

(n)

(principal ideal generated

by n ) . (c) (d)

b b

e,fEE: e t f = f t e . e , f s E : efEE A e + f - e f E E .

Ramakotaiah ( 1 ) showed t h a t e a c h commutative b i r e g u l a r n e a r - r i n g i s isomorphic t o a s u b d i r e c t product of f i e l d s and hence a b i r e g u l a r r i n g . More i n f o r m a t i o n can be o b t a i n e d i n C o u r v i l l e ( I ) , C o u r v i l l e H e a t h e r l y ( I ) , M i r o n - S t e f a n e s c u ( 1 ) a n d Ramakotaiah ( 3 ) .

3f More on minimality

95

f) MORE ON MINIMALITY

W e conclude this paragraph with some results concerninq minimality o f non-nilpotent N-subqroups and left ideals of N. A s w e will s e e , considering minimality does not imply that t h e results c a n be reached by minimal efforts. However, w e first reap the fruits of previous sections. 3.50 DEFINITION M SN N

(L 4, N) is called a minimal non-nilpotent N-subgroup (left ideal) if it i s minimal in t h e s e t of all N-subgroups (left ideals) of N which a r e not ni 1 potent.

Clearly i f L A, N = No and L is a minimal non-nilpotent N-subgroup then L is d minimal non-nilpotent left ideal. 3.51 THEOREM Nc7l0, DCCN.

M sN N i s a minimal non-nilpotent (a) (Scott (5)). N-subgrcup => M contains a right identity e with Ne = Me = M (see also Beidleman (6)). ( b ) If

L 4, N

is a minimal non-nilpotent left ideal then L contains a non-zero idempotent.

(c) (Beidleman (1)). I f L A, N is a minimal non-nilpotent N-subgroup then L is a direct summand o f ". Proof. (a) If mEM is not nilpotent then m 2 EMmEM is M and by 3.13 M contains not nilpotent, s o Mm a r i g h t identity e. By t h e minimality o f M, N e (not nilpotent!) = M = Me. (b) By t h e minimum condition in N , L c o n t a i n s a minimal non-nilpotent N-subgrour, M. M has a right identity e by (a) and s o L has a non-zero idempotent e. (c) L contains a riqht identity e by (a) with L e = L . and L i s a direct summand o f ,,,N.

By 1.13, N=L+(r):e)

96

3 3 ELEMENTS OF THE STRUCTURE THEORY NEW,,

3 . 5 2 COROLLARY ( B l a c k e t t ( 2 ) ) .

DCCN,

N w i t h o u t non-zero

M

n i l p o t e n t N - s u b g r o u p s . Then e a c h m i n i m a l N - s u b g r o u p

is

g e n e r a t e d b y an i d e m p o t e n t e w h i c h i s a r i g h t i d e n t i t y o f M. We now t u r n t o m i n i m a l i d e a l s .

3.53

PROPOSITION ( S c o t t

M nilpotent,

L e t I b e a m i n i m a l i d e a l , rl

(4)).

M c I.

sFT N ,

I M = (0).

Then

Mk be = t o ) , k 2 2 , M k - I p {O). T h e n M ~ - ~ . M= {OI, s o M k - l i s contained i n the ideal Mk-1 i s (0:M). Hence t h e i d e a l J g e n e r a t e d b y c o n t a i n e d i n (0:M). S i n c e (0) J E I, J = I a n d

Proof. Let

I M = {Ol. See a l s o K a a r l i

( 2 ) and Scott

(16)

NE~,,

3.54 THEOREM ( S c o t t ( 6 ) ) .

DCCN,

I a minimal i d e a l .

Then I i s a f i n i t e d i r e c t sum o f N - i s o m o r p h i c m i n i m a l l e f t ideals o f

N

(and t h e r e f o r e c o m p l e t e l y r e d u c i b l e

NI).

when c o n s i d e r e d a s

-Proof.

We n e e d 3 l e m m a t a a n d k e e p t h e a s s u m p t i o n s o f t h e

theorem.

Nr b e f a i t h f u l a n d

Lemma 1. L e t

Nr.

ideal of

{Of

Let

A be a minimal

L c (A:r)

be a

r

v Ly = {oI. T h e n L i s a f i n i t e d i r e c t sum o f N - i s o m o r p h i c m i n i m a l l e f t i d e a l s o f N. l e f t i d e a l such t h a t

Proof.

3 If

Nr

(a) Since

L$(O:Y~),

YIEr: (

0

“(o:Y1)L

:

PI

~

~

){Ol ~

yEr:

Ny =

i s f a i t h f u l and

so then

L

$. {Ol,

( 0 : ~ ~= )(o:yl)nL=L. ~

3

Y ~ E ~( :0

:

~

= ~

(o:Y*)L.

P r o c e e d i n g i n t h i s m a n n e r , b y t h e D C C N we e v e n t u a l l y o b t a i n elements

yl,y2,.

..,yn€r

)

~

97

3f More on minirnality

Anyhow, we g e t a n o n - e m p t y s u b s e t C o f Cyl,.

. . ,ynl

of minimal o r d e r f o r t h e ( o : C ) ~ = EOI.

property that Set

C = : Cul,

(b) Define

...,o k } .

L1:=

L

We now show t h a t

hi:

=

k>l Li

morphisms.

k

k = 1

if

= (O:C\CU~})~ if

Li:

11

and

(iEC1,

are N-iso-

A

+

...,k l ) .

&ai

-+

101 $. L = L 1 a n d ( e ~ : u ~ ) ~ = C O f . Lol = L l u l Col a n d Nol = r .

1: T h e n Thus Since

LE(A:r),

Llol 9,i Nul

=

LlolcA.

By 3.4(a),

r.

Since A i s minimal,

Llul

and

= A

hl i s surjective. Also, K e r h l = ( 0 : ~ ~= )EOI; ~ hence hl i s an N - i s o m o r p h i s m . k > 1: S u p p o s e t h a t

3

jc11,

...,k l :

L-o ~

-

j L =(o:o.), so jJ L =(o:C), = COI, a contradiction t o jt h e m i n i m a l i t y o f E. = to}.

Then

Hence a l l

Lioi

Lioi

Also,

= A.

=+

Col

and ( a s a b o v e )

K e r hi

= (o:oi)n =

COI,

(0:UCuil)n a n d a g a i n hi

= (o:ui)n

( 0 : ~ )-

L = L i s an N-iso-

m o r p h i sm.

...,

( c ) L e t i be ~ { l , k l . Li i s a m i n i m a l l e f t i d e a l o f N: BY 3 . 4 ( e ) ,

Liui Thus

= A

c\r

N/(o:oi)

-N

=

r.

BY ( b ) ,

i s minimal.

L i + ( o : u i )/(o;oi)

of t h e N-group Li n ( o : o i )

Nui

= Col

i s a minimal i d e a l

N/(o:oi). (by (b)),

-

Since 2.8 gives

Li

=

98

53 ELEMENTSOF THE STRUCTURE THEORY

(d) Since a l l isomorphic.

Li

% aN

A,

t h e L i ' s a r e N-

...

tLk. We may ( e ) We show t h a t L = L 1 t assume t h a t k > l . I f R c L , V i c { l , ...,k l : RuicA. By ( b ) , L .1u . 1 = A , S O 1 R i € L i : Rui = Riai.

S e t El:= R 1 t , . . t R k . I f i += j, ~ ? . c L ~ ~ ( o : us ~o ) , 2 . 0 0. J J i So f o r a l l i c { l , k l Riai = R i u i . T h e r e f o r e (I?-J!')E = (01, so R-R'c(o:E)hL= = {I-)}, Hence R = I?' = e l + ...+R k .

...,

( f ) The sum i n ( e ) i s d i r e c t : 'Pk I f 11 = R 1 t . . . + R k = P I +

...

Then

V

iE{l,

...,k l :

Lai = piui

( R i, p i E L i ) = Riui.

...,

Thus ic{l, kl: R i - p i ~ ( o : C ) nL t h e proof i s complete.

and

Lemma 2 . I f I N {Ol a n d M SN N i s m i n i m a l f o r I M =/= ( 0 1 then (a) M contains a right identity. (b)

In M

i s minimal amongst a l l n o n - z e r o i d e a l s

of the nr. M which a r e a l s o N-subsroups.

(c) In 4 I

i s t h e sum o f m i n i m a l i d e a l s o f

NM.

P r o o f . ( a ) We s h a l l show t h a t MM i s m o n o g e n i c . I f kf m e M : Mm 3. M, I M M = ( 0 ) s i n c e V m c M : I ( M m ) = {Ol b y t h e m i n i m a l i t y o f M . D e n o t e t h e i d e a l g e n e r a t e d b y It4 b y J. S i n c e It4 i s c o n t a i n e d i n t h e i d e a l (O:M), Js(0:M) a n d J M = 101. But I M (01, I M c I a n d so J = I a n d we a r r i v e a t t h e c o n t r a d i c t i o n I M = (01.

a

99

3f More on rninimality

So M i s m o n o g e n i c a n d t h e r e s u l t ( a ) f o l l o w s f r o m 3.13.

I M p I01 and IMcInM, I n M i s a n o n - z e r o i d e a l o f M. L e t I01 K 9 M b e s u c h t h a t K sN N and K c I n M . S i n c e KMEKEI, Kc(K:M)n I a n d ( K : M ) n I p {OI, s o ( K : M ) n I = I a n d Ic-(K:M). I t f o l l o w s t h a t IM'(K:M)MGK. By ( a ) , M c o n t a i n s a r i g h t i d e n t i t y e , s o ( I n M ) e = I n M. Thus I n M!(! I n M ) M E I M c K (b) Since

and ( b ) f o l l o w s .

9 IOI,

In M

(c) Since

minimal i d e a l W o f T a k e some

there exists a

NM

in

I n M .

mEM,

= I01 o r NM ( s i n c e Wm i s an N-endomorphic image o f W ) . I f Mm $. M t h e n I M m = {Ol. But M contains a r i g h t i d e n t i t y e , s o We = W a n d t h u s Wm = WemrIMm = i O l .

Mm = M

If

then

Wrn

i s either

a minimal i d e a l o f

Hence

mcM: Wm =

ideal o f

I01

or

i s a minimal

Wm

NM.

L

W = WeC-L a n d

9

(01.

S o L i s t h e sum o f m i n i m a l i d e a l s o f

L d R M.

O f course,

1 WmiTieL,

=

so

Also

L sl M,

;EM:

NM.

LiTi =

and L i s a non-

mcM

zero ideal o f M contained i n L

5:

In M

I n M.

By ( b ) ,

a n d ( c ) i s shown.

A l t h o u g h t h e r e a d e r m i g h t b e g a s p i n g f o r b r e a t h , we n e e d a t h i r d Lemma, w h i c h w i l l b e u s e d i n t h e p r o o f

o f 4.47.

$ 3 ELEMENTS OF THE STRUCTURE THEORY

100

Lemma 3 .

I N dp I 0 1

If

t h e r e e x i s t s an N-group

and a m i n i m a l i d e a l (a)

Nr

such t h a t

( o : r ) n I = (0).

( b ) I+(o:r)

(c)

r

A SIN

b

YEr:

C

(Azr).

M, W ,

Proof. Let

e

InM

NM.

minimal i d e a l s o f

I nM,

i n lemma 2. B y p a r t

be as

( c ) o f t h i s lemma, W =

r.

IY = ( 0 1 v NY =

i s t h e sum o f

By t b e f a c t t h a t

we c o n c l u d e f r o m 2 . 4 8 ( e ) t h a t W

i s a d i r e c t summand.

3

SO

U

Define

43

In

InM:

r:=

M/U

M = WU ;.

and

‘L

=N W .

InMIU = W;U/,

A:=

So A i s a m i n i m a l i d e a l o f

NI’.

We n o w p r o v e t h e lemma.

(o:r)n I

(a)

= CO1

or

= I.

If

I e ( o : r ) . So I r = { o } a n d ( I n M)e = I A M y s o I n Ms( I But Thus A = ( 0 1 , a c o n t r a d i c t i o n .

(o:r)nI = I IMEU.

then

A

M)McIM~U.

( c ) I f Y E r , 3 mEM: y = m t U . I f Nm=M, the m i n i m a l i t y o f M gives us I ( N r n ) = (0).

So

INe(o:y).

t h e l e f t i d e a l generated by (IN)%5(o:y).

By 1 . 5 2 ,

generated by I N So

I y = Col,

(IN)I1

If

i s

(IN)I1

IN

then i s the ideal

and t h e r e f o r e e q u a l s I . completing t h e proof.

T i r e d , b u t h a p p y we a r e r e a d y f o r t h e P r o o f o f t h e theorem,

Suppose t h a t

I N = COI

and L i s a

m i n i m a l l e f t i d e a l o f N c o n t a i n e d i n I . So and

L A N.

Thus

L = I

LN =

and t h e theorem h o l d s .

COI

101

3f More on minirnality

If

p

IN

N/

(0:

{O),

r1

let

r,A

N ur

=:Nu,

be as in Lemma 3 . If is faithful and has A as a

minimal ideal. Set I & : = I t ( o : r ) / ( o : r ) . lemma 3(c), tl y c r : ( N a y = r v I l y = c o ) ) . By lemma 3(a), I I/(o) 'b 1 ' . Thus by % lemma 1, I -N 1' is the direct sum o f minimal N - i somorphic 1 eft i deal s . By

zN

nN

The p r o o f is now complete. Note that if NcnO i s simple and has the DCCN then N N is completely reducible and 2.50 i s applicable. Cf. 4 . 4 6 and 4 . 4 7 . 3.55 COROLLARY (Scott ( 4 ) , ( 6 ) ) . NEW,, DCCN, I a non-nilpotent minimal ideal o f N; O ( N ) : = sum o f all nilpotent left ideals. Then O ( N ) r l I = ( 0 ) and Q ( N ) is nilpotent. Proof.

See S c o t t ( 1 ) o r ( 4 ) f o r t h e p r o o f t h a t Q ( N ) i s n i l -

potent. of

V

By 3 . 5 4 a n d 2 . 4 8 ,

NI. L e t L iE1

v

2

Q ( N ) n I i s a d i r e c t summand I be s u c h t h a t I = Q ( N ) n I ; L . By 2 . 2 2 .Q.

qeQ(N)n I

t, R E L : i ( q + R ) : i q + i R (mod Q ( N ) n I n L ) .

Since ( Q ( N ) n 1 ) n L = {O},

i ( q + k ) = i q + i R . Hence

I' = I ( Q ( N ) n

I+L)

I ( Q ( N ) ~ I)

COI, since Q(N) i s n i l p o t e n t . so

=

S I(Q(N)nL)+IL.

B u t b y 3.53, I~SILSI

and t h e l e f t i d e a l g e n e r a t e d by 1 2 , t h e i d e a l g e n e r a t e d b y 1'

(by 1.52)

and

I (by minimality) coincide.

So I i s c o n t a i n e d i n t h e l e f t i d e a l g e n e r a t e d b y I L C L , I = L and Q ( N ) n I = { O } . 3.56

REMARK There a l s o e x i s t r e s u l t s c o n c e r n i n g n e a r - r i n g s w i t h

ascending chain conditions. Oswald ( 2 ) . (4),

For "Goldie-type"

ones,

F o r more r e s u l t s , c o n s u l t S c o t t ( l ) ,

D i Sieno- D i Stefan0

and Zand ( 1 ) .

see

Kaarli

(2),

(4), R a m a k o t a i a h - S a n t h a k u m a r i ( 1 )

102

$ 4 PRIMITIVE NEAR-RINGS

This paranraph presents a discussion o f t h e " b u i l d i n q stones, near-rinqs

a r e made o f " ,

Similar t o r i n q theory,

the so-called t h e "atoms"

"primitive near-rinqs".

a r e n o t t h e simple near-

r i n g s as one m i o h t e x p e c t a t a f i r s t g l a n c e . an i m p o r t a n t c o n n e c t i o n (4.47).

g i v e n a n e a r - r i n q N,

f r u i t s (= N-groups)

r e c o q n i z e them b y

we l o o k a t a l l o f i t s

and ask, w h e t h e r t h e r e a r e f a i t h f u l and

' ' e n o u g h s i m p l e " o n e s amonq t h e m . N " p r i m i t i v e on t h i s N-group". p r e c i s e we f i x

however,

The i d e a t o c o n s i d e r p r i m i t i v e

n e a r - r i n g s comes f r o m t h e b i b l e ( " Y o u w i l l their fruits"):

There i s ,

If t h i s i s t h e c a s e , we c a l l

S i n c e "enouqh s i m p l e "

i s not

i t s menninq i n w a n t i n g N-qroups o f t y p e v.

The r e s u l t i n q c o n c e p t i s t h a t o f " v - p r i m i t i v i t y " . We g e t t h e h i e r a r c h y 2 - p r f r n i t i v i t y < T > l - p r i m i t i v i t y < Z ' mitivity,

discuss conditions,

O-pri-

w h i c h f o r c e some o f t h e s e c o n c e p t s

t o c o i n c i d e a n d make a l o t o f w o r k t o w a r d s a d e n s i t y t h e o r e m which i s c o m p a r a t l e t o t h e c e l e b r a t e d one i n r i n o t h e o r y due

t o N . J a c o b s o n . We r e a l l y q e t o n e f o r 2 - p r i m i t i v e n e a r - r i n q s w i t h i d e n t i t y (4.52).

A d d i n q a c h a i n c o n d i t i o n , we a r r i v e a t

a Wedderburn-Artin-like

s t r u c t u r e theorem (4.60).

Before that,

we g e t " b e t t e r a n d b e t t e r " d e n s i t y - l i k e s t r u c t u r e t h e o r e m s f o r 0-,

1- a n d 2 - p r i m i t i v e n e a r - r i n g s .

theorems on v - p r i m i t i v e symmetric v - p r i m i t i v e

I t comes o u t t h a t m a n y

n e a r - r i n q s c a n be d e r i v e d f r o m z e r o -

n e a r - r i n q s w h e r e t h e y a r e much e a s i e r

t o o b t a i n s i n c e t h e s e ones behave more l i k e r i n q s . many p r o o f s c o n c e r n i n g e v e n z e r o - s y m m e t r i c

However,

near-rinqs

differ

t o t a l l y f r o m t h e comparable ones i n r i n g t h e o r y . Anyhow,

t h e " b u i l d i n g stones" mentioned above ( 2 - p r i m i t i v e

n e a r - r i n g s w i t h i d e n t i t y ) a r e shown t o b e d e n s e i n or

Maff(r) ( i f

MGo"k51 ( r ) t M c ( T ) fixed-point-free

No

i s a ring) or i n

( i f No

MGo,{al

i s a non-ring),

automorphism group

where

(r).

AutN 0

HomD(r,r)

(r)

or

Go

i s the

I n particular,

4a General

103

i f Go = ( i d ) , t h e l a t t e r t w o o n e s a r e Mo(r) a n d M ( r ) . F i n a l l y , t h e d e n s i t y p r o p e r t y i s s e e n t o be a k i n d o f a n i n t e r p o l a t i o n p r o p e r t y and a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t w i l l be o b t a i n e d . R e c a l l a g a i n ( p . 1 ) t h a t r * = r \ { o ) , a n d so o n .

a 1

G E N E R A L

1 . ) DEFINITIONS A N D E L E M E N T A R Y RESULTS 4.1

CONVENTION I n a l l w h a t f o l l o w s , w w i l l be a n y n u m b e r unless otherwise specified.

E I O , ,Zl ~ 4.2

DEFINITION

( a ) 1.4 i s c a l l e d u - p r i m i t i v e on

Nr:

<->

Nr

is f a i t h f u l

o f type w.

y:

( b ) ri i s v - p r i m i t i v e : <=> 3 N r E N N i s w - p r i m i t i v e on ( c ) I A N i s c a l l e d a u - p r i m i t i v e i d e a l o f N : <-> N/I w - p r i mi t i ve . 4.3

Nr. is

T h e n the following

PROPOSITION L e t ' I be an i d e a l o f N . conditions are equivalent: ( a ) I i s w-primitive. (b)

3 NrENq:

I =

(o:r)

( c ) 3 L s$, N : I = ( L : N ) Proof.

( a ) ->

Nr is

A A

o f type w.

L i s v-modular.

( b ) : I i s w - p r i m i t i v e -> N/I i s u - p r i m i t i v e N / I r -> Nr ( a s i n 3 . 1 4 ( b ) ) i s o f t y p e w a n d

o n some I = (o:r).

( b ) -> ( c ) : L e t r be = Ny p I o l . ( 0 : ~ )= : L . Then 'L L i s m o d u l a r . By 3 . 4 ( e ) , N / L r , so L i s w-modular. Finally, I = ( o : r ) = (o:N/L) = (L:N). nN

( c ) => ( a ) : T a k e N / L = : r . T h e n ( a s above) I = (L:N) = ( o : r ) .

Nr

i s o f type w and

8 4 PRIMITIVE NEAR-RINGS

104

4.4

COROLLARY

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

(a) N i s w-primitive.

4.5

(b)

I01

(c)

3

i s a w-primitive ideal.

L 3% N :

t w-modular A (L:N)

IOI.

=

REMARKS ( a ) Observe t h a t ( c ) i n 4.3 and 4 . 4 q i v e " i n t r i n s i c " characterizations o f primitivity

-

t h a t w i l l be

f o r i t e n a b l e s one t o r e c o g n i z e

extremely helpful,

p r i m i t i v i t y " w i t h i n N". (b) I f

N i s v - p r i m i t i v e on r t h e n IA N

if

i s a w-primitive

(c) 2-primitivity

Nv

( d ) The n e a r - r i n g s

(e)

ideal then

r

Io?

1 4

N.

and

i m p l i e s 1 - p r i m i t i v i t y and t h i s i n t u r n

implies 0-primitivity

n e a r - r i nqs.

N $. f 0 1 ,

( a l w a y s o n t h e same g r o u p ) .

o f 3 .8 a r e e x a m p l e s o f w - p r i m i t i v e

(on Z4).

If N i s v-primitive

on

r

then

M(r)

N 4

(1.48).

( f ) See 95 o f B e t s c h ( 3 ) f o r a d i s c u s s i o n o f t h e s p a c e s o f w-primitive 4.6

PROPOSITION L e t i d e n t i t y e.

( v = 1,2)

ideals

of

NEWo.

N contain either a l e f t or a riqht

Then

(a) Every u - p r i m i t i v e i d e a l I o f N i s modular.

(b) I f e i s a l e f t i d e n t i t y o f N t h e n N i s 1 - p r i m i t i v e iff N i s 2-primitive

(and i n t h i s case e i s a two-

sided identity). Proof.

( a ) I f e i s a l e f t i d e n t i t y i n N t h e n (because

N/I

i s w - p r i m i t i v e o n some

of

N/I

by 3.4(c).

So

N/Ir) e + I

\f nEN: e n

f

i s an i d e n t i t y

n e E n {mod I ) .

If e i s a r i g h t i d e n t i t y , t h e a s s e r t i o n i s t r i v i a l . (b) L e t N be I - p r i m i t i v e on 3.4(c),

Nr

e i s a two-sided

i s unitary.

Nr.

By

i d e n t i t y f o r N.

Flow a p p l y 3 . 7 ( c )

By 3 . 4 ( b ) ,

and 3 . 1 9 ( a ) .

105

4a General

4.7

PROPOSITION L e t FI be s i m p l e a n d N i s v - p r i m i t i v e on r. P r o o f . (o:r) 4 N , so the c o n t r a d i c t i o n

4.8

Nr

(o:r) = to1 Nr = { o l ) .

b e o f t y p e w . Then

(for

(o:r) = N

gives

PROPOSITION ( B e t s c h ( 3 ) ) . L e t t h e r i n g N be w - p r i m i t i v e on r . T h e n N i s a p r i m i t i v e r i n g on t h e N-module r ( ( N . J a c o b s o n ) , p. 4 ) .

Proof, I f

r

= Ny

abelian. If nEN:

then

% r -N N/(o:y)

nl+(o:$),

and

(r,+)

is

n 2 + ( o : y ) ~ N / ( o : y ) then

n(nl+(o:y)+n2+(o:y))

= n ( n , t ( o : y ) ) + n ( n 2 + ( o: y ) )

= nnlt(o:y)+nn2t(o:y)

=

.

v

Hence tf y l , y 2 ~ r nEN: n(y1ty2) nyltny2, and N r i s a ( r i n g - ) module. Each N-submodule o f N r i s a n i d e a l , s o = l o ) o r = r . F i n a l l y , Nr p { o ) by a s s u m p t i o n , s o Nr i s i r r e d u c i b l e a n d N i s p r i m i t i v e on r . 4.9

C O R O L L A R Y (Ramakotaiah ( 1 ) ) . I f N i s commutative and u - p r i m i t i v e then N i s a f i e l d .

LEX~(N):

(L:N) = {OI. L 9 N , s i n c e P r o o f , BY 4 . 4 ( ~ ) , 3 N i s c o m m u t a t i v e . By 3 . 2 5 , (L:N) is the greatest i d e a l i n L, s o L = (01 a n d N c o n t a i n s a r i g h t i d e n t i t y . By 1 . 1 0 7 ( c ) , N i s a r i n g , h e n c e a p r i m i t i v e r i n g by 4 . 8 a n d by ( N . J a c o b s o n ) , p . 7 a f i e l d . In ( 3 ) , R a m a k o t a i a h s h o w s t h a t i f I q N = N o a n d I €g0(t\1) t h e n I is a 0 - p r i m i t i v e ideal. Near-rinas N with a f a i t h f u l , s i m p l e , n o n t r i v i a l N-group a r e c a l l e d ? - p r i m i t i v e a n d a r e s t u d i e d in H a r t n e y ( 4 ) , M e l d r u m ( 7 ) , ( 1 3 ) . S e e a l s o B e i d l e m a n (7),(8),(9). H o l c o m b e blalker ( 1 ) s t u d y 3 - p r i m i t i v e n e a r - r i n q s PI, w h i c h m e a n s t h a t N h a s a f a i t h f u l N-group o f t y p e 3 ( s e e t h e l a s t l i n e s o f p. 80).

8 4 PRIMITIVE NEAR-RINGS

106

2 . ) T H E CENTRALIZER

4 . 1 0 DEFINITION E n d N ( r ) = H m N ( r , r ) =: C N ( r ) = : C i s c a l l e d t h e c e n t r a l i z e r o f Nr ( c f . K a a r l i ( 2 1 , R a m a k o t a i a h ( 3 ) ) .

A u t N ( r j =: G o .

A u t N ( r ) =: G N ( r ) =: G;

0

Go:

if

=

~ E C ; ;

l i k e w i s e .:G

otherwise 4 . 1 1 REMARKS ( a ) (C,.)

C Q = {o} ( b ) ~ E <=> (c) If

(ci,o)

. N 4 M C N ( r ) (l-1 5

is f a i t h f u l then

NT

4 . 1 2 NOTATION

a n d (Go,") are q r o u p s , ( " g r o u p s w i t h z e r o " ) a r e rnonoids.

i s a monoid, a n d (Go,,")

(GO,")

If

fn:

nEN,

r

-+

'I

Y

-c

ny

;

FN(r):=

M&9.

ffnlnENl =: F .

4 . 1 3 PROPOSITION ( M l i t z ( 3 ) ) .

(a) If

= ImcM(r)I

(b)

tl

hECN(r):

(c) If Proof.

tl

CN(r)= fom) = : MF(r).

Nr i s m o n o g e n i c t h e n

Q =

r

(a) If YET:

WfEF: mof = h/n = id.

then

CN(r) = { i d } .

hECN(r)

and

fnEFN(r)

then

(hofn)(y) = h(ny) = nh(y) = ( f n o h ) ( y ) ;

so

hEMF(r).

sMF(r). I f y 1 , y 2 & r = Ny a n d y 1 = n l y A y 2 = n 2 y . Then

C o n v e r s e l y , l e t f be

n&N

then

f(nv1) =

3

nl,n2EN:

(fof,)(vl)

=

(fnof)(vl)

= nf(Y1)

and

107

4a General

Hence

feCN(r).

(b)V

heCN(r)

noc:R: h ( n o ) = n h ( o ) = n o .

( c ) f o l l o w s from ( b ) . 1.14 N O T A T I O N

eo:

=

e o ( N r ) := { y c r

el:

=

el(Nr):

Ny = NO =

Rl.

ri.

= C Y E ~ NY =

1 . 1 5 REMARKS ( B e t s c h ( 6 ) ) .

(a)

oEeo,

e l $. 0 ( c ) eon e l (b)

P 0.

so

eo

<->

Nr i s m o n o q e n i c . n $. r .

= 0 <->

r

( d ) Nr i s s t r o n q l y m o n o g e n i c -> ( e ) Nr

eo

i s u n i t a r y ->

and

=> y = l y c R

( f ) G ( B o ) = Bo g r o u p s on

=

A G(B1)

eo

and

(for

= R

w =

= Bow

el,

y e e 0 ->

Ny

R ==>

nocR => Nu = NnocNo = R =I>

so G i n d u c e s p e r m u t a t i o n

el, (if

WE^^)

+ 0)

el

on

el.

The n e x t p r o p o s i t i o n i s a " S c h u r - t y p e lemma". 4 . 1 6 P R O P O S I T I O N ( B e t s c h (S), M l i t z ( 3 ) ) . ( a ) Nr i s s i m p l e (hEC A 2

(b)

n

A

yEB1:

= { o } ->

h(y)EOl)

Nr i s N - s i m p l e

->

C = Colu M o n N ( r )

he6.

->

C = Epi,,,(r,Q)u E p i N ( r , r )

EpiN(r,n) = {el!). (c)

Nr i s N o - s i m p l e

=>

C = G0

and

.

(if

NEW,,

84 PRIMITIVE NEAR-RINGS

108

Proof. ( a ) follows from the f a c t t h a t h s C : Ker h so e i t h e r Ker h = Col ( t h e n h s M o n N ( r ) ) o r Ker h = r ( t h e n h = 0 ) . We may assume t h a t I f hEC A ] yeel: h(y)sel then h 9 6 , so h s M o n N ( r ) . Now h ( r ) = h(Ny) = Nh(y) = r . ( b ) V hsC: Im h = r .

Im h sN

r,

so e i t h e r

Im h = R

SIN

r

r,

+ Iol.

or

( c ) f o l l o w s from ( b ) . i n which We a r e mainly i n t e r e s t e d i n t h e c a s e t h a t C = G o , e v e r y n o n - z e r o N-endomorphism o f r i s an N-automorphism. 4.17 PROPOSITION (Betsch ( 6 ) ) .

( a ) G i s f i x e d - p o i n t - f r e e ( 1 . 4 ( b ) ) on e l . (b)

Nr i s s i m p l e t h e n

If

P r o o f . ( a ) Assume t h a t f o r

'd

6Er

3

= ny = 6 .

( b ) ->:

Then

nEN:

So

6 = ny.

C = Go <-> qEG

Then

and

1

A
yeel

r : r iN A. ~ ( y )= y .

g ( 6 ) = g(ny) = ng(y> =

g = id.

Assume t h a t h i s an N-isomorphism r + b c N h s C = G 0 c {6lu A u t N ( r ) , a c o n t r a d i c t i o n .

r.

I f hEC, h 6 then Ker h 9 r . s o Ker h = I o l . ?r T h e r e f o r e h i s a monomorphism and r Irn h . So Im h = r , a n d h s A u t N ( r ) . <=-:

-

4.18 C O R O L L A R Y ( B e t s c h ( 6 ) ) . I f Nr i s o f t y p e 1 o r i f i s s i m p l e a n d f i n i t e t h e n C = Go.

Nr

P r o o f . I f N r i s of t y p e 1 t h e n Nr i s s i m p l e . Assume t h a t h i s a n N-isomorphism r * A < N r. R e p r e s e n t r a s r = Ny and c a l l h ( y ) = : 6 . N6 = Nh(y) = h(Ny) = = h(r) = A. I f & E e l t h e n N6 = r . s o r = A , a c o n t r a d i c t i o n . I f beeo t h e n N6 = R = I o l , s o A = { o l a n d t h e r e f o r e r = C o l , which a q a i n i s a c o n t r a d i c t i o n .

109

4a General

Now a p p l y 4.15(d) a n d 4 . 1 7 ( b ) .

Nr

If 4.19 NOTATION

i s s i m p l e and f i n i t e ,

For

apply 4.17(bj.

y , 6 ~ ~ we r define (o:y)N

<->

6:

y

=

?r

y

4.20

%

;

(0:a)

No

0

Go(y) = G o ( 6 ) .

6 : <->

REMARKS ( B e t s c h ( 5 ) ) .

(a)

%,?

( b ) The e q u i v a l e n c e c l a s s e s o f of (c)

r.

are equivalence relations i n

y,6Er:

=> (0:y)

?

are exactly the o r b i t s

r.

on

Go

%

y%6 -> NO

=

(for

p 6

(o:q(y))

%

% %,%

0

via

No

i n s t e a d o f N stems

f r o m 4 . 1 3 ( c ) : i n t h e f r e q u e n t case t h a t R = o t h e r w i s e be t h e a l l - r e l a t i o n i n a n y c a s e . 4.21 P R O P O S I T I O N (Betsch ( 6 ) ) . % t h e n ?, a n d % c o i n c i d e o n Proof.

If

nly

y%6

If

h : 'I ny

+

+

r

= ( o : 6 ) =>

N = No

nl,n2~N n16

i s well defined.

= n26.

h turns out

n6

t o be an N-automorphism,

Now

would

el.

= n 2 y ==> nl-n2E(o:y)

Therefore

r, ?

Nr i s u n i t a r y a n d

then f o r a l l

(y,6Ee1)

g ( y ) = 6 ==>

gEGo:

=> y a ) .

= (0:s) NO

The r e a s o n f o r d z f i n i n g

3

y%6 ->

so

h ( y ) = h ( 1 y ) = 1 6 = 6,

hEG. hence

y

5

6.

84 PRIMITIVE NEAR-RINGS

110

3 . ) INDEPENDENCE A N D D E N S I T Y

An a p p r o p r i a t e frame f o r o u r

n e x t c o n s i d e r a t i o n s i s g i v e n by

4 . 2 2 D E F I N I T I O N ( M l i t t ( 9 ) ) . L e t M be a n a r b i t r a r y s e t a n d t h e s e t of a l l f i n i t e s u b s e t s o f M . A map f(M) r : f ( M ) + INo i s c a l l e d a r a n k map i f

(a) r(0) = 0 (b)

1

(c)

ti F€f!(M)

FE$(M)

mEM:

ti

r(F u E m l )

m,ncM:

= r(F)+a

with

aEI0,l)

[ r ( F u ( m l ) = r ( F W I n 1 ) = r ( F ) ->

r ( F w { m , n I ) = r(F)].

->

F i s then c a l l e d r-independent i f

r(F) = IF[.

4.23 R E M A R K

With r e s p e c t t o r - i n d e p e n d e n c e , S t e i n i t z ' s t h e o r e m i s f u l f i l l e d ( s e e A . Kertesz, "On independent s e t s of

e l e m e n t s i n a l g e b r a " , Acta S c i . M a t h . 260 - 2 6 9 ) . S e e a l s o K a a r l i ( 2 ) .

(Szesed) 21, 1960,

4.24 EXAMPLES

r ( F ) : = I F [ . Then r i s a r a n k f u n c t i o n a n d every ( f i n i t e ) subset i s r-independent.

( a ) Define

( b ) Take a v e c t o r s p a c e M o v e r a f i e l d K .

Set r(F): = = dim L ( F ) ( l i n e a r h u l l ) . r i s a rank function and r-independence i s j u s t l i n e a r independence.

( c ) Take a n N - g r o u p r a n d d e f i n e f o r e a c h

r(@) a s t h e number o f non % - e q u i v a l e n t g e n e r a t o r s ( i . e . r ( Q ) = l @ n 6 1 / Q l ) . Then r i s a r a n k f u n c t i o n a n d #

= {y l y . . . , y n l

i s r-independent

if

oEf(I')

#eel

tr i S j : v l h j . This independence i s c a l l e d %-independence. The same c a n be d o n e f o r 5.

and

111

4a General

In the theory of rings each primitive r i n g R i s isomorphic t o a " d e n s e " s u b r i n g T'? o f a r i n g HomD(I',I') f o r some i r r e d u c i b l e R-module I' a n d w i t h D = HomR(I',I') ( t h e c e n t r a l i z e r ) making Dr i n t o a v e c t o r s p a c e ( s e e ( N . J a c o b s o n ) , p . 26 3 1 ) . D e n s i t y means h e r e ( i n o u r n o t a t i o n ) t h a t SEN ( y l , ...,y s l lin. indep. i n r 1 61 6 5 ~ r 3 FeR j i E : ( l , sl: r(yi) = 6i.

-

...,

,...,

( I t i s c l e a r t h a t o n l y v a l u e s o f i n d e p e n d e n t e l e m e n t s c a n be arbitrarily prescribed.) We a r e g o i n g t o p r o v e s i m i l a r t h e o r e m s f o r n e a r - r i n g s . B u t b e f o r e d o i n g s o we h a v e t o t a k e a l o o k a t t h e d e n s i t y c o n c e p t ( s e e a l s o Adler ( 1 ) and Ramakotaiah-Rao ( I ) ) . 4 . 2 5 N O T A T I O N L e t M be a s u b s e t o f some a t o p o l o g y i n M as i n B e t s c h ( 7 ) :

We i n t r o d u c e

M(T).

I f mcM a n d y c r , d e f i n e S ( m , y ) : and q : = fS(m,y)[mcM A y E r 1 .

= Im'EM(m'(y)

= m(y)l

4 . 2 6 P R O P O S I T I O N ( B e t s c h (7),(11)). (a)

yis

t h e s u b b a s e o f some t o p o l o g y t o p o l o g y " ) on M.

i s dense i n M

( b ) NEM

<->

V

SEIN

tf m e M

w.r.t.

7

yl, . . . , y

7

(the "finite

<==>

s ~ r3

nEN

V

ie(l,

...,$ 1 :

: n ( q ) = m(v+

P r o o f . s t r a i g h t f o r w a r d and hence o m i t t e d . I n a l l t h a t f o l l o w s , " d e n s i t y " means " d e n s i t y w i t h r e s p e c t t o o f 4.26". 4.27 REMARKS ( a ) I f M and N a r e s u b n e a r - r i n g s o f M ( r ) t h e n i t i s easy t o see t h a t No i s d e n s e i n Mo i f f No+Mc(r) i s d e n s e i n Mo+Mc(r). N o t e t h a t N o t M c ( r ) a n d M o + M c ( r ) a r e no n e a r - r i n g s i n g e n e r a l ( s e e 4 . 5 3 ( e ) ) , e x c e p t i n some i m p o r t a n t s p e c i a l c a s e s . ( S e e 4 . 5 4 and 4 . 6 0 . )

112

$ 4 PRIMITIVE NEAR-RINGS

( b ) If N i s d e n s e i n i n Mo.

M then

9 {id)

( c ) Observe t h a t i f H

r

automorphism group of

tf

b

mEMH(r)

= NnMo(r)

i s dense

i s a fixed-point-free

MH(r) r

then

(since

Mo(l')

h(m(o)) = m(o)).

hEH:

If H = { i d )

No:

then

M(r).

MH(r) =

( d ) We w i l l b e m a i n l y i n t e r e s t e d i n n e a r - r i n g s w h i c h a r e d e n s e i n M (r) a n d 77 (I-):= M (r)+Mc(r) G: G: G: ( 4 . 5 2 and o t h e r s ; . 4 . 2 8 T H E O R E M ( R a m a k o t a i a h (2), B e t s c h ( 7 ) ) .

L e t H be a f i x e d -

r.

p o i n t - f r e e g r o u p o f a u t o m o r p h i s m s o f some g r o u p

(a)

(b)

W

YET*

A

v

ti

3

6Er

mEMH(r):

~ ' E P \ H ~ m: ( y ' ) =

tf S E N b

y1 y

0).

., Y ~ E ~ *

. .

,

Hyi

6 1 y . . . y 6 s ~ r 3 mEMH(r) (c) If

H

6

<->

b

SEN

tf

{id),

W

Mc(r)

(d) I f

3

61y

Proof.

W

nEN

c RH(r),

N

v

V SEN

<->

v

E

iE{l,...,sI:

, Hri

Y~,....Y,E~*

Hyi

...,d S € r 3

nEN

Hyj

tj i c { l , . . . , s I :

I n a n y c a s e we may assume t h a t RH(r) = MH(r)

otherwise (a)

aEHy

-fl

haEH:

Define

mEM(r)

by

a = h,(y)

m(y)

= 6

and

0

mEMH(r);

d e t e r m i n e d by t h e c o n d i t i o n s ( v y ' ~ r \ H y : m(y')

9j

= bi.

HH(r) <-> for

i

+j

= 6i.

{id),

for

(since H i s fixed-

haV)

m(a):= I

clearly

n(yi)

n(yi) H

i

for

M(r).

point-free).

A

=+

= 6i.

m(yi)

Hyj

=/=

N i s dense i n

ylY...,ysEr,

j

M H ( r ) <->

... ,s?:

i E { l ,

i

for

Hyj

i s dense i n

NrMH(r)

61y...y6s~r

= 6 A

(m(y)

= 0).

m(y)

aEHY

.

Then a&HY m i s uniquely = 6 A

113

4a General

( b ) D e f i n e maps

mi€MH(r)

b

= o

mi(y')

y'4Hyi:

m: = m l +

Then

with

...+m S

w i l l do t h e j o b .

By ( b ) a n d 4 . 2 6 ( b ) .

<-:

If

Hyi

p

Hyj

If

Hyi

= Hyj,

V

n(yj)

and

= tii

(as i n ( a ) ) .

( c ) =+:

->

mi(yi)

for

i

9

j,

b

mEMH(r)

the result i s clear.

= m(yi)

nEN: n ( y i )

a>

and t h e r e s u l t f o l l o w s a g a i n f r o m

= m(yj)

4.26( b ) . ( d ) +: By 4.27,

I f one

( s a y yl)

yi

= bi-6

V

fulfills

ncEMc(r) Take

= 61.

...,sl.

for

i ~ { 2 ,

ic{l,.,.,s):

Two o r m o r e

take

= 0,

map w h i c h i s c o n s t a n t no(yi)

n(yi)

=

MH(r).

t o be t h e

n0ENo

with n: = notn

Then

C

= tii.

c a n n o t be z e r o .

yi

(R,,(r))o

i s dense i n

No

If all

the

0,

yi

r e s u l t follows from ( c ) . <==: I f

and

yl,

SEN,

... , y S c r *

61,...,6s~r,

Then

3

n(o) =

nEN 0,

define

iE{l,

nENo

so by 4 . 2 7 a )

,

Y ~ + ~= :o

..., s + l l :

and b y ( c ) ,

M i s dense i n

4.29 THEOREM ( B e t s c h ( 7 ) ) .

No

i

and

6s+1:

=

= tii.

H 5 Aut(T)

and

r

Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

(a)

i s discrete i n

RH(r).

(b)

i s discrete i n

MH(r).

( c ) H h a s o n l y f i n i t e l y many o r b i t s o n

( i n t h i s case, H

+ {id}.

( a ) ->

Then

r

t r i v i a l l y hold f o r

i s finite).

since

MH(r),

as i n

r. H = {id).

S o we assume t h a t

MH(r)Em0.

(b): Trivial,

0.

Because o f

i s dense i n

4.26.

Proof. Again the r e s u l t s

j

g,,(r).

rev,

Let

n(yi)

for

Hyj

Hyi

MH(r) f

FIH(r).

I14

§4 PRIMITIVE NEAR-RINGS ( b ) ->

( c ) : Assume t h a t H h a s i q f i n i t e l y many o r b i t s

r.

on

3

Then

If

msMH(r) a n d a n e i g h t o u r h o o d U o f m.

Take

3

SEN

y1

= HyJ

Hyi

assume t h a t

,..., y S ~ F :U

then Hyi

S(m,yi)

+ Hyj

for

S 3

fl

i = l

S(m,yi). So we w i l l

= S(m,yj).

i

p

j.

S i n c e H has i n f i n i t e l y many o r b i t s ,

3

y S t l E r \ ( { o l u Hyl u

\

Then

...,s t l l

i t C l ,

ml:

Define

...

=

0

3

Hy,).

...t e s )

m (el+

\y'

eiE:MH(I')

and

jE.Il,

...,s t l l :

= mltestl.

m2:

S

Then

and

ml

If m ( Y s t l ) m1

P0

are

E

fl S(m,yi) i=l

then

ml(Ys+l)

then

m2(yStl) = Y

= 0

5 U.

=I= m(yst,),

= m.I

If m ( y S t 1 ) = so

m2

m2

Anyhow,

0

~

+

+ 0 ~= m

so

(

~

dp m . U c o n t a i n s an e l e m e n t

+m

and

c a n n o t be

discrete. ( c ) ->

(a):

I f H h a s o n l y f i n i t e l y many o r b i t s o n

t h e n each element o f

MH(r)

and o f

flH(r) i s

u n i q u e l y d e t e r m i n e d b y i t s e f f e c t o n f i n i t e l y many s u i t a b l e elements o f So

7is

discrete.

r.

r

~

~

115

4b 0-primitive near-rings

b l 0-PRIMITIVE N E A R - R I N G S

Now we s h a l l p r o v e a " d e n s i t y - l i k e "

s t r u c t u r e theorem f o r

0 - p r i m i t i v e n e a r - r i n g s , Gle s t a r t w i t h z e r o - s y m m e t r i c o n e s . We may a s ~ u m e ( 1 . 4 6 ) t h a t i f N i s 0 - p r i m i t i v e o n r t h e n N Q M ( T ) . G e n e r a l i z a t i o n s c a n b e f o u n d i n M l i t z (4),(3),(12) a n d K a a r l i ( 6 ) .

4 . 3 0 T H E O R E M ( B e t s c h ( 6 ) ) . L e t NewO be 0 - p r i m i t i v e o n r . I f N i s a r i n g t h e n N i s a p r i m i t i v e r i n o on t h e N-module

r and Jacobson's d e n s i t y theorem i s a p p l i c a b l e . I f N i s a n o n - r i n g t h e n we q e t a k i n d o f a d e n s i t y property:

...,y , ~ r , i c { l , ..., s ? : nyi

(D): \ scIN

3

ncN

Proof.

61,...,6s~r

6i.

N i s a r i n g we o n l y h a v e t o a p p l y 4 . 8 .

If

Now l e t l e t

V

%-indep.

yl,

N

s be

I n t h e t e r m i n o l o g y o f (D),

be a n o n - r i n g . > 1

and f o r

...,s - 1 1

tc{l,

let

S(t)

be t h e statement

v Lemma.

t

...,S I : fl

kc{ttl,

(0:Yi)

$ (0:Yk)

,

i = l

Vtc{l,

...,s - 1 1 :

S(t).

P r o o f . By i n d u c t i o n o n t . Since f o r

(o:y)

YEel

V

i d e a l o f N,

i,jE{l,

i s a maximal l e f t

...,s l :

(o:yi)~

E ( o : ~ . ) =>

( o : y . ) = ( o : y . ) => yi%yj -=> i = j . J 1 J Particularly: S(1): k~(2, s l : (o:yl)$

v

...,

9(O:Yk). Now assume

S(t),

s 2 3

and

...,sl.

k~ftt2,

t

Then Since

fl

(O:Yi)$(O:Yk) i=l (o:yk)

and

(o:Yt+l)$(o:Yk)*

i s maximal,

t

[I

(o:yi)+(o:Yk)

i=1

= (o:Yt+l)+(o:Yk)

= N*

$ 4 PRIMITIVE NEAR-RINGS

116

t

n

Since N i s not a r i n q ,

n (o:yttl)+

(o:vi)

i=l by 3 . 4 ( i ) , $(o:yk) than S f t t l ) .

which i s n o t h i n q e l s e

Now r e t u r n t o t h e p r o o f o f 4.28 and l e t yl, . . . , y , , be a s i n (D). A g a l n we u s e i n d u c t i o n o n 61,.,.,6 S

tc{l,

...,s l .

t = 1

If

then

b

Now assume t h a t

b

iE{1,

..., t l :

3

ylEel

nlaN:

tE{l,,..,s-ll

nlyl

3

= 61.

ntEN

= 6i.

ntyi

t

By t h e lemma,

+ {ol.

Lyttl

3

Therefore

Now we t a k e

: nttlyi

n

L: =

i=l

Since

LYttl

RcL: Lyt+l nttl:

= 6i,

(o:vi)$(o:y

$N

t+l),

hence

(3*4(a)),

= 6ttl-nt6t+l

=

LYttl

r.

.

and g e t \1 i E { l , and t h e p r o o f i s c o m p l e t e . = Ltnt

...,t t l l :

4.31 R E M A R K S

(a)

(D)

i s no " r e a l " d e n s i t y p r o p e r t y s i n c e t h e r e i s no

n e a r - r i n g i n s i g h t i n which f i n i t e topology). ( b ) From

and

(D) i t

N i s dense ( w . r . t t h e

+)

f o l l o w s (Ramakotaiah ( 2 ) ) t h a t ,

yl,...,yscr

if

SEIN ?r

a r e %-independent,

i=l

= W(Yi))). ( c ) The c o n t e n t o f 1Bli

= 1,

(0)

m i g h t be v e r y t h i n :

i f e.g.

(D) i s t r i v i a l . So i t i s n o t t o o s u r p r i s i n g

t h a t t h e c o n v e r s e o f 4.28 does n o t h o l d :

+)

( B e t s c h ) : If o n e c h a n g e s o f k.25 t o Y':={S(m,y)ln&:Mn A y€e,(r)] t h e n o n e g e t s a " r e a l " d e n s i t y t h e o r e m w.r.t. t h e r e s u l t i n g c o a r s e r topology.See also 9.230.

4b 0-primitive near-rings

117

L e t N be t h e n o n - r i n q {fEMo(iZ4)lf(2)E{0,21Af(3) 3f(l)I. In z4, O 1 = { 1 , 3 1 , 1 % 3 %( D ) i s f u l f i l l e d , b u t I0,21 QN Z4, so Z4 i s n o t s i m p l e a n d t h e r e f o r e N i s n o t 0 - p r i m i t i v e on Z 4 .

( d ) ( D ) i s equivalent t o t h e following property: (0’):

3

v

yl,. . . , y ,cr,

SEN

iEI1,

nEN

%-indep.

mE:M(r)

.,.,sI: nyi = m ( y i ) .

Now we t u r n t o a r b i t r a r y n e a r - r i n n s . 4.32 T H E O R E M

( a ) Let N be 0 - p r i m i t i v e o n r. Case 1: Nor -/= { 0 1 . Then N o i s 0 - p r i m i t i v e on so 4.30 i s applicable ( f o r r ) , and NC Case 2 :

5

NO

Mc(r).

Nor

(01.

r,

Then

Mc(r)

N =

and

r

i s a

non-zero simple g r o u p i s 0 - p r i m i t i v e on r a n d ( b ) Conversely, i f e i t h e r N o N, E Mc(r) o r i f N = Mc(r) where r (01 i s s i m p l e t h e n N i s 0 - p r i m i t i v e on Proof.

( a ) Anyhow,

If

Nor

I01

N,

e

Mc(r).

r

then NO

and

No

i s 0 - p r i m i t i v e on

(01,

r.

i s o f t y p e 0 by 3 . 1 8 ( a )

r

(3.15(a)).

“by f a i t h ” , s o R = r a n d N = N, = Mc(r) by 1 . 5 0 ( b ) . S i n c e Ncr i s s i m p l e i f f r i s s i m p l e , ( a ) i s shown ( o b s e r v e t h a t Ncr Col!). If

Nor

=

No

= {01

+

( b ) A g a i n b y 3 . 1 8 ( t h i s t i m e by ( b ) ) , i f

N r i s o f t y p e 0 . S i n c e NccMc(r), N o a n d N, ( a n d hence N ) a c t f a i t h f u l l y o n r , s o N i s 0 - p r i m i t i v e o n I’. I f N = Mc(r), r $. Iol and s i m p l e , t h e r e s u l t i s clear.

0 - p r i m i t i v e then

118

4.33

§ 4 PRIMITIVE NEAR-RINGS

R E M A RK

(D) w o u l d

and i n

No

= (0:6)

r,

( i n

if

n o t n e c e s s a r i l y mean t h e same i n would be d e f i n e d by

%

Nr

(o:y)

y%fi:<->

=

il). Cf. 4 . 1 9 .

4 . 3 4 THEOREM ( R a m a k o t a i a h ( 1 ) ) . E a c h 0 - p r i m i t i v e prime i d e a l

ideal i s a

dp N.

P r o o f . L e t I b e a 0 - p r i m i t i v e i d e a l o f N. L e t Nr b e o f t y p e 0 w i t h g e n e r a t o r yo s u c h t h a t I = ( 0 : r ) (4.3). Assume t h a t

Ji $ (o:r),

J1J2

Jir SO

9 N: J1J2tI

= JiNy

(o:r) = I,

=

c Jiyo

+ {ol.

= Jiyo

Jiyo

0

r.

Now

J1 $ I

A 5

A J2$I.

Jir.

Since

By 3 . 4 ( a ) ,

r,

= Jlr =

JlJ2r

a contradiction.

R E MA R K I n 5 . 4 0 we w i l l s e e t h a t t h e c o n v e r s e o f 4 . 3 4

holds i f 4.36

r.

dN

Jiyo

4.35

J1,J2

i ~ { 1 , 2 1 , Jir

For

so

3

N = No

h a s t h e DCCN.

THEOREM ( R a m a k o t a i a h ( 1 ) ) . E v e r y m a x i m a l m o d u l a r i d e a l

I of

Nsno

Proof.

i s a 0 - p r i m i t i v e one.

L e t I b e a m o d u l a r m a x i m a l i d e a l . By 3 . 2 2 ,

I is

c o n t a i n e d i n a modular maximal l e f t i d e a l L . Since (L:N)

i s the largest ideal of N contained i n L

( b y 3.25), we g e t

by 3.21(a). By 4 . 3 ( c ) ,

I?(L:N)

and

(L:N)

By t h e m a x i m a l i t y o f I,

I i s 0-primitive,

i s modular

I = (L:N).

s i n c e b y 3.29 L i s

0-modul a r . By t h e w a y ,

if N i s 0 - p r i m i t i v e on

n o t necessarily simple (K.

r

a n d Ny = : A t h e n A i s

Kaarli).

F o r t h e r e s t o f t h i s s e c t i o n , we g i v e a d e s c r i p t i o n o f a c l a s s o f 0 - p r i m i t i v e near-rinqs which are n o t 1 - p r i m i t i v e . This d i s c u s s i o n i s d u e t o H o l c o m b e (5), w h e r e t h e p r o o f s c a n b e found,

too.

119

4b 0-primitive near-rings

4.37

4.38 DEFINITION I f

:'G

=

EFr, H < A u t ( T ) ,

(r,+)E(a,

t h e t r i p l e (T,B,H)

be the s e t of

r\el

NOTATION I f N r E N q , l e t A : = generators". If A sN r , let (cf. 3.14(a)!).

-

"non-

AutN/(o:A) ( A )

H ( B ) ~ B we , ,-all

c o m p a t i b l e i f a t l e a s t one o f t h e

following conditions i s satisfied:

r.

( a ) B i s no n o r m a l s u b g r o u p o f

3 YEAB 2 (3 h ' c H 3

(b) (c)

(3

A

4.39

BEB

tl

y~r\B

+

hEH : y+B h(y). R E B : y+B = h ' ( y ) ) h

1

Y ' E ~ \ B: h ' ( y ' ) - y ' & B ) .

be 0 - p r i m i t i v e on r , N a n o n - r i n o w i t h NE% i d e n t i t y a n d DCCL, a n d l e t A ( a s i n 4 . 3 7 ) b e an N - s u b g r o u p

THEOREM L e t

of

r

Then

such t h a t

i s not f a i t h f u l , b u t o f type 2.

NA

N i s n o t 1 - p r i m i t i v e on A,

many o r b i t s o n el,

G ( 4 . 1 0 ( b ) ) has f i n i t e l y

i s c o m p a t i b l e and

(T,A,G)

(where A i s a f i n i t e dimensional v e c t o r space o v e r the d i v i s i o n r i n g

GAvta)).

Conversely:

4.40

THEOREM L e t subgroup.

r

Let

b e an a d d i t i v e g r o u p and A be a n o n - z e r o

GA

be a r e g u l a r group o f

automorphisms

o f A w h i c h h a s o n l y f i n i t e l y many o r b i t s o n A. L e t H b e a subgroup o f (a)

( b ) each

Aut

(r,+)

such t h a t

i s compatible.

(r,A,H) hEH

i s r e g u l a r on

AA.

( c ) H h a s o n l y f i n i t e l y many o r b i t s o n (d)

hEH: h/,EG*.

Then

N = IfEMHu{a) ( r ) I f / A E M G A ( A ) l

0-primitive, a n d t h e DCCL.

b u t n o t 1 - p r i m i t i v e on

nA.

i s zerosymmetric,

r,

has an i d e n t i t y

5 4 PRIMITIVE NEAR-RINGS

120

r9

I f moreover

and A i s a f i n i t e d i m e n s i o n a l v e c t o r

A

tJ

s p a c e o v e r some d i v i s i o n r i n g D a n d i f then

N = If€M,,"~,)(r)lf/,EEndD(r))

r,

b u t n o t 1 - p r i m i ti ve on

"(

NE

hEH:

h/AED

i s also 0-primitive,

no,?i!z 3, , a n d

moreover

i s a ring.

o :A)

4 . 4 1 R E M A R K See a l s o H o l c o m b e ( 4 ) f o r t h e m o r e q e n e r a l c a s e t h a t A i s o n l y a f i n i t e u n i o n o f N-subgroups o f t y p e 2 w i t h zero intersection. 4.42

t h e n i n t h e n o n - r i n g case o f 4.30

GA = { i d )

REMARK I f

we g e t n e a r - r i n g s o f t h e f o r m N = ( f c M o ( r ) l f ( A ) F A ] ( s e e e . g . N o i n 3 . 8 ) . C f . R a m a k o t a i a h - R a o (1),(3),(4). Conversely,

(r,t)

if

i s a f i n i t e group and A a non-

t r i v i a l subgroup then

N:= { f E M o ( r ) l f ( A ) F A }

i s a finite

n e a r - r i n g w i t h i d e n t i t y , z e r o - s y m m e t r i c a n d 0-, b u t n o t 1 - p r i m i t i v e o n r. A i s j u s t t h e s e t o f n o n - g e n e r a t o r s and i s an N-subgroup s u c h t h a t N / ( o : A ) i s a non-ring if

lAI

> 2.

C\

1 - P R I M I T 1 V E N E A R - R I FIGS

Now l e t N b e 1 - p r i m i t i v e o n

r r

r.

C = Go

Then

(by 4 . 1 8 ) ,

i s n o t N-isomorphic t o a p r o p e r subgroup (4.17(bl),

= eouel

R = Col

(by 4.15(d)),

+

Ak N , L EOI 3 Y E r : We s t i l l a s s u m e t h a t N E

t~ L

4.43

Ly =

or

r

R =

r

(3.2)

and

(by 3.4(a):.

M(r).

THEOREM

r . Then Nor f ( 0 ) A R = r . T h e n N o r , Nc = Mc(r) a n d e l = r .

(a) L e t N be 1 - p r i m i t i v e on C a s e 1:

on

If

No

i s a r i n g then

i s 1-primitive

N i s dense i n

Maff(r)

where r i s a v e c t o r space o v e r t h e d i v i s i o n r i n g D : = HomN (F,r). 0

121

4c 1 -primitive near-rings

If

No i s n o t a r i n g t h e n applicable.

(D)

Case 2 : Nor ( 0 1 A R = ( 0 1 . Then m i t i v e on r a n d 4.30 h o l d s . Case 3: Nor = { o l . Then N = N, a s i m p l e g r o u p dp (01.

=

o f 4.30 i s

N = No

Mc(r)

i s 1-priand

r

is

( b ) C o n v e r s e l y , i f a n e a r - r i n g NEM(r) i s such t h a t N o i s 1 - p r i m i t i v e on r w i t h Nce(IO1, M c ( r ) l o r i f N = Mc(r) (r { o } a n d s i m p l e ) then N i s 1 - p r i m i t i v e on r .

+

+

P r o o f . ( a ) I f Nor (01, N o i s 1 - p r i m i t i v e on r by 3 . 1 8 ( a ) . S i n c e e a c h s t r o n g l y monogenic N-group has e i t h e r R = ( 0 1 o r n = r, t h e r e s t f o l l o w s from 1 . 5 0 , 3.9, 3 . 1 5 ( a ) , 4 . 2 7 ( a ) a n d 4 . 3 2 . ( b ) I f No i s 1 - p r i m i t i v e on r and N, = {O) o r 'L N, Mc(r) t h e n e i t h e r R = ( 0 1 o r R = r ( 1 . 5 0 ) , s o N i s 1 - p r i m i t i v e on r by 3 . 1 8 ( b ) and 3 . 1 5 ( a ) . 'L If N Mc(r), r s i m p l e and (01, then c l e a r l y

-

-

N i s 1 - p r i m i t i v e on

r.

+

4.44 R E M A R K 4.43 i s t h e main r e a s o n f o r d e f i n i n g " s t r o n g l y monogenic N-groups r" as i n 3 . l ( b ) a n d n o t by t h e c o n d i t i o n s "monogenic" and ycr: (Ny = R v Ny r)", f o r 4 . 4 3 would n o t be t r u e i n t h i s c a s e : Take 'l = H8, N o : = { f ~ M ~ ( r ) l f ( =P )f ( 6 ) c 1 0 , 2 , 4 , 6 1 A N,: = {fcMc(r)lf(0)c{0,2,4,611. A f ( 4 ) ~ { 0 , 4 ) ) and

'v

T h e n one can show t h a t N : = N o t N c i s a subnear-ring of M ( r ) enjoying the following properties:

Nr

r

and

a r e f a i t h f u l , s i m p l e a n d monogenic. Moreover,

NO

ycr: (Ny = R = { 0 , 2 , 4 , 6 ) v Ny = r). B u t (01 f R r, and N o i s n o t 1 - p r i m i t i v e on r ( i t i s n o t e v e n t r u e t h a t f o r a l l y c r Noy i s e i t h e r = { o l , = R o r = r, s i n c e N04 = { 0 , 4 1 ) .

122

8 4 PRIMITIVE NEAR-RINGS

From 4 . 3 0 a n d 4.45

r

= e0u

el

we g e t w i t h a s t r a i q h t f o r w a r d p r o o f

N€TIO

THEOREM L e t t h e n o n - r i n a

r

be 1 - p r i m i t i v e on

but

without %-equivalent generators.

N

Then

i s dense i n t h e n e a r - r i n q

€370

For 1-primitive near-rinqs

{ f c M o ( r ) \ f ( e o ) = {oil.

w i t h D C C we g e t a w h o l e b u n c h

o f i m p o r t a n t r e s u l t s ( c f . Rarnakotaiah ( 3 ) , B e t s c h ( 1 0 ) ) : 4 . 4 6 THEOREM ( B e t s c h (3)).

Let

e n d o w e d w i t h t h e DCCL.

be 1 - p r i m i t i v e on 7 and

NEWO

Then

( a ) T h e r e a r e o n l y f i n i t e l y many % - e q u i v a l e n c e c l a s s e s i f N i s a non-rinq. (b)

3

S E N: ,,N

s. =

N-isomorphic

1 Li,

r)

l e f t i d e a l s and N-groups o f t y p e 1 ( s o

2.50 i s a p p l i c a b l e ! ) ; s =

f i n i t e l y many p a i r w i s e ( t o

Li

i = l

i f N i s a non-rinq then

Ir/%l-i.

( c ) A l l N - g r o u p s o f t y p e 0 a r e N - i s o m o r p h i c a n d o f t y p e 1. (d) N contains a r i g h t i d e n t i t y (not necessarily two-sided). (e)

N is s i m p l e .

( f ) N i s e i t h e r 2 - p r i m i t i v e o n 'I o r t h e r e i s n o N - g r o u p

o f t y p e 2, Proof.

If N i s a ring,

(b)

-

( f) are e i t h e r well-known

o r t r i v i a l . So we w i l l a s s u m e t h a t

No

i s a non-rinq.

( a ) S u p p o s e t h a t t h e r e a r e i n f i n i t e l y many - - e q u i v a l e n c e assume t h a t

i 2 1,

y o ~ e o . Then

hence

( b ) Now l e t

Y

~

~

representatives of

YOEeOl

yl,

(o:yo)

Y

.~'yS~

= N

p

We may

(o:yi)

for

So b y ( D ) o f 4 . 3 0

y1yy21...~f31.

(o:yo)=(o:y1)=(o:~y1,y2))3'. d i c t i o n t o t h e DCCL.

.

v,,vl.y2,...

classes w i t h representatives

.

which i s a contra-

. b e. a c o m p l e t e s y s t e m o f

t h e %-equivalence classes w i t h

...,y S c e 1 .

S

Then

n

( 0 : ~ ~= ) I 0 1

i= 1

,

but

123

4c I-primitive near-rings

minimal l e f t i d e a l s . Now a p p l y 2 . 5 0 ( p )

tf

j E { l ,

...,sl:

to get

N =

'.1 L

Since

r

by 3 . 1 0 .

j = lj * ,-b

L j -N

L.$(o:Y~), J

( c ) Holds by t h e p r o o f o f ( b ) and 3 . 1 1 ( a ) . ( d ) By ( b ) a n d 3 . 2 7 ( d ) ,

N1 e. sided. (e)

o f 3.8

shows t h a t e i s n o t n e c e s s a r i l y t w o -

3

If I Q N,

minimal, IE(0:L.) J

N contains a riqht identity

. . . ,s l :

jc(1,

L j n I = C O l . =

{O}

But %

(for

L j -N

LjL)I.

Lj

Since

ILjcInL

F), w h e n c e

j

=(O), I = {Ol.

is SO

( f ) By 4 . 7 o r b y ( c ) .

Note t h a t 4.46(a) i s not v a l i d f o r r i n g s : I f 2 s p a c e IR , c o n s i d e r e d a s a n H o m ( r , r ) - m o d u l e , (xELP)

are pairwise inequivalent w.r.t.

n e a r - r i n g which i s p r i m i t i v e on 4.47

=/=

i s the vector all

(1,x)

Hom(r,T)

i s a

a n d h a s t h e DCCL.

N contains a l e f t identity;

COROLLARY N E ' ~ ' ) ~ . D C C N , PI

r

%,

r

I 9 N,

{Ol. T h e n

(a) N !s

1 - p r i m i t i v e c=+

N i s 2 - p r i m i t i v e <->

N i s simple.

( b ) I i s 1 - p r i m i t i v e c->

I i s 2 - p r i m i t i v e <->

I i s maximal.

Proof.

( a ) By 3 . 4 ( c )

and 3.7(c),

2-primitivity coincide.

N i s simple then

If

by 3.4(b)

1 - p r i m i t i v i t y and

I n t h i s case,

I = N

N i s simple.

i s a m i n i m a l i d e a l and

a n d Lemma 3 i n t h e p r o o f o f 3 . 5 4

I : = N; t h e n

A =

r)

(with

N has a f a i t h f u l N-group

Nr

o f t y p e 1, s o N i s 1 - p r i m i t i v e . (b)

follows from (a).

K a a r l i ( 2 ) showed t h a t if N = No i s s i m p l e a n d U i s a m a x i m a l N subgroup of

N w i t h NU

{O} t h e n N i s I - p r i m i t i v e .

K a a r l i ( 4 ) and A d l e r ( 1 ) .

Cf.

also

124

9 4 PRIMITIVE NEAR-RINGS

d ) 2-PRIMITIVE N E A R - R I N G S

A g a i n we assume t h a t i f

N i s 2 - p r i m i t i v e on r then

NCM(r).

1 . ) 2-PRIMITIVE N E A R - R I N G S The s t r u c t u r e o f 2 - p r i m i t i v e n e a r - r i n g s c a n b e d e s c r i b e d as follows.

4.48 THEOREM ( a ) L e t N be 2 - p r i m i t i v e on Case 1:

Nor

I01 A R

r. Then = r. Then

No

i s 2-primitive

r,

N, = M c ( r ) a n d e l = r . No i s a r i n g t h e n N i s d e n s e i n M a f f ( r ) (as i n 4.43); i f No i s a n o n - r i n g t h e n (0) o f 4 . 3 0 i s a p p l i c a b l e ( f o r No).

on

If

Nor 9 { o }

A

R = {ol.

p r i m i t i v e on

r

and 4 . 3 0

Case 2 :

Case 3 :

Nor

=

(01.

Then

Then

N

= No

i s 2-

i s applicable.

N = Mc(r)

and

r

is a

c y c l i c group o f prime order. Conversely,

(b)

if

Nce{{O),Mc(r))

No

i s 2 - p r i m i t i v e on

or if

N = Mc(r)

(r

r

o f prime o r d e r ) then N i s 2 - p r i m i t i v e on The p r o o f i s s i m i l a r t o t h e o n e o f 4 . 4 3

4.49

r

a n d if I A

unless order).

I = Mc(r)

r.

and t h e r e f o r e o m i t t e d .

Betsch ( 7 ) ) . I f N i s 2 - p r i m i t i v e { 0 1 , t h e n I i s 2 - p r i m i t i v e o n r, (where r i s n o t a c y c l i c qroup o f prime

T H E O R E M ( c f . F a i n (1) a n d

on

with

a c y c l i c group

N, I

p

125

4d 2-primitive near-rings

( a ) We f i r s t s h o w t h i s t h e o r e m f o r

Proof.

Ir

Ev d e n t l y , Assume t h a t If

IA =

3

i s faithful.

sI r.

A

Iol

then consider

{01.

~ E A : N6

Ir =

But

NEQ.

Therefore

INACIA =

i s f a i t h f u l . Hence

Iol,

N6 =

NA =

(01,

r

A

.b

(01, and N A =

{Ol,

I

so

NA

If

NA.

since

sN f ,

r. Ir

whence

A = {o}.

If

I A

+ {ol

3

then again

6fA:

I6

=+

Io1.

16 sN r , s o 16 = Consequently A = r , f o r I6cA. T h e r e f o r e I i s 2 - p r i m i t i v e o n r. ~ ( 1 6 )= (NI)~FIG,

Since

r.

( b ) Now l e t N b e a r b i t r a r y . We may assume t h a t

N

No,

=/=

I s excluded.

so case 2 o f 4.48

I f N f a l l s i n t o c a s e 1, No i s 2 - p r i m i t i v e o n r . BY 2.18 , I , = ~n ri, 2 N 0 ' I f I , p (0) t h e n I, i s 2 - p r i m i t i v e o n r , hence

I i s 2 - p r i m i t i v e on r by 3.18(b). I f I, = {Ol t h e n I e N c = Mc(r). S i n c e el = r , 10 AN r . 10 = { o l i m p l i e s t h a t f o r a l l y = n o c r and f o r a l l i E I i y = ino = 0 , so I = I o ) .

Hence

10

T a k e any

3

+ Io)

and so

mceMc(T);

is1 : io = yfr:

10 =

m o =:p. C

r. Because o f

10 = ,'I

u.

= i o = p = mco = mcy, h e n c e i = mc % I = Mc(r). If 3 P E P : r Zp, I is 2 - p r i m i t i v e o n r; i f not, I i s not 2-primitive. Now

iy

-

a n d we g e t

If N i s i n c a s e 3, 4.50

REMARK

I i s t r i v i a l l y 2 - p r i m i t i v e on

4 . 4 9 c a n n o t b e t r a n s f e r r e d t o 0- o r 1 - p r i m i t i v i t y ,

n o t even f o r f i n i t e ,

abelian, zerosymmetric near-rings.

I t i s e a s y t o show t h a t i f N i s e . g . O - p r i m i t i v e o n I 9 N t h e n I r i s f a i t h f u l and monogenic. B u t n o t necessarily simple: T a k e r: = z 8 , A : = IO,2,4,61 a n d E: = I 0 , 4 l . N:

r.

= {ffMo(r)lf(A)rA

A f(5)=f(l)

h

f(7)=f(3)),

r

and

I:= (0:A).

9 4 PRIMITIVE NEAR-RINGS

126

N i s 0 - p r i m i t i v e on

r,

but

E d1

r.

Moreover,

s t r i c t l y monogenic and I has a r i q h t i d e n t i t y . e v e n b e 0 - p r i m i t i v e o n some o t h e r g r o u p Assuming t h a t ,

take

and ( 0 : 3 ) 1

I"

Ir

is.

I cannot

=:IyA:

Ir

i n

and p u t

L : = ( 0 : ~ ; ) ~Then . L i s a maximal l e f t i d e a l o f I ( 3 . 4 ( f ) ) L (0:1)1 , L ( 0 : 3 ) 1 ( s i n c e ( 0 : 1 ) 1 and (0:3)1 c a n n o t be m a x i m a l ) . T h e r e f o r e ( o : l ) I + L = ( f l : l ) I+ L = I . b u t ( 0 : l ) n (0:3) = E ~ I c - L , s o I w o u l d h a v e t o b e a r i n g

and

by 3.4(i),

a contradiction.

So p o i n t e d o u t , N N =

A s Y.S.

I

(0:l) n(0:3).

S e e m i n g l y t h e r e i s no "smal l e r " c o u n t e r e x a m p l e t h a n t h a t above w i t h 4096 e l e m e n t s . See a l s o 5 . 1 9 ( a ) . By t h e way, case

o n e c a n u s e Z o r n ' s lemma t o show t h a t i n a n y

I 9 N (I9 IO), N w - p r i m i t i v e )

h a s some I - g r o u p s

o f t y p e v. 4 . 5 1 COROLLARY ( F a i n ( 1 ) ) . L e t P b e a 2 - p r i m i t i v e i d e a l o f NET,. L e t I b e a n o t h e r i d e a l o f N c o n t a i n i n g P . Then P i s a 2-primitive

Proof. I

9 P,

A NIP,

ideal of I . and

NIP

'Ip p I01

and

i s 2-primitive.

Since 'Ip i s 2 - p r i m i t i v e . Hence

P i s 2-primitive i n I.

2 . ) 2 - P R I M I T I V E NEAR-RINGS I.IITH IDENTITY

I n t h i s case,

=

Also,

then

if

N = No

(if %

=

% %

NE~,)

or

el

=

r

( i fN $ % ) .

(by 4.21).

Recall that a 1-primitive near-ring cfl0 with a l e f t identity i s a l r e a d y a 2 - p r i m i t i v e one w i t h i d e n t i t y ( 4 . 6 ( b ) ) . We a r e now i n a p o s i t i o n t o g e t a " r e a l " a n d f u n d a m e n t a l d e n s i t y theorem.

4d 2-primitive near-rings

N

N f N, (case 1

No a non-ring

No

(case 2 o f

o f 4.48)

No a r i n q

=

127

4.48)

-

Ma f f ( r )

HomD ( r ,r 1

Dr a v e c t o r s p a c e $. fol

l'fc (r)

M C ( r ) = MC(r)

Go f i x e d - p o i n t -

0

0

r

f r e e on

(b) Conversely,

e v e r y n e a r - r i n g w h i c h i s dense i n Maff(r) o r HomD(r,r) (where r i s a non-zero v e c t o r space o v e r some d i v i s i o n r i n g D ) o r d e n s e i n (r) o r

mc 0

(r)

MC

fixed-point-free

(Go

i s 2-primitive

on 7 )

0

on Proof.

r,

where Co = G

w

0

2 - p r i m i t i v e on note that

= MC 0

(r)

4.28(c)

r,

= MGoV{b}

(r)

since

Y I sN r .

then

(r)

My; 0

o(r)

= M

=

Go

(D)

o f 4.30,

and 4 . 2 7 ( a ) .

Mo(r),

then (D)

of 4.30

No,

implies that

N

which i s t r i v i a l l y dense i n ( s i n c e

(r)

MC

= MCal(r)

=

Mo(r).

0 =/=

i s not a rinq,

on

and t h e r e s u l t f o l l o w s f r o m

equal t o )

If N

No

i s therefore I f No i s a

0

Go = { i d }

in

If

i s fixed-point-free

Go

+ {id}

Go

No

and has an i d e n t i t y .

CvcrJs(Y) =

gE:Go:

If

r

and has an i d e n t i t y

cannot occur.

t h e s t a t e m e n t is c l e a r .

ring,

If

r

( a ) I f N i s 2 - p r i m i t i v e on

t h e n case 3 i n 4.48

1

{61.

apply again 4.27(a).

0

i s dense

8 4 PRIMITIVE NEAR-RINGS

128

( b ) Assume now t h a t N i s d e n s e i n H o m D ( r , r ) , where i s a n o n - z e r o v e c t o r s p a c e o v e r some s k e w - f i e l d D.

r

N i s a dense s u b r i n g and t h e r e f o r e a p r i m i t i v e

Then

r i n g o n r . F r o m t h i s we d e d u c e : I f N i s dense i n M a f f ( T ) then

i s dense i n

No

s o r h a s no n o n - t r i v i a l N o - s u b g r o u p s , HomD(r,r), a n d N i s 2 - p r i m i t i v e o n r. I f N i s d e n s e i n M O(r) GO

each dense s u b n e a r - r i n g of t h a t i s t r i v i a l l y 2 - p r i m i t i v e o n r. I f G o {id} t h e n M o(r) = FIG ( r ) a n d 0

GO

4.28(c)

Nr c a n n o t c o n t a i n n o n - t r i v i a l

shows t h a t

N-subgroups. Finally i f

in

(Tic 0

N i s dense i n

(r))o

a non-trivial

MC (r)

then

0

As above,

= MC ( I - ) .

r

No

i s dense

cannot c o n t a i n

0

No-subgroup (or one can use 3 . 1 8 ( b ) ) .

4.53 REMARKS (a) I t i s n o t t r u e t h a t each 2 - p r i m i t i v e n e a r - r i n q w i t h identity, N take

MC(r): and

N: =

=/=

r

No

and

a non-ring,

No

i s dense i n

f i n i t e w i t h I i d l f G s A u t ( r ) , G f i x e d - p o i n t free Nr h a s R = r, s o

F i G ( r ) . Then

CN(r) = { i d ) ( 4 . 1 3 ( c ) ) a n d t h e r e f o r e MC(r) = M(r). But N M ( r ) , s o N c a n n o t be d e n s e i n M ( r ) b y

+

4.29.

This i s a l a t e b u t convincing reason f o r

introducing t h i s crazy

N, ( r ) ,

w h e r e one f i r s t

0

s w i t c h e s down t o

No

(by forming

(r)

Co = End

and

NO

t h e n back up b y a d d i n g a l l o f t h e c o n s t a n t s :

MC(r)

would be t o o b i g i n general. ( b ) 4 . 5 2 ( a ) does n e i t h e r h o l d f o r 0 - p r i m i t i v e n e a r - r i n g s with i d e n t i t y nor f o r 2-primitive near-rings without i d e n t i t y ( n o t even f o r

N =

No

and N f i n i t e ) :

129

4d 2-primitive near-rings

r:

n4

N : = {fcMo(T)\f(A)!!A) i s 0 - p r i m i t i v e on r w i t h i d e n t i t y , b u t n o t dense i n M C ( r ) = Mo(r) ( 4 . 2 9 ! ) . M: = { f E M o ( r ) l f ( 3 ) = 0 1 i s If

=

and

A:

= (0,2),

0

2 - p r i m i t i v e on

MC (r)

in

r,

w i t h o u t i d e n t i t y and a g a i n n o t dense

Mo(r).

=

0

(c) A l l 2-primitive near-rinas with i d e n t i t y on w h e r e No ( d ) 4.32,

i s a non-ring,

4.43

and 4.48

Z4,

w i l l b e c l a s s i f i e d i n 4.63.

reduce t h e theorv o f p r i m i t i v e

near-rings t o those o f p r i m i t i v e zero-symmetri c nearr i n g s . We w i l l t h e r e f o r e m a i n l y d e a l w i t h t h o s e o n e s

i n t h e sequel. (e) Recall (4.27(a))

that

Kc

general. H e r e i s some e x a m p l e :

= -x)

r A!

=

i s "only a set" i n

(f) 0

G = {id,-idl

i s a fixed-point-free

IR.

= {o,id,-id}.

C:

XER:

f(-x)

(with

-id(x):

=

autsmorphism group on

Vc(IR)

= {fEM(IR)If(O)

= 0 A

-f(x)).

n If m c ( R ) = : N , t a k e nl: = sin+-pN and n2: = n C o n s i d e r n : = nlon2EM(IR ) . = id+pN. n no = s i n o ( i d t r ) - s i n ( % ) i s n o t an odd f u n c t i o n , t h u s n o t b e l o n g i n g t o MC(IR), w h e n c e ngN a n d N i s no n e a r - r i ng.

4.54

COROLLARY I f N i s 2 - p r i m i t i v e o n

r

with

(r)

AutN

= {id)

0

t h e n N i s dense i n e i t h e r one of t h e f o l l o w i n g n e a r - r i n o s ( n o t a t i o n as i n 4.52): H o m D ( r , r ) , Maff(r), Mo(r) o r

M(r) 4.55

(cf.

4.65).

THEOREM ( R a m a k o t a i a h non-ring on

r

classes w.r.t.

(2)).L e t

NET),

be a 2 - p r i m i t i v e

w i t h an i d e n t i t y . Then any two e q u i v a l e n c e

2

(except t h e zero class) are equipotent.

$4 PRIMITIVE NEAR-RINGS

130

T*/%

Proof. L e t E be i n

C o n s i d e r t h e map

and

a f i x e d element o f E.

E

f: G

+

E

9

+

q(E)

(with

Since G i s fixed-point-free

(4.52),

aqain).

f i s injective.

so f i s a b i j e c t i o n .

f i s surjective,

By d e f i n i t i o n ,

G=AutN(r)

3 . ) 2 - P R I M I T I V E Z E R O - S Y M M E T R I C N E A R - R I N G S WITH I D E N T I T Y A N D A M I N I M A L LEFT I D E A L . 4 . 5 6 THEOREM ( B e t s c h ( 7 ) , c f . D e s k i n s ( 2 ) ) . w i t h i d e n t i t y which i s 2 - p r i m i t i v e on

r

L e t N = N , b e a nr. and has a m i n i m a l

l e f t i d e a l L . Then %

(a) L = N r' ( b ) 3 e2 = eeL*:

L = Ne = L e

and

morphic t o Proof.

(a) Since

and

+ {ol,

Lr

3

W i t h y as a b o v e , and

m i ni m a l ) . k-%e

Pie

E

i s antiiso-

(eNe;). YEr:

+ {ol,

Ly

so

Ly =

r

Now we c a n a p p l y 3 . 1 0 .

yeel.

= ey

3

(C,,,(T),a)

Hence

(o:y)nL

= Le = L.

eEL":

e y = y.

=

and

e2 = e

{O} a n d L e

By ( a ) ,

Therefore

I01

(since L i s

Le

{Ol.

e2-eEL n ( o : y ) =

SN L,

Since

Le = L .

By L e C N e s L ,

C N ( r ) = CN(L) = CN(Ne)

( i t can

be e a s i l y v e r i f i e d t h a t N - i s o m o r p h i c N-groups have i s o m o r p h i c centralizer-semigroups). For

neN,

consider

d e f i n e d and

t n : Ne xe

+

+

Ne xene

.

i s well-

tn

E C ~ ( N ~ ) .

C o n s i d e r n e x t t h e map

h:

eNe ene

+ +

CN(Ne). tn

If

e n e = erne

so h i s w e l l - d e f i n e d . C l e a r l y , t h e n t n = t,, h i s an a n t i h o r n o m o r o h i s m . I f h ( e n e ) = h ( e m e )

t n = t,

and

Specializing

then

tj X E N : x e n e = t n ( x e ) = t m ( x e ) = xeme. x = :e

we g e t

e n e = eme

and h i s

shown t o b e i n j e c t i v e . Finally,

b

c€CN(IJe)

3

nEN:

c ( e ) = ne.

ene = e c ( e ) = c ( e 2 ) = c ( e ) = ne aet

c ( x e ) = x c ( e ) = xene

tn(xe),

Therefore

a n d f o r a l l X E N we

so c = tn.

4d 2-primitive near-rings

131

and h i s s u r j e c t i v e , hence an a n t i i s o m o r p h i s m . COROLLARY ( B e t s c h ( 7 ) ) .

4.57

N ~ r t ) h a s an i d e n t i t y a n d

If

a minimal l e f t i d e a l L t h e n a l l f a i t h f u l N-groups o f type 2 ( i f those e x i s t ) a r e N-isomorphic

(r,

determines t h e p a i r

I f e i s as i n 4.56(b),

t h e group

CN(r))

u n i q u e l y "up t o i s o m o r p h i s m " .

i s a group w i t h z e r o and L = Ne a s a f i k e d - p o i n t -

(eNe,.) a c t s on

(eNe\{O),*)

( t o L ) and N

f r e e a u t o m o r p h i s m g r o u p (by r i g h t m u l t i p l i c a t i o n ) . Hence e " b r i n g s b a c k " some i n f o r m a t i o n o n

4.58

r

REMARK F o r m o r e i n f o r m a t i o n o n t h e s e t o p i c s ( a p a r t i a l c o n v e r s e o f 4 . 5 6 , t h e u n i q u e n e s s o f (r, CN(r)), s e e B e t s c h ( 6 ) a n d 5 7 a ) , i n p a r t i c u l a r 7.5.

4.)

4.59

COROLLARY L e t 2.50

( f o r "),

(2.50(a)

be a 2 - p r i m i t i v e n e a r - r i n g w i t h

NEW,

5

i s applicable,

hence a l s o

a n d G h a s f i n i t e l y many o r b i t s o n

and 4.21),

i s d i s c r e t e "on

r,

w h i c h i s t h e same a s t o s a y t h a t

MG(r)

Z

(4.29).

7a) f o r the information t h a t i f a f i x e d - p o i n t - f r e e

automorphism group H o f

r

h a s f i n i t e l y many o r b i t s o n

MC(r) h a s t h e DCCL. See a l s o K a a r l i 4.60

etc.)

2 - P R I M I T I V E P!EAR-RINGS WITH I D E N T I T Y A N D MINIMUM CONDITION

DCCL a n d i d e n t i t y . T h e n 4 . 4 6

See

9.227.

out of NSM(T). C f .

r

then

( 2 ) and Oswald ( 1 0 ) .

THEOREM ( B e t s c h ( 7 ) ) . L e t N b e 2 - p r i m i t i v e o n r w i t h D C C f o r t h e l e f t i d e a l s o f No and w i t h a n i d e n t i t y . Then N i s equal t o one o f t h e f o l l o w i n g n e a r - r i n g s ( n o t a t i o n as i n 4.52):

NpNo

N - N o

No a r i n g

Maff(r) H o m D ( r , r )

No a n o n - r i n g

Mc ( r ) 0

MC(r)

dimDr f i n i t e Go h a s f i n i t e l y many

o r b i t s on

r

132

$ 4 PRIMITIVE NEAR-RINGS

P r o o f . f o l l o w s from 4 . 5 2 and 4.59.

Note t h a t

Rc (I') 0

i s a n e a r - r i n g i n t h i s case ( f o r i t equals N ) . 4 . 6 1 COROLLARY I f N h a s a n i d e n t i t y , i s 2 - p r i m i t i v e o n r a n d i f t h e n o n - r i n g No h a s t h e DCCL a n d A u t N ( r ) = ( i d ) 0

N = M(r) ( i f N No) or o t h e r w i s e N = Mo(r). I n b o t h c a s e s , r ( a n d t h e r e f o r e N, t o o ) i s f i n i t e . So t h e D C C i m p l i e s f i n i t e n e s s ! then e j t h e r

4.62

R E M A R K These r e s u l t s i l l u s t r a t e some r e m a r k s i n t h e

preface: w h i l e the "elements o f r i n q theory" a r e r i n g s

o f l i n e a r mappings on r , t h o s e ones f o r n e a r - r i n q t h e o r y a r e n e a r - r i n g s o f a r b i t r a r y mappings ( p e r h a p s w i t h some r e s t r i c t i o n s ) o n

r.

4 . 6 3 THEOREM ( K a a r l i ( 4 ) ) I f

-

then

I a-S z- N

and i f S / I

i s 2-primitive

I d- N .

Proof. Since I i s a 2 - p r i m i t i v e l e f t i d e a l o f S,

I = (L:S)s

h o l d s f o r some 2 - m o d u l a r l e f t i d e a l L o f S b y 4 . 3 . By 3 . 3 4 ,

L 2NS.

Consequently,

Hence

I

=

(L:SINn S

and ( L : S l N 2 N .

I i s an i d e a l o f N .

See a l s o K a a r l i ( 2 ) a n d R a m a k o t a i a h ( 2 ) .

I n t h e l a t t e r paper i t

i s shown t h a t i f N€Hl i s f i n i t e a n d 2 - p r i m i t i v e o n

r,

if N i s a

lrl-1 i s a p r i m e t h e n e i t h e r N=M(T) o r N=Mo(T) n o n - r i n g and i f or ~ N ~ = ~ i r f~ zr ; i s a b e l i a n , N z N rOr h o l d s i n t h e l a s t c a s e ( t h i s r e s u l t can be deduced f r o m 4.55

and 4.61).

133

4d 2-primitive near-rings

5.)

AN APPLICATION TO INTERPOLATION T H E O R Y

4.64

reg

DEFINITION I f

and

NEM(r),

N i s said t o f u l f i l l

the f i n i t e interoolation property I f

w

SE:nU

3

ncN

w

v

Y1#...*YSEr,

Yi

sl:

n(yi)

i c { l #...,

t

Yj

9

for

\I

j

61’..-’6sEr

= 6i.

T h e r e i s an o b t r u s i v e s i m i l a r i t y t o t h e d e n s i t y c o n c e p t s .

In

fact: 4.65

Let

THEOREM

N I M(r)

with

N

and

No

No

not a ring.

Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

r*,

( a ) No i s 2 - f o l d t r a n s i t i v e o n

r

( b ) N i s 2 - p r i m i t i v e on

Go = { i d ) .

with

( c ) N f u l f i l l s the f i n i t e i n t e r p o l a t i o n property.

Proof.

( a ) ->

(b):

Nr i s t r i v i a l l y f a i t h f u l , 2 - f o l d

t r a n s i t i v i t y implies 1-fold i n turn that

Nr

{ol.

If

t r a n s i t i v i t y and t h i s

I o l 9 A sN r ,

take

0

Some

6cA”

No6 =

r

and

Then A =

W

ycr

2

n,ENo:

no6 = y .

So

r.

+

9

If Goy 5: G o 6 but y 6 and ( s a y ) 6 o, take 0 . Then noEN 0 w i t h n o y = o A no6 (0:6) , s o y+6 i n (0:Y) NO NO Nor * h e n c e (4.20(c)), a contradiction, Therefore Goy 9 G o 6 some

+

Go = { i d } .

(b)

(c):

b y 4.54

( c ) ->

(a):

trivial.

K a i s e r ( l ) , Lausch ( 5 ) , Ramakotaiah ( 3 ) .

Cf.

and 4.28(d).

P l i t z (12),(13),

P i l z ( 2 5 ) and

134

5 4 PRIMITIVE NEAR-RINGS

4 . 6 6 REMARKS

( a ) So i f a n e a r - r i n g N o f m a p p i n g s o n r i n t e r p o l a t e s a t o and 2 o t h e r p l a c e s then N i n t e r p o l a t e s a l r e a d y on an a r b i t r a r y ( f i n i t e ) n u m b e r o f p l a c e s . Compare t h i s w i t h the corresponding "1 i n e a r " r e s u l t i n r i n g theory ( ( N . J a c o b s o n ) , C o r o l l a r y t o theorem 1 on p . 3 2 ) . This i s a " p u r e l y i n t e r p o l a t i o n - t h e o r e t i c " r e s u l t . ( b ) I t c a n be s h o w n t h a t i f N f u l f i l l s t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y a n d 1i-l > 3 t h e n N o i s a non-

ring. 4 . 6 7 C O R O L L A R Y T a k e r = (IR,+). Then a n y o n e o f t h e f o l l o w i n g n e a r - r i n g s and a l l n e a r - r i n g s c o n t a i n i n q o n e o f t h e m h a v e t h e p r o p e r t i e s t h a t Pi i s 2 - p r i m i t i v e on r w i t h N No, Go = { i d ) and No n o t a r i n g :

+

[XI,N 2 :

t h e n e a r - r i n g o f a l l s t e p f u n c t i o n s on IR the subnear-ring of M(1R) g e n e r a t e d by t h e t r i g o n o metric polynomials.

N1: N3:

= IR

F o r a l l o f them f u l f i l l t h e f i n i t e i n t e r p o l a t i o n p r o p e r t y w h i c h q u a l i f i e s them f o r 4 . 6 5 . The a u t h o r h o p e s t h a t n e a r - r i n g s o f i n t e r p o l a t i n g f u n c t i o n s become i n t e r e s t i n g f o r s p p r o x i m a t i o n t h e o r y ( b e c a u s e t h e s e f u n c t i o n s c a n be i t e r a t e d w . r . t . 0 ) . After a l l t h a t complicated s t u f f the r e a d e r will p o s s i b l y a g r e e w i t h t h e a u t h o r t h a t t h e p r i m i t i v e n e a r - r i n q s have s u c c e s s f u l l y r e v e n g e d t h e i r d i s c r i m i n a t i n g name.

,

135

One f i l l s t h e t r a s h i n t o some b a g s W i t h t h e s e one o n l y c a l c u l a t e s . The r u b b i s h w h i c h you s t i l l c a n s m e l l Is o f t e n c a l l e d t h e " r a d i c a l " . T h i s b e a u t i f u l poem d a t e s way back t o 1 9 7 5 . The a u t h o r i s s t i l l i n h i d i n g .

55

RADICAL THEORY

T h i s p a r a g r a p h e q u a l s on h a r v e s t : t h e s t r a i n s o f p r e v i o u s p a r a g r a p h s a r e h i q h l y r e w a r d e d by t h e f a c t t h a t many r e s u l t s o f t h i s 0 5 a r e easy consequences of previous ones ( c f . e.g. 5 . 4 8 o r 55 c ) , d ) ) . A n e a r - r i n q N m i q h t h a v e no f a i t h f u l N-qroup o f t y p e v . The n e x t q e n e r a l c a s e i s t h a t a l l N - g r o u p s o f t y p e v work t o o e t h e r t o be z e r o . N i s t h e n c a l l e d t o get the intersection fl(o:r) " v - s emi s i mp 1 e 'I . Any how, t h is i n t e r s ec t ion " m e a s u r e s 'I h o w f a r N i s away t o be v - s e m i s i m p l e a n d i s c a l l e d t h e v - r a d i c a l I t contains a l l disoustinq guys, f o r factorino out /dv(N). N ) gives a v-semisimple near-rinq N/

2"(N)'

2v(

F i r s t we g i v e s e v e r a l v-modular l e f t i d e a l s . and Between J O ( N ) object the Ne d i s c u s s , when gv

21(N)

21/2(N),

a l l v , ?,(&I Nil

c h a r a c t e r i z a t i o n s of ?"(N), using We q c t immediately. there i s another radical-like i n t e r s e c t i o n o f a l l 0-modular l e f t i d e a l s . i s " h e r e d i t a r y " a n d prove t h a t f o r

=

N i s v-semisimple

@qv(Ni).

30(N)~31(N)~J2(N)

Also f o r

v

1

$. 7 ,

2v(No)'2v(N).

i f f N i s a s u b d i r e c t product o f v-primitive n e a r - r i n q s . With c h a i n c o n d i t i o n s t h i s s u b d i r e c t p r o d u c t becomes a f i n i t e d i r e c t sum a n d we q e t ( 5 . 3 1 ) i n s p e c i a l c a s e s t h a t N i s v - s e m i s i m p l e i f f N i s a f i n i t e d i r e c t sum o f s i m p l e v - p r i m i t i v e n e a r - r i n o s w i t h D C C L . I n 5 . 3 2 we q e t a " W e d d e r b u r n Artin-1 i k e " s t r u c t u r e t h e o r e m f o r v - s e m i s i m p l e n e a r - r i n q s .

5 5 RADICAL THEORY

136

&(N)

c o n t a i n s a l l n i l N-subqroups,

i d e a l s and

qo(N)

r i n g case,

&(ti)

21,2(N)

all nil left

a l l n i l i d e a l s . However, i n c o n t r a s t t o t h e i s not necessarily n i l i f

N i s f i n i t e (cf.5.48).

F i n a l l y we c o n s i d e r t h e n i l a n d t h e p r i m e r a d i c a l o f a n e a r -

r i ng. a ) JACOBSON-TYPE RADICALS: COMMON THEORY

1.)

DEFINITIONS A N D C H A R A C T E R I Z A T I O N S O F THE RADICALS

As u s u a l , l e t N be a n e a r - r i n g and v ~ { 0 , 1 , 2 1 . Recall our c o n v e n t i o n a b o u t t h e i n t e r s e c t i o n o f an e m p t y c o l l e c t i o n o f s u b s e t s o n p a g e 1. 5.1

DEFINITION

Iv(N):

II

=

Nr o f

(o:r)

i s called the

type v

v - r a d i c a l o f N.

5.2

n

THEOREM * d v ( N ) =

I

I v-Dr.id.of

=

L

n

(L:N)

v-mod. l e f t id.of

N

.

N

Proof. 4.3. The r e l a t i o n s b e t w e e n t h e r a d i c a l s a r e e a s i l y d e s c r i b e d : 5.3

PROPOSITION ( a ) + ' J ~ ( N 5) c d l ( ~ ) c r d 2 ( ~ j ) . (b) I f

NE?$

then

Yl(N)

= q2(N).

( c ) I f N i s a r i n g t h e n 'Jo(N) ( J a c o b s o n - r a d i c a l o f N). Proof.

(a):

= Cal(N)

= )a2(N)

=

q(N)

by 3.7(a).

(b) : b y 3 . 7 1 ~ ) a n d 3 . 1 9 ( a ) . (c):

If

Nr

i s o f t y p e v and N i s a r i n g t h e n one i s a n N - m o d u l e . The r e s t i s

sees as i n 4 . 8 t h a t obvious.

Nr

137

5a Jacobson-type radicals: common theory

5.4

T H E O R E M ( B e t s c h (3)).

+0

qu(N)

P r o o f . By d e f i n i t i o n ,

Nr

u

If

=

n

Nr

i s s t r o n g l y monogenic,

o f type v

(0:r)

so

n

L. L u-mod. l e f t id.in N

*dv(N) =

then

=

(o:r). il

But

(o:y),

YCr

where each

i s

(o:y)

N

o r a v-modular l e f t i d e a l

(3.23). Conversely,

3

Then N/L

?, aN

l e t L b e a v - m o d u l a r l e f t i d e a l of N.

NrENlj

r

3

r

yO€r:

(by 3.4(e))

=

A

NYO

L = (o:y0)

i s o f t y p e u.

(3.23).

Hence t h e

a r e j u s t a l l u - m o d u l a r l e f t i d e a l s ( o r = N)

( 0 : ~ ) ' s

and t h e r e s u l t f o l lows. T h i s r a i s e s t h e q u e s t i o n what happens w i t h t h e i n t e r s e c t i o n o f a l l 0-modular (= maximal modular) l e f t i d e a l s o f 5.5

DEFINITION

21/2(N):

n

=

L

N.

*

L 0-mod. l e f t i d e a l o f rt 5.6

REMARK

%I/2(N)

i s o f t e n denoted by

l i t e r a t u r e (see e.g.

m o t i v a t e d by t h e f a c t t h a t

tdlIE(N)

o n l y " h a l f o f an i d e a l " ( a l e f t ,

"D(N)"

i n the

Our n o t a t i o n i s

Betsch ( 3 ) ) .

i s i n qeneral

but not necessarily

a t w o - s i d e d i d e a l ) and b y i t s l o c a t i o n :

Proof. "Jo(~) =

c

Nr o f

n

n

Nr o f t y p e 0

type 0

These (0:~)'s

n

(o:r)

=

Nr o f

n

type 0

n

ycr

(o:y)

(0:y).

ycel(r)

are (as i n 5.4) e x a c t l y a l l 0-modular

l e f t i d e a l s . Hence

';to(N)

c Pl12(N).

i s a t r i v i a l consequence o f 3 . 7 ( a )

alI2(N) and 5.4.

c al(N)

138

5 5 RADICAL THEORY

The f o l l o w i n g r e s u l t comes f r o m F a i n ( 1 ) . 5.8

PROPOSITION [ L 4, N Hence a l ( N ) and

3

h

22(N)

k L 5'dY(N) A vE{~,~)]=+LE]~(N). a r e semiprime i d e a l s .

k E M :

P r o o f . I f Lk2Ldy(N), b u t L $ ' d V ( N ) , then 3 N r E N 9 : Nr i s o f t y p e v a n d Lr f C o ) . S o 3 y c r : Ly $. l o ) . Hence y & e o , s o Y E e l a n d Ly dN r by 3 . 4 ( a ) . Thus Ly = r a n d Lr = r. T h e r e f o r e r = Lr = = 2~ r = = L k r = { o ) , a c o n t r a d i c t i o n .

...

5.9

REMARK I f L

v = 2

i n 5.8, t h e r e s u l t remains v a l i d i f

s Nr .

5.10 C O R O L L A R Y j l ( N ) c o n t a i n s a l l n i l p o t e n t l e f t i d e a l s and c o n t a i n s m o r e o v e r a1 1 n i l p o t e n t N - s u b g r o u p s .

I2(N)

C f . 5 . 3 7 a n d 5 . 4 5 f o r more r e s u l t s i n t h i s c o n n e c t i o n . 5 . 1 1 E X A M P L E S The f o l l o w i n g e x a m p l e s s h a l l show t h a t no two of a(1,2y g e n e r a l l y c o i n c i d e , n o t even f o r

70,

*dl,

a2

f i n i t e zero-symmetric near-rings.

See Betsch ( 3 ) .

G e n e r a l i z a t i o n s c a n be f o u n d i n Meldrum ( 1 3 ) . Nl i s I-primitive ( a ) N1: = r f E M o ( z 4 ) ! f ( 2 ) = f ( 3 ) = 0 ) . on Z 4 h e n c e Adl(Nl) = {O), b u t n o t 2 - p r i m i t i v e , hence by 4 . 4 6 ( f ) = N1. So =k in general.

"d(N1)

gl(N) q2(N)

( b ) L e t N 2 : = t f ~ ~ l O ( ~ , ) ~ f ( 2 ) ~ { O By , 2 }3 }. 3. we know t h a t N 2 i s 0 - p r i m i t i v e , b u t n o t 1 - p r i m i t i v e on r. S i n c e e a c h map € N 2 i s d e t e r m i n e d by i t s e f f e c t on 1 , 2 , 3 , N 2 i s t h e sum o f t h e l e f t i d e a l s L 1 : = ( 0 : 2 ) n ( 0 : 3 ) , L2: = (O:l)n(O:3) and L3: = ( 0 : 2 ) n ( 0 : 3 ) . Since ( 0 : l ) c\ ( 0 : 2 ) n ( 0 : s ) = { a ] , N 2 = L1;L2;L3. The map

q, * Y

*

I 0

L1

with

fy

N 2 - i s o m o r p h i s m . Hence

fY(x): = 2r

L 1 -N2

q.

Y

x f 1

x = l Similarly,

i s an

139

5a Jacobson-type radicals: common theory fb

B4.

L 3 -N2

and L 2 i s an N2-qroup o f t \ / p e 2 .

lL21 = 2 L2+L3,

Therefore

Ll+L3

and

L1+L2

l e f t ideals. Their intersection f s S i n c e N c o n t a i n s an i d e n t i t y , 5.3(b)).

are 0-modular

31/2(N2)

11(N2)=

q2(N2)

B u t each N2-group o f t y p e 2 i s

3.11(a).

Hence

g2(N2) = (0:L2)

(0). (by

kN L 2

by

$. {Ol =

= (0:2)

= Ca1/2(N2). g1(N2)

Observe t h a t

-

=

g2(N2) =

(0:2)

i s not nilpotent

i n s t r i k i n g contrast t o the s i t u a t i o n in r i n g theory!

Compare 5 . 4 5 !

-

( c ) N3

:=

CfEMo(Z,xZ,)If(A)E

A and

(a,O)-(b,O)

E

A

--',

f ( a , O ) - f ( b , O ) c A 1 w i t h A : = {(0,0),(2,0)1 h a s i d a s i d e n t i t y . A l l ( a , 2 ) w i t h a €2Z4 g e n e r a t e t h e N 3 - Q r o u p r : = Z4xZ2, r h a s o n l y Z4x{Ol and A a s n o n - t r i v i a l N3-

20(N3)

s u b g r o u p s , and they a r e n o t i d e a l s . Hence = {(0,0)1. Now A a n d Z,xEOI/A a r e N 3 - g r o u p s o f t y p e 2 . The a n n i h i l a t o r s o f ( a , O ) ( a r - Z n ) , o f ( 2 , O ) and o f ( 3 , O ) + A are maximal m o d u l a r l e f t i d e a l s o f N 3 , t h e i r i n t e r s e c t i o n 0 3 contains /d,,2(N3). B u t D ' - = C O I . I n 5 . 3 7 ( b ) we w i l l s e e t h a t t h i s implies D C ( N 3 ) , whence = D

f COI 5 . 1 2 EXAMPLE

=

Yo(N3).

If

N = Nc

21 / 2

then

2,12(N3)

Cd,(N) = Cdll2(N)

s e c t i o n o f a l l maximal normal subgroups o f radical" o f

(N,+)),

while

'a2(N)

normal maximal subqroups o f

(Apply 3.21(c), ideal i n

2.)

3.29,

N = Nc

= gl(N)

(N,+)

= inter-

('Baer-

= intersection o f a l l

(N,+).

t h e f a c t t h a t each 0-modular

left

i s 1-modular and 5.2).

RADICALS O F RELATED NEAR-RINGS

20,

a2

To b e a b l e t o t r e a t j1a n d j o i n t l y (at least for a w h i l e ) we i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n w h i c h comes f r o m u n i v e r s a l a l g e b r a ( s e e (Hoehnke) and M l i t z ( 6 ) ) .

$ 5 RADICAL THEORY

140

5.13 DEFINITION A map % w h i c h a s s i q n s t o e a c h n e a r - r i n q N a n

NN)

ideal

o f N i s c a l l e d a r a d i c a l (map) i f f o r e v e r y

N ,N ' ~ r ) : (a) @N/a(N)) (b) I f

1°)

=

heHom(N,N')

5.14 D E F I N I T I O N

(b) 2 - r a d i c a l :

I A N

h(@(N)) c

a(h(N)).

If% i s some r a d i c a l map t h e n

(a)@-semisimple:

If

then

K

and

5.15 P R O P O S I T I O N

E

g(N) =

c->

<->

N

@(N) denote

I f @,is

NEr)

i s called

(0).

= N. {k+IlkEK)

by

a r a d i c a l map a n d

K+I/I.

N,N'

are

E?)

then (a) If

h: N

radical (b) I f (c) (d)

.

9

N'

and N i s n - r a d i c a l

then

i s $2-

N i s a - r a d i c a l t h e n ! IsN: N / I i s R - r a d i c a l .

b I4N: \ IsN

aL(N/I)3z(N)tI/I.

KEN:

R(N/*) =

K/r

a>

KtIz

8(N)).

(e) I f N i s simple then e i t h e r N is%-radical simple. Proof.

N'

(a):

orR-semi-

b y 5.13(b).

(b): b y ( a ) . (c):

Consider the canonical epimorphism

n:

N

+

N/I

a n d a p p l y 5.13(b).

(d): by (c). ( e ) : t h i s h o l d s because

R(N)A

N.

3"'s

I t w o u l d have been s i l l y t o i n t r o d u c e 5 . 1 3 i f t h e would n o t I n f a c t , B e t s c h ( 3 ) h a s shown t h e f o l l o w i n g

be r a d i c a l s .

141

5a Jacobson-type radicals: c o m m o n theory

5.16 THEOREM

For

, N

vc~0,1,21

-+

2v(N)

Proof. Clearly dv(N) A N. Let n + T v ( N ) b e

,.dv'N/.a

(N) )

i s a r a d i c a l map.

and l e t

be a n N -

r

V

g r o u p o f t y p e u . By 3 . 1 4 ( a ) ,

r

is an

N/

a,(N)-

g r o u p o f t y p e v s i n c e E.dV(N) F ( o : r ) . Hence nr = (n+'d,(N))r = l o ) . S i n c e r was a r b i t r a r y , n&n(O:A), where A r a n g e s o v e r a l l N-groups o f type v. T h u s nECaV(N) and n+av(N) = a v ( N ) , s o 5 . 1 3 ( a ) i s shown. T o s e e 5 . 1 3 ( b ) , l e t h be

and

EHom(N,N')

nc~v("l).

Im h = : N " . L e t I' be a n N " - g r o u p o f t y p e u. 2r Since N" = N/Ker h , r c a n be c o n s i d e r e d a s N-group o f t y p e v ( s e e 3 . 1 4 ( b ) ) . T h e r e f o r e nr = { o } . T h i s i m p l i e s t h a t h ( n ) r = n r = { o l e Again,

r

i s a r b i t r a r y , so

h(n)c2v(h(N)).

5 . 1 7 R E E A R Y , F o r >,,(!I) ( v c { 0 , 1 , 2 } ) , 5 . 1 5 ( e ) c a n be i n p r o v e d i f N i s simple then e i t h e r N i s a v - r a d i c a l or v - p r i m i t i v e (since a l l ( o : r ) 9 N). l h e n e a r - r i n g N 1 o f 5 . 1 1 ( a ) i s an example o f a s i m p l e g 2 - r a d i c a l near-rinq. 5.18 T H E O R E M Let I A N be a d i r e c t summand o f -

Then

N.

Y , , ( N ) ~ I E . ]vv(I) h o l d s f o r a l l v ~ ~ O , 1 / 2 , 1 , 2 1( l e t N = N o f o r v = 1 ) .

S u p p o s e t h a t N = I i J . Then N / J I. Take vcIO,1,2}. Each I - o r o u p o f t y p e v i s a n N - g r o u p o f t h i s t y p e b y 3.14 ( b ) . Let be t h e c l a s s o f t h e s e PI-groups. Now

Proof.

? v ( ~ ) = Nr o f A t y p e v( o : r )

1,'~)0 I

Finally, i f w

n o f type =

1/2

A

W

v (O:r+

=

(o:rlb,

and

v

$pi.

t h e n we g e t w i t h 3 . 2 8 a n d 3 . 3 3

3 1 , 2 ( ~ ) In (fl c( N 1 l n L =

G

I

=

-

A

L E .Yo ( N 1

(r,,I)C;-nin1 L do

=

$ 5 RADICAL THEORY

142

EXAMPLES ( a ) L e t N b e t h e n e a r - r i n g N 2 o f 5 . 1 1 ( b ) .

5.19

(=N1 o f 3 . 8 ) .

Since I i s 1 - p r i m i t i v e on

31(N)= ( 0 : 2 ) =I ( 5 . 1 1 ( b ) ) ,

so

b e f i n i t e and A a n o n - t r i v i a l N:={fEMo(r)If(A)cA} a l s o Ex.

5.32

T1(I)c'&(N)n subgroup.

has ~ l ( ~ l ( N ) ) c J 1 ( N )

lo).

I l v y c

r:

r

Then ( K a a r l i ( 9 ) ) . See

i n Meldrum ( 1 3 ) .

j o (N )

i s 0-primi tive, E

But

I. A l s o l e t

( b ) L e t t h e n o t a t i o n and s i t u a t i o n b e as i n 4 . 5 0 .

I f

LetI:=(O:Z)

Z q ,&=)'I(

f(y)

E

=

{O}.

Since N

I contains the n i l potent ideal

I 0 , 4 1 1 . By 5 . 3 7 ( d ) ,

70(I)# { O l ,

whence

~ o ( I ) ' ~ o ( N )A 1 . 5 . 2 0 T H E O R E M L e t Ni ( i E I ) b e a f a m i l y o f n e a r - r i n g s , ~ ~ ! 0 , 1 , 2 } -____a n d l e t N b e t h e d i r e c t sum o f t h e N i l s w i t h N = N o i f v = 1 . Then P roof. LetvdO,1,21. j e c t i o n s , 5.13

. .1(

I f n.:N+Ni 1

denote the canonical

( b ) g i v e s us f o r a l l

g " ( " q v ( N j ) .

Hence

&(N)E.

pro-

i s 1 the inclusion

mi - J " ( N )

G

i E I

4

0

iE1

?"(Nil.

Conversely,

for

each

l e f t i d e a l s Li

o f Ni

3.33.

l e t

For

i E I ,

i d e a l s i n Ni. Li's,

LEJ"(N)

which are associated w i t h L v i a

ii denote

S i n c e L c o n t a i n s t h e d i r e c t sum o f i t s

I t i s n o t known t o t h e a u t h o r i f 5 . 2 0 = O

1'

the s e t o f these l e f t

we g e t

p r o d u c t s and f o r v = 1 / 2 .

N = N

t h e r e e x i s t v-modular

Anyhow,

i s also true f o r direct

one c a n deduce f r o m 5 . 2 0

that

I a i m p l i e s t h a t f o r a l l v ~ t O , 1 , 2 1 we g e t ?"(N)

=

UEA

1' ?"(Ia). I t i s easy t o see t h a t t h e corresponding r e s u l t UEA does n o t n e c e s s a r i l y h o l d i f t h e Icr a r e " o n l y " l e f t i d e a l s . =

143

5a Jacobson-type radicals: common theory 5. -

( 4 ) ) L e t S be an i n v a r i a n t s u b n e a r - r i n g

2 1 THEOREM - (Kaarli (see 1.31)

o f "No.

Then

120)= In particular, particular,

&(N)n s. i f S i s an i d e a l

this holds

we g e t f o r

e v e r y n e a r - r i n g N=N

72( 1 2 ( N ) ) Proof. -__

Sr

Let

be of

2-modular

o f S/L

2 2 2 ( N ) n S.

Then

is,

as a n S-subgroup,

X ( N ) = N <=>

If{O}

as i n 5.13

radical

z(M)=M,

if

s e m i s i m p l e c l a s s i f I A-N ,

R-semisimple.

5.22

(I),

I( K ( N ) ) =

Kaarli

(a)

(2))

(11,

be zero-symmetric

HolcombeWiegandt

and

( v ~ 1 0 , 1 , 2 } ) i s Kurosh-Amitsur t h e sense t h a t

holds f o r every N t

Amitsur,

Holcombe ( 7 ) , ( 1 5 ) ,

(1).

be as

no.

?,,

is"idempotent"in

( b ) By 5 . 2 1 ,

x(I)=I,

i s said to

~-semisimple,implies

(2),(4),(8),(9), M l i t z ( 7 ) , ( 1 1 ) ,

d e f i n e d on

(Kaarli

N

.

N=N0

The m a i n p a p e r s i n t h i s a r e a w h i c h c o n c e r n

THEOREM L e t a l l n e a r - r i n g s i n 5.13

IqN,

R ( N ) a n d if

X(N)=N f o r every nr.

us h e r e a r e Betsch-Wiegandt Walker

radical

H h a s some

homomorphic image M of

t h e o r y i t i s k n o w n t h a t t h i s h o l d s i f f &(N)=N

implies

hereditary have a -

i s

be

n(I)=I.

with

and h(N)=M i m p l i e s

that I

i s c a l l e d a Kurosh-Amitsur

every non-zero

z(N/I)=N/I

radi-

( 7 ) ) o r even o f c a t e -

(e.g.Mlitz

( 7 ) ) . L e t a l l n e a r - r i n g s u n t i l 5.23

(see e.g.Holcombe

From g e n e r a l and

since

results hold f o r other radicals

universal algebras

zero-symmetric.

ideal

2

)2(N)

I n t h i s a r e a we g e t i n t o c o n t a c t w i t h t h e q e n e r a l

c a l t h e o r y of

if

trivial.

t y p e 2 i s an S-group o f t y p e 2.

of

T h i s makes u s c u r i o u s i f s i m i l a r

gories

&(N).

The c o n v e r s e i n c l u s i o n a l s o h o l d s ,

e v e r y N-group

as well.

:

0

t y p e 2 , f r o n i w h i c h we g e t

Hence Nr i s o f

Again i n

r ks S/L, w h e r e L i s a o f S . B y 3 . 3 4 , L zNS. E v e r y

type 2.

l e f t ideal

N-subgroup

=

i n N.

no.

Tv(

2,,(N))

Ad2

=

iff it TV(N)

/63

i s Kurosh-Amitsur. Also, i s KuroshT 3 ( N ) i s d e f i n e d t o b e t h e intersection where

o f t h e a n n i h i l a t o r s o f a l l N-groups l a s t lines of

p . 80).

r

o f type 3 (see the

35

144

RADICAL THEORY

( c ) By e x a m p l e N 1 ) i n t h e " N e a r - r i n g s o f l o w o r d e r " ( A p p e n -

31

i s n o t Kurosh-Amitsur; in Kaarli ( 9 ) i t i s dix), shown t h a t i s n o t Kurosh-Amitsur e i t h e r . ( d ) If i s K u r o s h - A m i t s u r s u c h t h a t R(N)=N=> N={O} for a l l z e r o - n e a r - r i n g s b u t w i t h K(N)=Nf o r some N = N o = k { O ) t h e n

7o

%? c a n n o t have a h e r e d i t a r y s e m i s i m p l e c l a s s

(BetschBut and have h e r e d i t a r y semisimple c l a s s e s ( K a a r l i ( 4 ) , Holcornbe-Walker ( I ) ) .

T3

72

Wiegandt ( I ) ) .

We c o n c l u d e o u r t r o u b l e s o m e t r i p t o r e l a t i v e s o f N b y a consideration of t h e behaviour o f

2,(N)

on t h e one hand

on t h e o t h e r hand. Our f i r s t r e s u l t i s and '&(No), ],,(Nc) an i m m e d i a t e consequence o f 2.18: 5.23 COROLLARY

b

v ~ C 0 , 1 , 2 ) : >,,(N)

T h i s i s n o t v e r y much, compute

indeed.

I t would be f i n e t o be a b l e t o

and /dv(Nc) (5.12!), a s > " ( N ) = ~ , , ( N ~ ) ~ ~ , , ( N ~s)i ,m i l a r t o 5 . 2 0 . T h i s i s n o t t h e case (see a l s o 9 . 7 7 ) : 5.24

via

= (~,(N)),t(~,(N)),.

>,,(N)

Cdv(N0)

perhaps

If N = IfcM(Z4)(f(0) = f(2) = f(3)l. One c a n show t h a t & ( N ) = N, a 2 ( N o ) = No (5.11(a)!), but o n l y o f t h e maps w h ich are constant c o n s i s t s 12(Nc) EXAMPLE

= 0

= 2.

or

F r o m t h a t o n e s e e s t h a t t h e r e i s no o b v i o u s s i m p l e c o n n e c t i o n between ~v(N),&,(No)

IvC

and JV(Nc).

But av(No)

i s always i n

N) :

5.25 PROPOSITION

N))o;

i n particular,

It suffices t o prove the " i n particular",

?"(No)

See 5 . 6 7

>,(No)%(),(

-

],(No)"),(N) Proof.

V W E C O ,1,21:

(t) for

i s t r i v i a l l y contained i n

(22("1))c

No.

for

145

5a Jacobson-type radicals: common theory

3 . ) SEMISIMPLICITY

T h r o u g h o u t t h i s number, l e t v b e

~{0,1,2),

unles

other wi

indicated. 5.26

DEFIfIITION

N i s v-semisimple:

5.27

EXAMPLE N i s v - p r i m i t i v e ->

5.28 -

COROLLARY

<->

N i s $,,-semisimple.

N i s v-semisimple.

( a ) E a c h d i r e c t sum o r d i r e c t p r o d u c t o f v - s e m i s i m p l e near-rings (b) I f

<=>

i s v-semisimple.

b

i E 1 : Ni€?),

tl

icI:

then

8

Ni

i s v - s e m i s i m p l e <->

ic1 Ni

i s w-semisimple.

P r o o f . 5.20. 5.29 T H E O R E M ( B e t s c h ( 3 ) ) .

N

N i s w - s e m i s i m p l e <->

i s isomorphic

t o a subdirect product o f v-primitive near-rings. Proof. Consider the s e t o f v - p r i m i t i v e 1 . 5 8 and 4 . 2 ( c ) .

5.2,

N have t h e

5.30 T H E O R E M L e t c->

i d e a l s and a p p l y

D C C I (DCCL). Then N i s v - s e m i s i m p l e <->

N i s isomorphic t o a subdirect product o f f i n i t e l y

many u - p r i m i t i v e n e a r - r i n g s w i t h D C C I (DCCL). P r o o f . =I>: of

The f a m i l y

N has

n

(Io)aEA o f all v-primitive

I a = {Ol.

ideals

We c l a i m t h a t i t s u f f i c e s

UEA

t o t a k e f i n i t e l y many section

I f not,

1"'s

t a k e some

t o get a zero i n t e r I (aEA). Since aO

n aEA

=

CO),

t h e r e i s some

al€A:

I

fi aO

Con t in u n g i n t h i s w a y we g e t a c h a i n

I =I O1 a.

.

I 31 n I a . a. "1

=...

8 5 RADICAL THEORY

146

w h i c h d o e s n o t t e r m i n a t e a n d we a r r i v e a t a c o n t r a d i c t i o n . Hence N i s i s o m o r p h i c t o a s u b d i r e c t p r o d u c t o f f i n i t e l y many v - p r i m i t i v e n e a r - r i n g s . The r e s t f o l l o w s by 2 . 3 5 . f o l l o w s from 5 . 2 9 .

<-:

5 . 3 1 T H E O R E M ( 8 e t s c h ( 3 ) , B l a c k e t t (2)). Let N = N o a n d v ~ { 1 , 2 1 . Then N i s v - s e m i s i m p l e w i t h D C C L <-> N i s a f i n i t e d i s t r i b u t i v e s u m o f i d e a l s which a r e v - p r i m i t i v e s i m p l e n e a r - r i n g s w i t h a r i g h t i d e n t i t y and D C C L . P r o o f . ->:

by 5 . 3 0 , 4 . 4 6 ( e ) a n d ( d ) a n d 2 . 5 2 ( b ) .

<==: By 5 . 2 9 , N i s u - s e m i s i m p l e .

Using 2 . 3 5 ( b ) r e p e a t e d l y ( b y i n d u c t i o n ) one s e e s t h a t N has t h e DCCL.

5 . 3 1 h a s many i n t e r e s t i n g c o r o l l a r i e s ( w h i c h m o s t l y s t e m f r o m Betsch ( 3 ) ) , which r e p r e s e n t t h e " n o n - l i n e a r " v e r s i o n o f t h e c e l e b r a t e d Wedderburn-Artin-structure theorem. 5.32 THEOREM I f then

N = No

i s v-semisimple

(vEC1,21)

with DCCL

( a ) N has a r i g h t i d e n t i t y ( s o 3 . 4 3 i s a p p l i c a b l e ) . ( b ) N i s c o m p l e t e l y r e d u c i b l e , so a l l o f 2 . 5 0 i s a t hand (for N). ;L );(L2,; 3L ); :(Lkl; ;L ), ( c ) N = (Lll; lnl 2n2 knk where f o r i c C 1 , k l , L i l ,...,L i n i a r e pairwise

...

...

...

...

...,

Ni:

=

"i l'Lij-isomorphic

l e f t i d e a l s o f N i and Ni-groups

j=1 o f t y p e v.

a r e a l s o simple l e f t i d e a l s o f v . Each N - g r o u p o f t y p e v i s N - i s o m o r p h i c t o o n e o f them; 2 . 5 0 c a n be a p p l i e d f o r All

Lij

N and N - g r o u p s o f t y p e

"

( d ) N h a s o n l y f i n i t e l y many c l a s s e s o f n o n - ! I - i s o m o r p h i c M-groiips o f t y p e v . ( e ) Every i d e a l o f N i s a g a i n w-semisimple.

147

5a Jacobson-type radicals: common theory

_P- r o o f .

( a ) and ( b ) f o l l o w d i r e c t l y f r o m 5.31.

( c ) : by 5.31,

4.46(b), 2.49,

3 . 4 1 ( b ) and

2.48(e),

3.11 ( a ) . ( d ) : by ( c ) . ( e ) : by 5 . 3 1 a n d 2 . 5 5 ( a ) . See a l s o Deskins

(Z),

following result cf.

__ 5.33

Kaarli

(2),(4) a n d Chao ( 1 ) . F o r t h e

5 . 1 8 a n d E x . 6 . 4 0 i n Meldrum ( 1 3 ) .

THEOREM ( K a a r l i ( 4 ) ) . Let and ( i f N has t h e DCCN)

Pt.oof.

N€nO a n d

I4N.

~ O ( I ) 2 ~ O ( N I) . n

We o n l y show t h e s t a t e m e n t f o r

--

(N

"9.1

1

i s I-semisimple with DCCL.

simple by 5.32(e);

by 2 . 8 .

21(1)+(21(N)A 5

i f N has t h e DCCL

Since

t h e same a p p l i e s t o

and 5.16

5.15(c)

Then + J , ( I ) G ~ ~ ( N ) ~ I

')'ll(N)

I/jl(rr) n I

t e l l us t h a t

1 = (01,

5.34 THEOREM ( B e t s c h ( 3 ) ) .

v ~ { 1 , 2 1 and

If

and i s v-semisimple and i f

Nr

5

2*.(N)n1.

!,r:

Some d e c o m p o s i t i o n s o f N i n d u c e d e c o m p o s i t i o n s o f

(a)

'a1C')

whence

k

1 ' ~w i t~h

=

ai

i = l

( b ) Each

(o:y)

Nr

+r

i s either

N =

No

h a s DCCL

i s monogenic t h e n

N ~ i o f t y p e v.

and

= N

or a f i n i t e intersection

o f v-modular l e f t i d e a l s .

(c) If Proof.

,.,r

i s o f t y p e 0 t h e n i t i s o f t y p e v.

( a ) By 5 . 3 2 ( c ) ,

s.

1 Li

N =

where each

Li

9%N

i = l

i s an N-group o f type v. L e t

r

S

= (

= Ny O

1 Li)yo

i = l

r

be

= Ny,.

Then

S

=

1

(Liyo).

i=l

Each

Liyo

q,r

8 5 RADICAL THEORY

148

by 3.4(a).

$. ( 0 1

Liyo

If

then

%

Li

=iN

Liyo

and

Liyo

i s

o f t y p e v , h e n c e s i m p l e . A s i n ( b ) -> ( c ) o f 2 . 4 8 one can choose some s u b s e t S o f {l, s }

...,

r

with

(b) If

~ ' ( L ~ Y ~ ) . iE S

=

(ycr)

nE(o:y)

k

1 6i

y =

and

then

i= 1 k

k

1 6i)

o = ny = n(

,I n 6 i

=

i=l of t h e

Ails

b y 2.30.

S i n c e t h e sum

1=1

V

i s direct,

i E { l ,

...,k l :

n6i

= 0.

k

Hence

nc

fl

(0:6~).

i = l If 6iEf30(Ai) If fiicBl(Ai)

then then

(o:ai) Ai

aN c\,

=

N.

N/(o:6i),

hence

(0:6~)

i s v-modular. ( c ) Use 5 . 3 4 ( a ) (SO

S e e C h o u d h a r i (1) ( n o . a,(N)

and t h e f a c t t h a t

Nr

i s simple

k = 1). 3.34)

f o r more c h a r a c t e r i z a t i o n s o f

(especially via quasi-regularity).

If

Nc'TI0n ??, i s

2 - s e m i s i m p l e w i t h DCCL a n d i f simple ideals

-

5.31)

N = @ I k ( I k f i n i t e l y many t h e n t h e c e n t e r o f (N,.) i s isomorphic

t o t h e d i r e c t p r o d u c t o f t h e c e n t e r s of

I n t h i s case, i s a group,

Ik =

M o(r)

(4.60), Go isomorphic to t h e c e n t e r of if

(as semigroups). (Ik,') the center o f (Ik,.)

Go

(see Holcombe ( 2 ) ) .

149

5b Jacobson-type radicals: special theory

b ) JACOBSON-TYPE RADICALS: S P E C I A L T H E O R Y

5.35 T H E O R E M (Ramakotaiah ( 1 ) ) . 'Jo(N)

=

5.36 THEOREM (Ramakotaiah ( 1 ) ) . a o ( N ) o f N contained in 21/2(N).

N).

(Jli2(N):

i s the greatest ideal

P r o o f . Let I be a n o t h e r i d e a l o f N w h i c h i s i n 2 1 / 2 ( N ) . I f $, i s t h e s e t o f a l l 0 - m o d u l a r l e f t i d e a l s o f N t h e n tf L E ~ : I S L . Hence b L&: 1 5 ( L i : N ) , so I 5 n ( L : N ) = >,(N). LE

5

Observe t h a t i n t h e f i n i t e c a s e t h i s a l s o f o l l o w s from 3 . 2 7 ( b ) and 3 . 2 5 .

5.37 T H E O R E M ( R a m a k o t a i a h ( l ) , c f . K a a r l i ( 4 ) , Chao ( I ) ) . L e t N

=

(a) tdli2(N)

i s the g r e a t e s t q u a s i r e g u l a r l e f t i d e a l of N.

( b ) lll2(N)

cdntains a l l nil l e f t ideals.

(c)

20(N)

i s t h e g r e a t e s t q u a s i r e g u l a r i d e a l o f N. contains a l l nil ideals.

(d) ao(N)

P r o o f . ( a ) l ) We show t h a t ) 1 / 2 ( N ) i s quasiregular. Let z b e E > ~ / ~ ( N ) . I f t+L, ( 3 . 3 5 ) then Z o r n ' s lemma ( 3 . 2 2 ! ) g u a r a n t e e s t h e e x i s t e n c e o f a ( b y z) m o d u l a r l e f t i d e a l L , maximal f o r h a v i n g z&L*

i s another l e f t ideal containing L t h e n since b nEN: n - n z E L ' , L' = N. Hence L i s a maximal l e f t i d e a l , so > 1 / 2 ( N ) ~ L a n d we arrive a t a contradiction. Therefore )l,2(N) i s quasiregular. If

L'

ZEL';

No.

85

150

RADICAL THEORY

2) Now l e t Q b e a q u a s i r e q u l a r l e f t ideal. If Cdl12(N) = N then clearly Qe)1/2(N). SO assume

$. N.

that

L e t L be a m o d u l a r (by e , s a y )

maximal left ideal. Q')L then QtL = N. S o 3 q E Q 3 LEL: e = q+L. neN: ne-nq = n(q+L)-nqEL. Hence b neN: n-nq = n-netne-nqeL+L = L. T h i s s h o w s that L (p N) is m o d u l a r via q. By 3.38(c), q c a n n o t be q u a s i r e g u l a r , a contradiction. So QeL.

If

Hence

Q e

n

L 0-mod. in N

31/2~~).

L =

(b): by (a) and 3.38(b). ( c ) : by (a) and 5.36.

(d): by (c) and 3.38(b). Another intersection o f big things i s contained in 2 * ( 1 1 ) : N = N o , the intersection M o f all maximal N-subgroups is q u a s i r e g u l a r and contained i n

5.38 T H E O R E M If

g2(N).

Proof. Take Z E M . Assume that tBL,. Then t h e r e i s s o m e T4 < N N containing L, which i s maximal w.r.t. n o t c o n t a i n i n a z. A s i n part 1 ) o f t h e p r o o f o f 5.37(a) o n e s h o w s that is a maximal N-subgroup, s o ,EM, a contradiction. T h e r e s t will be obvious from 5.44. 5.39 THEOREM (Betsch (3)). L e t (a) >l,2(N)

= ( 0 1 <=>

N =

NE~)O have the DCCL. Then S

l * L i w h e r e the L i ' s a r e m o d u l a r

i=l

l e f t i d e a l s a n d I \ i - g r o u p s o f t s o e 0.

(b) In this c a s e , N c o n t a i n s a r i g h t identity a n d 2.50 is applicable f o r ".

151

5b Jacobson-type radicals: special theory

Proof.

( a ) ->:

&/2(N)

If

t h e 0-modular

= tUI,

the intersection o f

to}.

l e f t ideals =

As i n t h e p r o o f

o f 5 . 3 0 i t s u f f i c e s t o t a k e f i n i t e l y many o f them, s a y K1, Kk, m i n i m a l f o r h a v i n g i n t e r s e c t i o n = {O}. k Apply 2 . 5 0 ( 9 ) t o see t h a t N = l'Li, where i= 1

...,

Li= j$i <-:

If

K.

J

?,

N

N =

a r e N - g r o u p s o f t y p e 0.

N/

=

K~

k l'Li i = l

I Kl,...,Kkl

as i n d i c a t e d ,

(as

k

above) a r e 0-modular

n

l e f t ideals with

IOI.

Ki

i = l Hence

Cdl12(N)

COI.

=

(b): by 3.27(d)

and ( a ) .

5 . 4 0 THEOREM ( R a m a k o t a i a h ( I ) ) , L e t

NsV0

ha:e

t h e DCCN.

Then (a) (b)

2 1 / 2 ( ~ ~i s) n i l p o t e n t . N i s 0 - s e m i s i m p l e <=> N h a s n o n o n - z e r o q u a s i r e g u l a r i d e a l <=> N h a s n o n o n - z e r o n i l i d e a l <==> N h a s n o non-zero n i l p o t e n t i d e a l .

( c ) Each p r i m e i d e a l Proof.

$. N

i s 0-primitive.

( a ) and ( b ) a r e i m m e d i a t e consequences o f 5.37

a n d 3.40.

(c): Let

P

(1

N

By 2 . 1 0 4 ,

be prime.

?T: = N / P

has

n o n o n - z e r o n i l p o t e n t i d e a l s a n d i s t h e r e f o r e 0semisimple by (b).

has DCCN ( 2 . 3 5 ) with (since

ni =k lFi ID1

=

Since

N

+ P,

and 0 - p r i m i t i v e

CDl

N

+ ID},

ideals

( 5 . 3 1 ) . Hence P , B 2 . . . T k

i s p r i m e ) some

Pi =

lU1.

So

N

V1,...,Vk =

So

{GI,

{Ul

so

i s

0 - p r i m i t i v e and t h e r e s u l t f o l l o w s .

I n ( 4 ) , K a a r l i g e n e r a l i z e d ( a ) t o a w i d e r c l a s s of near-rings.

85

152

5 . 4 1 REMARKS

If

NE??~,

RADICAL THEORY

contains o f course a l l n i l ,

21(N)

n i l p o t e n t a n d a l l q u a s i r e q u l a r l e f t . i d e a l s o f N (5.37), b u t n o t n e c e s s a r i l y a l l n i l N-subgroups o f N ( b u t *d2(N) has t h i s p r o p e r t y - see 5 . 4 5 ) : I n N = ifsMo(Z,)lf(2) = 01 we h a v e (E0,21:Z4) S N N &(N)

IO).

gl(N)

nilpotent,

but

i n general

( s e e 5 . 1 1 ( b ) ) , On t h e o t h e r hand,

=

Also,

i s not n i l

>l(N)

c o n t a i n s one more i t e m :

PI = N 0 ' DCCL. L e t M ( N ) d e n o t e t h e i n t e r s e c t i o n o f a l l m a x i m a l i d e a l s o f N ( c f . M l i t z ( l ) , ( 2 ) ) . Then

5 . 4 2 THEOREM

(a) al(N)

WW.

( b ) If N € v 1 Proof.

ql(N)

h a s t h e DCCN t h e n

(a) L e t

P.4 N

=

g2(N) =

M(N).

b e a 1 - p r i m i t i v e i d e a l . Then

i s 1 - p r i m i t i v e w i t h DCCL,

N/P

h e n c e s i m p l e by 4 . 4 6 1 e ) .

Hence P i s maximal. (b):

by 4 . 4 7 ( b ) .

I n K a a r l i ( 4 ) t h e r e i s an example 1 9 N

I1(N)I # A

{a}.

=

N0 w i t h

7, : I )

=

{O}, b u t

See a l s o E x a m p l e N 1 ) i n t h e A p D e n d i x ( n r . ' s o f

low o r d e r ) .

3*) 2 2

5.43

PROPOSITION I f

NE??~

has a minimal No-subqroup t h e n N i s

not 2-radical. Proof.

Let

M SN N

i s o f t y p e 2.

be a minimal No-subqroup. If

N M = I01

then

NEVI, a c o n t r a d j c t i o n . Hence q2(Y)=N.

Now we look w h o i s c o n t a i q e d

in

g2(N).

M =

{O}

Then

NM

since

153

5b Jacobson-type radicals: special theory

5 . 4 4 THEOREM ( R a m a k o t a i a h ( 1 ) ) .

If

N = No,

rd2(N)

contains

a l l q u a s i r e g u l a r N-subgroups. Proof.

We p r o c e e d s i m i l a r t o 5 . 3 7 ( a ) .

r d 2 ( N ) =/- N .

Q

Let

SN

N

We may assume t h a t

be q u a s i r e g u l a r .

Q$22(N)

If

t h e n t h e r e i s some 2 - m o d u l a r l e f t i d e a l L w i t h L e t e be a r i q h t i d e n t i t y modulo L. that

L + Q SN N

group (3.29),

If

nEN

and

then

L=L+Q.

L t Q = N.

so

b y q . By 3 . 3 8 ( c ) ,

we s e e

L i s a maximal N-subLet = L

EL, q E 0 ) .

e = :E+q

ne-nq = n(L+q)-nqEL.

nEN: n - n q = n - n e t n e - n q E L + L

By 2.15,

Q+L.

Hence

and L i s m o d u l a r

q i s n o t q u a s i r e g u l a r a n d we a r r i v e

a t a contradiction. Cf.

Chao ( 1 ) .

From 3 . 3 8 we d e d u c e

5 . 4 5 COROLLARI I f N i s z e r o - s y m m e t r i c

then

??(N)

contains

a1 1 n i 1 ( p o t e n t ) N - s u b g r o u p s . R e c a l l f r o m 5.11(b)

that

2*(N)

i s not necessarily n i l (not

even f o r f i n i t e zero-symmetric n e a r - r i n g s w i t h i d e n t i t y ( n o t even u n d e r t h e a s s u m p t i o n o f d i s t r i b u t i v e g e n e r a t i o n L a x t o n (4))).

5.48 w i l l c h a r a c t e r i z e t h e case t h a t

-

see

a2(N)

is

n i l p o t e n t . See a l s o B e i d l e m a n ( 2 ) . i s the interse'ction of a l l 2-primitive l e f t ideals, J2(N) while

rdl12(N)

i s

g2(N)

2 - m o d u l a r 1e f t i d e a 1 s

i n t e r s e c t e d w i t h a l l 0-,

. g2( N)/21/2(t,)

but not

swallows up a l l 0-modular

ones ( L a x t o n ( 6 ) ) : 5.46

PROPOSITION Let

N = No

h a v e t h e DCCL. Then

i s z e r o o r a f i n i t e d i r e c t sum o f M - g r o u p s o f w h i c h a r e n o t o f t y p e 2.

'a2(N)/ 2 1 / 2 (N) t y p e 0.

85

7 54

Proof.

IT:

RADICAL THEORY

k i s a f i n i t e d i r e c t sum

= N/

rdZ(N)

-

k

N-groups o f t y p e 2 by 5 . 3 1 .

I*'Ci

i=l

of

m a.1 ria...-rk a { O }

1=2 i s a p r i n c i p a l s e q u e n c e . Assume t h a t ' J 2 ( N ) is c o n t a i n e d i n some 0 - , b u t n o t 2-modular l e f t i d e a l L . = C O ) c a n be r e f i n e d o f N . Then

N / 9 2 ( ~ )L / a 2 ( ~ ~ ) a

t o a n o t h e r p r i n c i p a l s e q u e n c e ( 2 . 4 0 ) and t h e f i r s t ZN N / ~ i s s i m p l e , s o factor

"22(Nj/L/22(N)

ri.

R - i s o m o r p h i c t o some B u t Ti i s o f t y p e 2 and i s n o t , s o we a r r i v e a t a c o n t r a d i c t i o n . N/*d2 ( (N )

N f i /2 *

Hence f o r each 0 - , b u t n o t 2 - m o d u l a r l e f t i d e a l L we g e t L t > 2 ( N ) = N and c o n s e q u e n t l y

N I L N'

L + ~ ~ ( P I ) / p'iL

This snows t h a t

(rj

Lnq2(N)

32(N).I n

Of

''2 1 / 2

>~(N)/L,,~~(N).

1

's

( = ]1,2(N))

i s a 0-modular l e f t 22(N)/31,,(N)* a11 these = :L'

a r e 0-modular l e f t i d e a l s w i t h " t r i v i a l " i n t e r s e c t i o n , s o by 2.50

a

Q2(N)/

1/2 ( N )

i s a f i n i t e d i r e c t sum o f N-groups o f t y p e 0 , b u t

not type 2 . From t h i s we deduce

5.47 T H E O R E M ( L a x t o n ( 6 ) ) . L e t b e a l e f t i d e a l . Then

L5I2(N) <->

3

NE?&

have t h e D C C N a n d l e t L

K dQ N : KFL, K n i l p o t e n t a n d

L/K

i s zero

o r a f i n i t e d i r e c t sum o f N-groups o f t y p e 0 , b u t n o t o f type 2 .

5b Jacobson-type radicals: special theory

P r o o f . =>:

Lfi21/2(N)

K

=:

% 22(N)/a,/2(N) .

155

i s n i l p o t e n t by 5.40(a).

By 5.46 a n d 2.55,

L/K

is a finite

d i r e c t sum o f N - q r o u p s o f t y p e 0 , b u t n o t o f t y p e 2 . <-:

L+22(N)/92(N) 1',

L/a2(")" 2,

Hence

L'$)2

(N 1n L)/K

L

L'22(N)"

a

BY 5.45,

L'

KSf2(N)

L/K

A

L.

is a finite

d i r e c t s u m o f N-groups of t y p e 0, b u t n o t o f t y p e 2 , s o t h e same a p p l i e s t o L/22(t4)n by 2.55, a n d hence a l s o t o

L+22(N)/a2(N) AL N / > 2 ( N ) .

But

i s a d i r e c t s u m o f t i - g r o u p s o f t y p e 2 , so L+>*(N)

=

Q2(N)

and

L5I2(N).

22(N)

From t h i s r e s u l t one can c o n s t r u c t a d e c o m p o s i t i o n of i n t o a n i l p o i e n t a n d a " t o t a l l y n i l p o t e n t " p a r t (Laxton ( 6 ) ) i n s p e c i a l c a s e s . F o r more q e n e r a l c a s e s , s e e L a x t o n - M a c h i n ( 1 ) and S c o t t ( 1 ) . N o w we c h a r a c t e r i z e t h e c a s e t h a t

72(N)

i s nilpotent.

5.48 T H E O R E M ( R a m a k o t a i a h (1)). L e t N E ~ ) , h a v e t h e D C C r l . Then the following conditions a r e equivalent: (a) 'J2(~~) is nil. (b) Ed2(N)

is nilpotent.

( c ) ' J 2 ( ~ils) q u a s i r e g u l a r . ( d ) >2(N)

= 21(N) = & / 2 (

N ) = ?,(N).

( e ) Each N-group o f t y p e 0 i s o f t y p e 2 . (f)

I 4 N : I i s 0 - p r i m i t i v e <==> I i s I - p r i m i t i v e <-> 2 - p r i m i t i ve.

( 9 ) Each 0 - m o d u l a r l e f t i d e a l o f N i s 2 - m o d u l a r . (h)

{O}

I Q N: I i s p r i m e ->

I i s 2-primitive.

I is

156

$ 5 RADICAL THEORY

Proof.

( b ) <->

( a ) <=>

( c ) h o l d s b y 3.40.

( c ) <->

(d): by 5 . 3 7 ( ~ ) .

( d ) ->

(e):

If

Nr

i s o f type 0 then

i s o f type 0 by 3.14(a) Since

N/J2(N)

i s 2-semisimple

( e ) =>

(f):

i s immediate.

( f ) ->

(d):

trivial.

( f ) <->

/g2

(9): by 4.3.

( f ) <==> ( h ) : b y 5 . 4 0 ( c ) also 5.61(b),

'J2(N) = 2 0 ( N ) e ( O : r ) ) . (5.16), N (X 1

Nr i s o f t y p e 2 b y 3 . 1 4 ( b ) .

so

i s o f t y p e 2 b y 5.34,

Cf.

(since

"/I2 ( N ) lr

and 4.34.

Chao ( l ) , K a a r l i ( 4 ) , S c o t t ( 8 ) , ( 1 2 ) .

F i n a l l y , we a r e g o i n g t o d e s c r i b e 2 - s e m i s i m p l e n e a r - r i n g s w i t h c h a i n c o n d i t i o n more c l o s e l y . 5.49 THEOREM ( C h o u d h a r i - T e w a r i

( 1 ) , Oswald ( 3 ) ) .

Let

NEn,

h a v e a r i g h t i d e n t i t y . Then t h e f o l l o w i n g s t a t e m e n t s are equivalent: ( a ) N i s 2 - s e m i s i m p l e w i t h DCCL. ( b ) N i s a d i r e c t sum o f f i n i t e l y many N - s i m p l e l e f t i d e a l s . A sN r 3 E AN r : r = A + E A ( c ) I f Nr i s m o n o g e n i c t h e n A AnE =

M

(d)

SN

N

101.

3

L d, N : N = M+L A M n L = 1 0 1 .

( e ) Each e x a c t s e q u e n c e

101

-L

M1

+

M2

(M1,M2

sN

N) s p l i t s

( d e f i n i t i o n s as u s u a l ) .

(f)

M1sM2 SN N,

3

K€HomN(M2,r):

MlcM2

W Nr

EN?

hEHomN(Ml,r)

F/Ml = h .

( 9 ) N s a t i s f i e s t h e DCCN and has no

non-zero n i l p o t e n t

B l a c k e t t ( 2 ) . where n e a r - r i n g s w i t h t h i s c o n d i t i o n are c a l l e d "semisimple").

N-subgroup ( c f .

157

5b Jacobson-type radicals: special theory Proof.

( a ) ->

( a ) : By 2 . 5 0 ,

( b ) => Li

by 5.32.

(b):

k If

N h a s t h e DCCL.

N-simple l e f t i d e a l s ,

then

N =

Ki:

n

2 - m o d u l a r l e f t i d e a l s and ( a s i n 2.50(9)) hence

22(N)

=

( b ) ->

(c):

If

= COI,

Ki

i = l

101.

N

k =

l'Li i = l

then by (a)

as a b o v e ,

...,k I

there i s a subset S o f

and 5 . 3 4 ( a )

l*Li, i = l

{l,

with

t y p e 2.

r,

If

A sN

$:=

{E AN T I A n E = { o ) ) . But

$Atr.

A n(Liyotr)

F.

maximality o f ( c ) =>

(d)

{ol

=

Hence

{O)

L gL 14:

f(M1)+L

+

M1

f +

i n

LiyO'$

Liyon(AtT)

= Io).

a n d ( c ) i s shown.

NN

i s monogenic (by e ) . be exact.

M2

= N A f(M1)nL

the "projection"

iES:

which c o n t r a d i c t s the

A t ' t T = I'

i s t r i v i a l since

( d ) ==> ( e ) : L e t

3

A+F $; r, 3

If

i s o f t y p e 2, h e n c e

Liyo

Therefore

'E: ( Z o r n ! )

t a k e some m a x i m a l e l e m e n t

= (01.

By ( d )

Then

f(M1) defined by i s an N-homomorphism. p(n) = p(f(ml)te):= f(ml) p/ =:p. Then f'lop: M 2 -+ M1 i s an N-homomorphism

p: N

+

M2

with

= idM ,

f-lopof

Hence

1 ( e ) =>

(f):

and l e t

{Ol

MI

+

Let

M1

1: 1 4

( f ) => map h: M2

(e):

id +

-+

f(M1).

N-homomorphism

M2

splits.

b e as i n t h e s t a t e m e n t ,

+

M1).

Then Then

f M1 + M 2 b e e x a c t . The i d e n t i t y c a n b e e x t e n d e d t o a n N-homomorphism

Let

f(M1)

-+

does t h e r e q u i r e d j o b .

T

+

Ml

b e t h e i n j e c t i o n map.

M2

s p l i t s ( s a y b y g: M 2

M2

5 : = hog: M2

M1,M2,r,h +

{O)

{Ol

+

Then c l e a r l y M2

+

MI.

f-loh

i s a splitting

9 5 RADICAL THEORY

158

( e ) ->

( 9 ) : F i r s t we show t h a t N h a s t h e D C C N . be a c n a i n o f N - s u b g r o u p s o f N . N = f40d~1=42=,..

Let

tl

L

i s m : 10)

+

M~

iM

( L ~t h e

~

-

~

i n j e c t i o n maps)

spl i t s .

Mi be c o r r e s p o n d i n g s p l i t t i n g Nhomomorphisms. Then hl: = g 1 i s a s p l i t t i n g N MI + N, h2: = g20g1 one f o r homomorphism f o r C O I ( 0 ) * M e + N, e c e t e a. I f Li: = K e r h t t h e n Li A, N a n d ( a s e a s i l y s e e n ) Furthermore, N = MitLi with Mi" L = Io).

Let

gi:

Mi-l

+

+

... .

L1"L2" But

i s comp e t e l y r e d u c i b l e ( 2 . 4 8 ( e ) ) ,

NN

eEN),

generable (since (2.50(e))

which causes

f i n i t e l y many s t e p s .

finitely

s o endowed w i t h t h e ACCL

Ll=L2=

...

to stop after

Hence t h e same a p p l i e s t o

.. .

Mo=M1=.

Now we show t h a t N h a s no n o n - z e r o n i l p o t e n t Ns u b g r o u p s . L e t M be such one. As b e f o r e , L n M = {Ol. L 4, N : N = L t M ,

3

Let

be = I l o t m o . ( & , E L ,

e

moe!1). A s i n 3.43,

mo

t u r n s o u t t o be a r i g h t i d e n t i t y f o r M, hence M c a n n o t be n i l p o t e n t and t h e p r o o f i s a c c o m p l i s h e d .

(9) -> ( b ) : N-subgroup.

N = Noeot(O:eo)

By 1.13,

Mon(O:eo) If

Mo = N

Hence Either

( B l a c k e t t ( 2 ) ) . L e t Mo b e a m i n i m a l 3 eo = eocN: 2 Neo = Moeo = Mo.

By 3 . 5 2 ,

= Mot(o:eo)

with

= (01.

t h e r e i s n o t h i n g t o p r o v e . So l e t

N=(o:eo) (o:eo)

Mo

9 N.

$. COI. i s a minimal N-subaroup o r i t c o n t a i n s

( b y a p p l y i n g t h e above c o n s i d e r a t i o n s t o (o:eo) i n s t e a d o f N) a n o t h e r s m a l l e r N - s u b g r o u p o f t h e form

(o:e,)nL

w h e r e L i s some l e f t i d e a l o f

N.

The D C C N a s s u r e s t h a t a f t e r f i n i t e l y many s t e p s we a r r i v e a t a m i n i m a l N - g r o u p intersection o f

(o:eo)

M1

which i s the

w i t h a l e f t i d e a l o f N,

h e n c e a n o r m a l s u b g r o u p . T a k e some i d e m p o t e n t

elcN

5b Jacobson-type radicals: special theory

with

Nel

groups).

= Mlel

Hence

M1.

159

(@:eo)=M,+(o:el)

Repeating t h i s procedure w i t h

(as

(o:el)

...

( i f necesszry) y i e l d s N = Mo: ;Mk w h e r e Mi a r e m i n i m a l (hence N - s i m p l e ) N-subgroups o f N.

Now by 5 , 4 0

21,2(N)

so N i s t h e d i r e c t s u m

IOI,

=

S

I*Li i = l

o f l e f t i d e a l s of type 0 (5.39).

o f above a r e N-groups o f t y p e 2,

The

Mi's

hence N-isomorphic

t o some L . ' s . J T h e r e f o r e N i s t h e f i n i t e d i r e c t sum o f l e f t i d e a l s w h i c h a r e N-groups of 5.50

t y p e 2.

REMARKS

(1) h a v e shown o n e c a n a d d t o t h e f o l l o w i n g c o n d i t i o n i f NE?:l

(a) As Choudhari-Tewari 5.49

( i )E a c h N - s u b g r o u p o f N i s m o n o g e n i c , p r o j e c t i v e ( d e f i n i t i o n a g a i n as u s u a l ) and g e n e r a t e d b y a n idempo t e n t

.

See t h e r e f o r t h e p r o o f (b) If N = N o

( c ) ->

i s 2-semisimple

(i) =>

from 4.46(d),

( 3 ) and S c o t t ( 1 ) .

.

Cf.

Chao ( 1 ) .

t h e n t h e DCCL o r DCCPI i m p l y

a l l o t h e r c h a i n c o n d i t i o n s o f ACCL, This follows

(e)

5.49(d)

ACCN,

DCCL,

and ( 9 ) . C f .

DCCN. Oswald

I t i s n o t known t o t h e a u t h o r i f a

2 - s e m i s i m p l e n e a r - r i n g w i t h ACCL o r A C C N h a s a l l o t h e r c h a i n c o n d i t i o n s as w e l l . ( c ) A s Mason ( 3 ) p o i n t e d o u t , i n j e c t i v e N=No-group.

( 3 ) and ( 4 1 ,

there e x i s t s no n o n - t r i v i a l

See m o r e o n t h a t i n h i s p a p e r s

i n Prehn ( I ) ,

Maxson ( 8 ) and Oswald ( 1 0 ) .

Banaschewski-Nelson I n particular,

(I),

see 9.264.

160

$ 5 RADICAL THEORY c ) THE N I L RADICAL

5 . 5 1 D E F I N I T I O N T h e sum o f a l l n i l i d e a l s o f N i s c a l l e d t h e

n i l r a d i c a l o f N and d e n o t e d by

V(N)

R a m a k o t a i a h (1) and P o l i n ( 2 ) ) .

Cf.

in

'J1(N)

(by

Gojan ( 1 ) .

5 . 5 2 THEOREM (a)

V(N)

i s t h e g r e a t e s t n i l i d e a l o f N. i s t h e s m a l l e s t i d e a l I o f N such t h a t

(b)?l(N)

N/I

has no n o n - z e r o n i l i d e a l s . Proof.

by 2 . 1 0 1 ( b ) .

(a):

n: N

Let

(b):

+

N/n(N)

N

=:

be the canonical

projection. If

If

T

4

IT, T

I : = n-l(f) 1 N. kEIN: n ( i ) = v ( i ) k = U ( z e r o i n

nil,

look a t

k

then 3 k i EKer u = "I(N). But i s n i l , so % S I N : (ikIE = 0 and i i s n i l p o t e n t . Hence I i s iE1

n(N)

hence

3

nil, therefore

It'?I(N)

Now ascurne t h a t By 2 . 1 0 3 ,

I+fl(N)/,

a n d we a r r i v e a t 5.53

COROLLARIES

N/I

a n d we g e t

T

=

{Ul.

i s w i t h o u t non-zero n i l i d e a l s . i s nil in

N/I,

so

I+?1(N)EI

T(N)SI.

(Rarnakotaiah ( 3 1 , Meldrurn ( 7 ) )

( a ) W i s a r a d i c a l map ( i n t h e s e n s e o f 5 . 1 3 ) . ( b ) V(N) 5 TI",) 5 ' d o ( N o ) t Cdo(N). ( c ) N i s n - s e m i s i r n p l e i f f N has no non-zero n i l i d e a l s .

(d) Each c o n s t a n t n e a r - r i n g i s 9 - s e m i s i m p l e . Proof.

(a):

by 2.100 and 5.52(b).

( b ) : b y 2.99, ( c ) : by

5.52(a),

55Z(b).

(d): by (b).

m),

5.37(d)

and 5.25.

161

5d The prime radical

It i s clear that f o r rings

coincides with the usual n i l

%(N)

radical o f rings.

? i sl s u b h e r e d i t a r y

5 . 5 4 THEOREM ( c f .

Maxson ( 1 ) ) . I f

then %(I) c

o n d i r e c t summands:

I 4 N

i s a d i r e c t summand

V(N) A I .

The p r o o f f o l l o w s f r o m 2.12. 5.55

REMARK?l(N)

i s a l s o i d e n t i c a l w i t h t h e "upper n i l r a d i c a l "

U = s(0)

o f Van d e r W a l t ( 1 ) . See t h i s p a p e r f o r a

V(N)

characterization o f

v i a "s-systems".

See a l s o

Beidleman ( 9 ) .

d ) T H E P R I M E RADICAL

5.56 DEFINITION The i n t e r s e c t i o n o f a l l p r i m e i d e a l s o f N i s

P(N)

c a l l e d t h e prime r a d i c a l o f N and denoted b y (other notations: >-*(N), Again,

L,(N),

m(0)).

Cf.

Gojan

(1).

t h i s i s j u s t t h e usual prime r a d i c a l i n the case of

rings. 5 . 5 7 PROPOSITION Proof.

pis

a r a d i c a l map.

( a ) P ( N ) 4 N.

(b) I f P:

h:N 3

showing t h a t

P

and

P 4 ??

i s prime then

i s p r i m e i n N b y 2.64

= h-l($)

(c) If

TT

h(P(N))

and 2.17(a),

cV(T).

i s a prime i d e a l o f

'IT: = N/aO(Nl

i ~ - l ( F )= : P i s p r i m e i n N. i f P 4 N i s prime then

i n (b),

Conversely, i n TI.

n(P)

then,

as

i s prime

H e n c e i f Ti i s i n e a c h p r i m e i d e a l o f TT t h e n e a c h xch-'({T)) and

X

i s i n each prime i d e a l o f

i s zero.

Therefore

?o(f4/aD(N))

i s zero.

N, s o

XERN)

1 62

6 5 RADICAL THEORY

The c o n n e c t i o n t o 2 . 9 3

i s q i v e n by and t h i s i s a semiprime i d e a l .

5.58 R E M A R K p ( N ) = / 8 ( { 0 } )

i s a n i l i d e a l and c o n t a i n s the sun

5.59 PROPOSITIONPP(N)

o f a l l nilpotent ideals. P r o o f : b y 2.105. F r o m t h i s we c a n l o c a t e P ( N ) : ~ ( N ) ~ T ? ( N ) ~ ~ o ( f i )I ! 2+ ()N, ) % j l ( N ) F ) 2 ( N )

5 . 6 0 COROLLARY

(and a l l i n c l u s i o n s can be s t r i c t ) . The f i r s t t w o i n c l u s i o n s c a n e v e n b e s t r i c t i n t h e c a s e o f r i n a s . 5 . 6 1 THEOREM

Let

(a)T(N)

...

-

Proof.

WN)

=

(b)Ea2(N)

N

E =

~h a v e t h e D C C N . T h e n

"d").

is nilpotent (cf. = &(N).

(a) f o l l o w s f r o m 4.34

(b):

"<="

"==>'I:

5.48)

<->P(N)='h(N)

=

and 5 . 4 0 ( c ) .

i s trivial.

f o l l o w s from ( a ) and 5.48(d).

5 . 6 2 THEOREM ( M a x s o n ( 1 ) ) . I f

'p(1) 5

IA

P(N)n

N

i s a d i r e c t summand t h e n

I.

T h i s r e s u l t f o l l o w s f r o m 2.63. 2.69 y i e l d s 5.63

EXAMPLE E a c h p r i m e ( e . g . semisimple.

More g e n e r a l l y :

each c o n s t a n t ) n e a r - r i n g i s

-

163

5e Concluding remarks

5.64 PROPOSITION N i s

iP-sernisimple i f f N i s i s o m o r p h i c t o a s u b d i r e c t p r o d u c t o f prime n e a r - r i n n s .

T h i s i s a d i r e c t c o n s e q u e n c e of 1 . 5 8 a n d 2 . 6 7 . 5 . 6 5 PROPOSITION Each p - s e m i s i m p l e n e a r - r i n g h a s no n o n - z e r o

n i l potent ideals. T h i s f o l l o w s f r o m 2 . 1 0 4 o r from 5 . 5 9 .

S e e more on t h a t i n S c o t t ( I ) , H o l c o n b e ( 2 ) a n d R a m a k o t a i a h - R a o ( 5 ) .

e ) C O N C L U D I N G REMARKS 5 . 6 6 SUMMARY Gle s u m m a r i z e some p r o p e r t i e s o f o u r r a d i c a l s (we i n c l u d e $1,2 a l t h o u g h i t i s n o t a r a d i c a l map).

Radical

‘8

R(N)q u a s i r e q u l a r ( N ) ? a l l quasi regul ar

’a2

’dl

21/2

I - I - 0

( N ) ?a 1 1 q u s s i r e g u 1 a r

i

%(N)

nil (N)”all n i l N-subgroups a(N)?all nil l e f t ideals (N)?all nil ideals (N)”all nilpotent ideals

-

-

(t)

-

(t)

(+)

t

t

+

t

+

t

( N ) i s the greatest

(N) i s t h e g r e a t e s t

1 - 1 - 1 -

8 5 RADICAL THEORY

164

II + II

me a n s

" y e s 'I

-

me a n s

'I

II

I1

no

"(+)"

stands f o r

NECI)~)"

I'

"yes,

if

( o t h e r w i s e un-

known t o t h e a u t h o r )

If

NE?,

has a minimal No-subgroup t h e n a l l r a d i c a l s a r e

NcWo

h a s t h e DCCN t h e n

=I= N* If

r2(N)

i s n i l iff a l l radicals

are equal.

5.67 SOME MORE REMARKS ( a ) S e e B e i d l e m a n (1),(3),(8),(11) between

rd2(N)

about t h e connection

and " ( s t r i c t l y )

small" l e f t ideals.

S i m i l a r c o n s i d e r a t i o n s can be f o u n d i n R i e d l ( l ) , C h a o and M l i t z ( l ) , ( 2 ) .

(1)

They a d o p t a l a t t i c e - t h e o r e t i c

p o i n t o f view ( t h e i n t e r s e c t i o n o f a l l maximal i d e a l s

(...)

= sum o f a l l " s m a l l " i d e a l s ( . . . ) ) . C f .

Oswald ( 5 ) .

( b ) R a m a k o t a i a h (1) s h o w e d t h a t e a c h b i r e g u l a r n e a r - r i n q

(3.49)

i s 0-semisimple.

( c ) The a u t h o r s u g g e s t s n o t t o u s e t h e n o t a t i o n s for

"I a n d

p,

respectively,

Jacobson-type

because these a r e not

radicals.

( d ) Ramakotaiah (1),(3) a l s o d e f i n e s a r a d i c a l "J-3(H)" c o n t a i n e d i n P(N), a s t h e i n t e r s e c t i o n o f a l l i d e a l s

I such t h a t

N/I

h a s no n i l p o t e n t i d e a l s .

f r o m T h e o r e m 8 o f Van d e r W a l t ( 1 ) t h a t

P(N)

a l l semiprime i d e a l s ( c f .

Maxson ( 1 ) ) and o f

( e ) See F r e i d m a n ( l ) , B h a n d a r i - S a x e n a

f o r a "Levi tzky-type"

2-3 = z-*.

( 1 ) f o r other charac(such as t h e i n t e r s e c t i o n o f

See a l s o t h e p a p e r b y C h o u d h a r i terizations o f

It follows

n(N).

( 2 ) and P l o t k i n ( 2 )

radical.

( f ) Ramakotaiah ( 4 ) a l s o d e f i n e d a r a d i c a l corresponding

t o t h e Brown-Mc-Coy

radical (?-radical)

t h e o r y as t h e i n t e r s e c t i o n modular i d e a l s .

Y(N)

in ring

o f a l l maximal

See a l s o C h o u d h a r i - T e w a r i

(3).

165

5e Concluding remarks

If ZEN is called "G-regular" i f the ideal generated {n-nz(nENl equals N then g(N) turns o u t t o be the intersection of all ideals I of N , such that N/I has no G-requl a r has no G-regular ideals. N/9(N) ideals and is a subdirect product of simple near-rinos with a right identity. by

(9) Laxton ( 3 ) defined o n e more "radical-like" ideal S ( N ) of N a s the intersection of all "s-primitive ideals". H e showed that lll2(N) F S ( N ) 5 a l ( N ) if NcV0

and gives a n example o f a dg. near-ring with 21/2(N) = S(N) = 21(N). See also Beidleman (7),(8),(9), Hartney ( 2 ) , ( 4 ) and Meldrum ( 5 ) , ( 1 3 ) . (h) Another radical w a s defined by Deskins (1) (see also Williams ( 1 ) ) . If N = N o has the DCCN then semisimplicity w.r.t. this radical is equivalent t o 2-semisimplicity, and this in turn to semisimplicity in the s e n s e o f Blackett ( l ) , (2) (see 5 . 4 9 ) . (i) Beidleman considered in (2) the "radical subqroup" RS(N) as t h e intersection of all maximal N-subqroups in near-rings By 5.38 w e know that in this case R,(N) 5 Beidleman proved e.g. that RS(N) = <-> 'J2(N) i s quasiregular (in his sense - s e e 3.37(c)). C f . 5.48(d).

€no.

12(N)

g2(~).

Q(N) = n L , where L ranges o v e r all maximal left ideals, w a s also considered (by various authors). If N has a right identity then (3.29) This and m o r e radicals c a n Q(N) = a l I 2 ( N ) be found in Choudhari 1).

(j) T h e "quasi-radical"

(k) Gorton (1) called an N group Nr t o be of c l a s s A ( A a non-zero cardinal number) if A E r . lAllA WfcM(r) 3 ncN V yEr: f(y) = ny. N i s called A-complete if N has a faithful N-group of class A . A radical C,(N) is defined as the intersection of all ( o : r ) , w h e r e r is a n N-group o f class a .

55

166

RADICAL THEORY ?*

He showed t h a t i f N i s A - c o m p l e t e on r t h e n N c a N r , and t h a t N i s 1-complete i f f N c is f a i t h f u l (a base 1.91). If N d 0 then E C1(N). of equality D e f i n i n g CX-modular l e f t i d e a l s a s t h o s e modular l e f t i s a n N-group o f c l a s s X i d e a l s L such t h a t N / L ( c f . 3.28) y i e l d s a r e s u l t s i m i l a r t o 5.2. A l s o , he g a v e s e v e r a l e x a m p l e s .

g2(N)

-

( 1 ) Maxson ( 1 ) p r o v e d t h a t t h e r e i s n o t such a f i n e c o n n e c t i o n between i n j e c t i v i t y o f N-groups ( d e f i n e d a s u s u a l ) and s e m i s i m p l i c i t y a s i n t h e r i n g - c a s e . He showed t h a t i f e a c h N-group i s i n j e c t i v e t h e n b u t g a v e an e x a m p l e t h a t t h e c o n v e r s e = COI, does not hold.

I2(N)

( m ) Ferrero developed a r a d i c a l theory f o r " p - s i n g u l a r n e a r - r i n g s " i n (18).

'a2)

(n) A radical (correspondinq t o f o r N - g r o u p s was c o n s i d e r e d by B e i d l e m a n i n ( 1 ) , ( 3 ) a n d ( 4 ) a n d by Choudhari i n ( 1 ) . (0)

V a n d e r W a l t ( 1 ) c a l l e d an i d e a l I o f N a n i l r a d i c a l i f I i s n i l , b u t N/I h a s no n i l i d e a l s a n y m o r e . He p r o v e d t h a t t h e sum o f a l l n i l r a d i c a l s o f N e q u a l s Q(M), w h i c h i s t h e g r e a t e s t n i l r a d i c a l o f N , w h i l e the i n t e r s e c t i o n of a l l n i l r a d i c a l s ( t h e smallest n i l r a d i c a l ) c o i n c i d e s w i t h P ( N ) . T h e r e f o r e he c a l l e d "I(?')t h e u p p e r ( l o w e r ) n i l r a d i c a l o f N.

( p ) M l i t z ( 2 ) , ( 3 ) and P o l i n ( 2 ) g e n e r a l i z e d t h i s r a d i c a l t h e o r y t o what t h e y c a l l e d "m-R-near-rinqs".

( 9 ) See a l s o o t h e r p a p e r s of M l i t z f o r a r a d i c a l t h e o r y i n u n i v e r s a l a l q e b r a s . However, t h e s e r a d i c a l c o n c e p t s turn o u t t o be " t o o l e s s n e a r - r i n q - s p e c i f i c " . ( r ) Another a t t e m p t t o g e t a r a d i c a l t h e o r y f o r r e r o s y m m e t r i c n e a r - r i n q s was made by S c o t t i n ( 4 ) . He u s e d a method s i m i l a r t o t h a t o f ( D i v i n s k y ) f o r r i n g s . As a n e x a m p l e he s t u d i e s t h e B a e r - l o w e r - r a d i c a l , w h i c h t u r n s o u t t o be = P ( N ) = > - 3 ( N ) for near-rinss w i t h D C C on N - s u b g r o u p s . C f . Holcombe ( 3 ) , ( 8 ) and K a a r l i (4) ,(7).

5e Concluding remarks

167

( s ) I t i s e a s y t o see t h a t i f N i s a z e r o - s y m m e t r i c n e a r -

r i n g w i t h i d e n t i t y a n d t h e D C C N a n d i f e i s some i d e m for all p o t e n t i n N t h e n J v ( N ) e = Ne n

Z,(N)

V&{O,

1,Zl.

I f Ne i s a m i n i m a l n o n - n i l p o t e n t N - s u b g r o u p o f N a n d if

z2(N)

i s n i l p o t e n t then

2,(N)e

i s the greatest

p r o p e r N - s u b g r o u p o f Ne, s o Ne/ ZZ(N)e

i s an N - g r o u p

o f t y p e 2 ( i t i s h a r d e r t o see t h a t a l l N-groups o f t y p e 2 a r i s e i n t h i s way).

,

(1)

See L a u s c h - F j o b a u e r

where t h e s e r e s u l t s a r e f o r -

m u l a t e d and p r o v e d f o r d g n r . ' ~ - b u t t h e y a r e v a l i d i n t h e general case. (t) Let,

denote the s e t o f a l l

f o r t h e moment,&

maximal" ( c f .

3.29)

ideals o f

F,oFlc

( i. e .

"strictly

those i d e a l s

of

N w h i c h a r e a t t h e same t i m e m a x i m a l N o - s u b No q r o u p s o f PIc). R o u t i n e a r n u m e n t s c r i v e t h e f o l l o w i n r r

g2(tI))c =

i n f o r m a t i o n on

If

L E

LnX C =

J , ( N )r i~ c: tic o r L nPIC€ 4

Ms?, N, c o n t a i n s L n N c 0 ?+I1 = N a n d Y = " ) .

(for if whence Conversely, Hence

( then

2 2(I.l)

M

if

E&

' d 2 ( N ) nP i c

See a l s o 5.12,

5.23,

n

then

N0tM

E

L+Ms

then

NO

N,

d2(N).

M.

M €4 5.24

and 9.77.

( u ) I n M e l d r u m ( 1 3 ) i t i s shown t h a t

&!(I)

=

mN)nI

if

I i s a d i r e c t summand o f N a n d t h a t

8?(

8 ieI

Ni)

=

8

R(Ni).

I n here,

?I?=$ o r @ = %.

ieI

( v ) In Angerer-Pilz

(1)

i t i s shown t h a t t h e r e e x i s t s a

n e a r - r i n g N o f o r d e r 32 w i t h .k,(N)c~l,2(N)c;\ll(N)C%,(N),

and 3 2 i s t h e s m a l l e s t

o r d e r such t h a t these f o u r r a d i c a l s a r e d i f f e r e n t . (See a l s o Meldrum ( 1 3 ) ) .

Also,

the following results

from Angerer ( 1 ) concerning r a d i c a l s o f " s m a l l " nearr i n g s are mentioned:

85

168

( a ) If ( N , + )

i s simple then e i t h e r

31(N)=

{OI,

3

N o r 1 / 2 ( N ) = C O I , " i j i ( ~ )= N . I N 1 i s t h e p r o d u c t of t w o primes o r i f

&(N)

( B ) If

RADICAL THEORY

=

(N,+)

i s c y c l i c o r n o n - a b e l i a n of o r d e r 8 t h e n

'k0(N)

=

Z,(N).

(Y) I f I N 1 i s t h e p r o d u c t of t h r e e d i f f e r e n t primes

2,(N) form a c h a i n t h e n e i t h e r Io(N)c y,/z(N)c Il(N) Zz(N) N o r Po(N)= T,/z(N)= % l i ( N ) & ( N ) o r 'Jo(N) zIjz(N) a,(H)c 22(N)o r a l l r a d i c a

then = %l/z(N). ( 6 ) I f t h e normal s u b g r o u p s o f ( T i , + )

=

=

=

=

=

S

coincide.

( w ) See t h e " N e a r - r i n g s of l o w o r d e r " i n t h e Appendix o r t h e r a d i c a l s o f n e a r - r i n g s o n m o s t g r o u p s of o r d e r 5 8 .

PART 111 SPECIAL CLASSES OF NEAR-RINGS § 6 D I STR IBUT IVE LY

G E N E RATED N EAR-R 1 NGS

8 7 TRANSFORMATION NEAR-RINGS

58 NEAR-FIELDS AND PLANAR NEAR-RINGS

59 MORE CLASSES OF NEAR-RINGS To keep this monograph within a reasonable size we will only cite, but not give proofs of some statements which lie a l i t t l e bit away from the main stream of discussion (but might be equally important)

170

56

DISTRIBUTIVELY GENERATED NEAR-RINGS

I n t h i s p a r a g r a p h we d i s c u s s t h e s e t y p e s o f n e a r - r i n q s w h i c h a r e s t i l l more " r i n g - l i k e " t h a n z e r o - s y m m e t r i c n e a r - r i n g s . I n f a c t , e v e r y d g n r . i s €3,. I f N i s a d g n r . t h e n t h e i d e a l s o f N T a r e e x a c t l y t h e normal N - s u b g r o u p s , b u t t h i s n i c e f e a t u r e d o e s n o t seem t o h e l p a l o t . F o r i n s t a n c e , a l l n e a r r i n g r a d i c a l s c a n s t i l l be d i f f e r e n t ( e v e n f o r f i n i t e d n n r . ' ~ ) . Abelian d g n r , ' s a r e r i n q s . We a l s o d i s c u s s t h e o p e n p r o b l e m o f e m b e d d i n g a z e r o - s y m m e t r i c near-ring i n t o a dgnr. I n t h e c a s e o f n e a r - r i n g homomorphisms t h o s e o n e s d e s e r v e p a r t i c u l a r i n t e r e s t which c a r r y t h e d i s t r i b u t i v e g e n e r a t o r s i n t o t h e ones o f t h e image. These "(N,D)-(N',D')-homomorphisms" a r e c h a r a c t e r i z e d . A l t h o u g h t h e d q n r . ' s f o r m no v a r i e t y , i t i s p o s s i b l e t o speak about " f r e e n e a r - r i n q s d i s t r i b u t i v e l y g e n e r a t e d by a g i v e n s e m i g r o u p " . N - q r o u p s r a r e s t u d i e d w h i c h have t h e p r o p e r t y t h a t t h e d i s t r i b u t i v e q e n e r a t o r s o f 4 I act " d i s t r i b u t i v e " ( = a s endomorphisms) o v e r r . F i n a l l y we s t u d y t h e s t r u c t u r e o f d q n r . ' ~ : 2 - p r i m i t i v e f i n i t e for a d g . n o n - r i n g s w i t h i d e n t i t y a r e j u s t t h e Mo(r)'s f i n i t e , n o n - a b e l i a n i n v a r i a n t l y s i m p l e group r . In t h e f i n i t e c a s e , Mo(r) = E(r) i f f r i s o f t h i s k i n d .

.

171

6a Elementary

a )

E L E M E N T A R Y

(dq.,b e t t e r :

N i s d i s t r i b u t i v e l y generated generable)

distributively

i f t h e r e i s a subsemigroup D o f

(Nd,.)

qenerating

(N,+). 6.1

NOTATION I f D g e n e r a t e s N we d e n o t e t h i s b y

6.2

EXAMPLES ( s e e ilolcomhe ( 3 ) f o r g e n e r a l i z a t i o n s ) (a)

If

(r,+).c'$l, Cuiei,

where

E(r)

i s a subnear-rinq o f

(H. Let

u i ~ { - l , + l l and

(End r,o)

Neumann ( l ) , ( 2 ) ;

(rn,+)

{ e l,...,enl

M(r),

ei.cEnd

r.

distributively

a n d c a l l e d t h e " e n d o m o r p h i sm

I"' ( s e e 1 . 1 5 ) .

near-ring on (b)

E(r) b y t h e s e t o f a l l f i n i t e

define

sums

generated by

(N,D).

Frohlich ( l ) ,

be a reduced f r e e

=:

E

(i.e.

(2)).

qroup w i t h generators

e a c h map

be e x t e n d e d t o an endomorphism on

E

+

Pn

can uniquely

rn; r n i s

then

t h e f r e e g r o u p i n some v a r i e t y o f g r o u p s ) . Define f o r the set

rnd

rn

two b i n a r y o p e r a t i o n s 8 , -

by

(61 8 6 2 ) ( e i ) :

(61

*

62)(ei):

@l(ei)+62(ei)

= ~ $ ~ ( $ ~ ( e ~( a) n )d e x t e n d f r o m E

96 DISTRIBUTIVELY GENERATED NEAR-RINGS

172

I n F r o h l i c h ' s p a p e r s , + i s r e f e r r e d t o a s the " a d d i t i o n o f t h e f i r s t t y p e " a n d 13 a s t h e " a d d i t i o n of t h e second type". ( c ) S i m i l a r t o ( a ) , the near-rinqs A(r) a n d I ( r ) , d e f i n e d a s t h e s u b n e a r - r i n g s o f M(r) g e n e r a t e d by the automorphisms ( i n n e r automorphisms) o f ( I - , + ) , are d g n r . ' ~ . 6.3

REMARKS A(r)

(a) E(T),

and

I ( r ) w i l l be s t u d i e d i n 9 7 c ) .

( b ) The E n d r n l s w e r e i n t r o d u c e d a n d s t u d i e d by H . Neuinann i n ( 1 ) and ( 2 ) . Her r e s u l t s on t h e s e t y p e s of near-rings include: E n d r n c o n t a i n s no i d e n t i t y , b u t a l l 0 f i x i n g some e i and s e n d i n g the o t h e r e ' s i n t o z e r o a r e j d i s t r i b u t i v e a n d c a n be v i e w e d a s " r z l a t i v e u n i t s " . T h e r e i s a 1 - 1 - c o r r e s p o n d e n c e rl, b e t w e e n t h e s e t f n of a l l f u l l y i n v a r i a n t subgroups o f r n and the s e t In o f a l l i d e a l s o f E n d r n by $:

Jn A

+

+

zn

bicfl,...,nl:

$(ei)cA1

A l l homomorphic i m a g e s o f End r n a r e a l s o some E n d rm's. Each E n d T n i s t h e homomorphic i m a g e o f End a n , where i s t h e f r e e g r o u p on n generators. S i m i l a r r e s u l t s hold f o r t h e near-rings of t h e kind 8 End r n . which a r e a l s o dg. ( s e e 6 . 9 ( d ) ) . n E IN ( c ) See F i t t i n g ( 1 ) f o r t h e problem, which automorphisms

o f a ( n o n - a b e l i a n ) group have the p r o p e r t y t h a t t h e i r s u m i s a n a u t o r n o r p h i s m a g a i n . C f . a l s o Heerema (1) and Robinson ( 1 ) f o r s i m i l a r q u e s t i o n s . ( d ) See P l o t k i n ( 2 ) f o r t h e c o n n e c t i o n between t h e r e p r e s e n t a t i o n s o f a group r and t h o s e o f E(!'). ( e ) See a l l papers of Dasic f o r g e n e r a l i z a t i o n s o f the conc e p t o f a d g n r . . C f . a l s o Meldrum ( 1 3 ) .

173

6a Elementary

Now we

s t u d y some e l e m e n t a r y p r o p e r t i e s o f d g n r . ' ~ . N o t e ,

i f N i s dg. b y D t h e n e a c h n = Caidi

6.4

with

ncN

= +1,

ai

that

i s a f i n i t e ( o r d e r e d ) sum

di€D.

PROPOSITION L e t N b e d o . b y D. (a) (b) (c)

tl n c N \1

dcD: d ( - n ) = ( - d ) n = - ( d n ) .

NEq).

\

n,n'EN

dsD:

= (-d)n't(-djn

(d) I f

n = laidi i

d(ntn')

= -dn'

and

-

2

dn+dn' A ( - d ) ( n + n ' )

=

dn.

n' = laid;

then

j

nn' = ~ a i ( ~ a ! d . d ! ) . 1

J

3 1 3

The p r o o f i s a c c o m p l i s h e d b y e a s y c o m p u t a t i o n s a n d t h e r e fore omitted.

6.5 PROPOSITION (Seth-Tewari ( I ) , Meldrum (13)). Let N be dg. by D a n d r an N-group with d ( y + y ' ) = dy+dy' for a l l deD, y,y't;D. If A S r then the N-ideal h generated by A is given by all finite sums of the form C ( y i + a i d i d - y i ) with yi", c r i ~ { l , - l l , dieD and 6eA. Proof.

The s e t o f a l l f i n i t e sums o f t h e f o r m X a i d i 6 i i s a

(r,+). h i s then closure o f A i n (r,+). To s e e t h a t A gN r , c o n s i d e r subgroup A of

n as

n = Caidi

a n d 6 as

j u s t t h e usual n o r m a l n(T+y)-ny,

decompose

= x(y.+u.d!G.-y.) 1 1 1 1 1

and

p r o c e e d a s u s u a l . (The n e x t r e s u l t shows t h a t i t s u f f i c e s t o show t h a t NEGK.)

See Meldrum (13) that 6.5 i s not v a l i d without the d(y+y')=dy+dy'assumption. Near-rings generated by a n inverse semigroup of distributive elements are treated i n M a h m o o d - M e l d r u m - O ' C a r o l l ( 1 ) a n d Meldrum (13). Examples of d.g. near-rings of low order c a n he found i n the appendix.

174

$ 6 DISTRIBUTIVELY GENERATED NEAR-RINGS

b)

6.6

SOME

AXIOIIATICS

P R O P O S I T I O N L e t N b e d q . by D a n d

r

( a ) I f A i s a normal s u b q r o u p o f

(I",+)

be a n N-qroup. then

r < = > ~ s

ASI

N N (This i s one s t e p t o w a r d s t h e s i t u a t i o n i n rinos,

Nr

since the ideals of qroups. )

a r e j u s t t h e normal N-sub-

2

( b ) H i s a b e l i a n <=> N i s d i s t r i b u t i v e . + ) ( c ) N i s a b e l i a n c-> N i s a r i n o . ( d ) I f N E ~t h e n N i s d i s t r i b u t i v e <=> N i s a b e l i a n <=> <-> N i s a r i n g . k

P roof. -

(a) If

1aidicN,

n = k

n(b+y)-ny = +...to

1aidi(6+y)-

i=l d ( b + Y )-okdkYk k

Since

dk(Gty)-dky

...-

and

A5

= -dky-dkb+dkyEA,

r

then

~

E

=

aldl(6+y)+

N

k

1aidiy

i=l 1d 1y .

= dk6tdky-dky = dk6Eh

(-dk)(G+Y)-(-dk)Y

6.4)

ycr

i=l

and (USinr:

= (-dk)Y+(-dk)b-(-dk)Y

=

we s e e t h a t i n a n y c a s e

k d ( 6 +y ) - 0 k d k y ~ A . P r o c e e d i n ? i n t h i s way y i e l d s A r. The c o n v e r s e f o l l o w s from 1.34(b) and 6.4(b). ( b ) =>: I f N 2 i s a b e l i a n t h e n f o r n , n ' , n " E N , n = X u i d i w e o e t n ( n ' + n " ) = C q 1. d 1. ( n ' t n " ) = = ~ a . ( d . n l + d . n " ) = ~ u ~ d ~ n ' t C a . d . n n" n ' t n n " . 1

<-:

N

=

1

1

1

1

Conversely, t h e proof o f 1.107(c) shows t h a t Pld i m p l i e s N~ to be abelian.

(c)

=>:

<=I:

trivial.

I f N i s a b e l i a n , t h e same a p p l i e s t o So N i s d i s t r i b u t i v e a n d t h e r e f o r e a r i n q .

+ )" N2

abelian"

N

2

s t a n d s f o r " V a , b , c , d ~ N : a b + c d = c d + ab".

.

175

6b Some axiomatics

(d) f o l l o w s f r o m (b) and (c). The n e x t r e s u l t e x a m i n e s t h e r 6 1 e o f i d e n t i t i e s i n d y n r . ' ~ . 6.7

TliEOREM ( i i g h (1)). Consider t h e dgnr. (a) If D contains a l e f t (right,

(N,D).

two-sided)

identity e

t h e n e s e r v e s a s t h e same f o r N. (b)

I f N c o n t a i n s e x a c t l y one l e f t ( o r r i g h t ) i d e n t i t y e then e i s two-sided.

Proof.

\r

If e i s a left identity of

(a)

n = CaidicN:

= Caidi

= n,

e n = eCaidi

D

then

= Ce(aidi)

= Cai(edi)

=

and s i m i l a r f o r r i g h t i d e n t i t i e s .

( b ) Assume t h a t e i s t h e o n l y l e f t i d e n t i t y o f N.

v v

nEN

hence

XEN:

(ne-nte)x

nEN:

n e - n t e = e.

= nex-nxtex = nx-nx+x

Therefore

b

nEN:

Then

= x,

ne = n

and e i s two-sided. S u p p o s e now t h a t e i s t h e u n i q u e r i q h t i d e n t i t y o f N . Again,

t a k e n and

x(en-n+e)

As a b o v e , 6.8

x = Caidi

= Coidi(en-nte)

en-n+e

arbitrary

= 1aidie

EN.

= ( 1 a1. d1 . ) ~ = x e .

so e i s a g a i n t w o - s i d e d .

= e,

REMARKS ( a ) Observe t h a t 6.7(b) the "left-case".

holds f o r general near-rinqs

i n

B u t s e e L i q h (1) f o r e x a m p l e s t h a t

i n a l l o t h e r c a s e s 6.7

does n o t h o l d f o r g e n e r a l

n e a r - r i ngs. ( b ) See L i g h ( 1 2 ) f o r a p r o o f o f " A f i n i t e d g n r .

c o m m u t a t i v e <->

N i s

all zero divisors are central". This

does n o t h o l d i n t h e i n f i n i t e case.

( c ) L i g h ( 1 0 ) !'lives with

( N , + ) = sn

group (d) I f (N,+)

Dep

characterizations o f a l l d g n r . ' ~N ( n 2 5)

or

(N,+)

a dihedral

(PchC21).

i s n i l p o t e n t and N dg.

then N/&(N)

(Gringlatz ( I ) ) . ( e l See F e i g e l s t o c k ( 1 ) , ( 2 )

f o r simple dgnr's.

i s a ring

8 6 D I STR I B UTI VE LY G EN ER ATED

176

C)

Dgnr.'s

NEAR-R INGS

CONSTRUCTIONS OF DISTRIBUTIVELY GENERATED NEAR-RINGS

so i t i s n o t q u i t e easy

have no p a r t i c u l a r s t e n c h ,

t o r e c o g n i z e t h e m amonq o t h e r n r . ' s .

The n e x t r e s u l t m i q h t

h e l p i n some c a s e s .

6.9

THEOREM

(a) If

M

(b) I f

M 5 N

and M i s dg. t h e n P i i s not do. i n general.

5 N

and N i s dg. t h e n M i s n o t dg.

( c ) E v e r y homomorphic imaqe o f a d g n r . i s dq. ( d ) E v e r y d i r e c t sum o f d g n r . ' s

( e ) E v e r y d i r e c t summand o f a d g n r . Proof.

( a ) T a k e some

dg.

by 6 . 4 ( b ) ,

M:

I01

=

.

i s i t s e l f dq..

N =?p N o .

N E ~w i t h

but

.

i s a dgnr.

i n qeneral.

N i s not

Then

i s

( b ) See L y o n s - M a l o n e (1) f o r an e x a m p l e o f a s u b n e a r -

E(S3)

ring of

(c) If

.

w h i c h i s n o t dq.

N'

h : N -I+

i s an e p i m o r p h i s m a n d i f PI i s d g , h(D)

by D t h e n a r o u t i n e c h e c k shows t h a t

h(D)

Ni

(d) Let the near-rings

Di E

=:

8

{dijljcJil.

component.

where

dij

I t i s e a s y t o show t h a t ni =

NA

and

n = (...,ni,...)cN,

1 uijidiji'

be dg. b y

,...

= (,..yO,dij,O

)E

stands i n the i - t h

r ?1fJ. . I i ~ IA

Put

(iaI)

Zij:

Define

= : N,

Ni

iE1

If

E

generates N ' .

D

jEJil 5

= : D.

Nd.

decompose e a c h

where each

jiEJi

ni

as

a n d t h e sum

Ji i s a f i n i t e one. Then

n =

1

1

i E I

J;

oiji

sum). C o n s e q u e n t f y ,

ai j i

( t h i s i s again a f i n i t e

D generates

(N,+).

6c Constructions

(e) From 6 . 9 ( b )

177

follows from (c).

we s e e t h a t t h e c l a s s o f

a l l dgnr,'s

i s no v a r i e t y .

H e n c e " d is t r ib u t iv e g e n e r a t io n " c a n n o t b e d e f in e d b y " e q u a t io n s " (see (Gratzer)).

T h i s b r i n g s up t h e q u e s t i o n about t h e s m a l l e s t ( t h e v a r i e t y " g e n e r a t e d " by

v a r i e t y which contains a l l dgnr.'s

t h e d g n r . ' ~ ) . We s t a t e w i t h o u t p r o o f ( i t u s e s a l o t o f u n i v e r s a l a1g e b r a ) 6.10

THEOREM ( M e l d r u m ( 1 ) ) . T h e f o l l o w i n q v a r i e t i e s c o i n c i d e : o f the type I(r).

( a ) The v a r i e t y g e n e r a t e d b y a l l n r . ' s

( b ) The v a r i e t y g e n e r a t e d b y a l l d g n r . ' ~ . (c)

no.

So e v e r y z e r o - s y m m e t r i c

n e a r - r i n g c a n b e e m b e d d e d i t i t o some

"descendant"

(see (Gratzer),

f i n i t e nr.'s 6.11

o f a dgnr.

NE"?,

THEOREM ( P l a l o n e ( 6 ) ) .

152/153).

Every f i n i t e zero-symmetric

I(r) =

r i n g c a n b e embedded i n t o

r

pp.

For

we g e t m o r e :

E(r)

=

Mo(r),

nearwhere

i s a c e r t a i n f i n i t e non-abelian simple group.

Proof.

Mo(r) ( 1 . 8 8 ( c ) ) w i t h

Embed N i n t o some

embed

r

Next,

embed S n

i n t o some

Sn

(e.g.

n23;

v i a Cayley's theorem).

i n t o the a l t e r n a t i n g group

An+2

in

t h e f o l l o w i n g way:

(1) I f

ncPn

i s even, l e t

n : = ( 1

2 n(1) n(2)

(2) If

-n : T h e map

nEPn

i s odd,

2 = ( 1 n(1) n(2)

h:

n

...n ( n )

nt2 ) nt2

.

let n

. . .n ( n )

3",n - c i Ai n t 2

ntl n+l

n t l nt2

nt2 n+l

1.

i s a g r o u p monomorphism,

+

a n e m b e d d i n q map.

Ant2

i s a f i n i t e s i m p l e non-

hence

$6 DISTRIBUTIVELY GENERATED NEAR-RINGS

178

a b e l i a n qroup c o n t a i n i n a p r o p e r l y a homomorphic imaoe of

i n 7.46

6.12

N c ~ M ~ ( ~ ) Q F I ~ ( A ~ We + ~ w) i. l l s e e

By 1.99,

Sn.

that

M0(An+*)

COROLLARY ( M a l o n e ( 6 ) ) .

= I(Ant2).

= E(Ant2)

NEW^

Every f i n i t e

can be

embedded i n t o a f i n i t e d g . n o n - r i n g w i t h i d e n t i t y . Several q u e s t i o n s remain open: I s e v e r y z e r o s y m m e t r i c n e a r - r i n g e m b e d d a b l e i n t o some d g n r . ? I s e v e r y d g n r . e m b e d d a b l e i n t o some d g n r . w i t h i d e n t i t y ?

E(r) e m b e d d a b l e i n t o some

I s every

I s e v e r y d g n r . e m b e d d a b l e i n t o some And s o o n .

6.13

Cf.

I(r)? E(r)?

a l s o Heatherly-Malone (2),

1.90 and 6 . 3 5 ( k ) .

R E M A R K I n ( 1 ) a n d ( 3 ) , H. L a u s c h d e v e l o p e d a n e x t e n s i o n For g r o u p theory ( v i a homological alqebra) f o r dqnr.'s. d g n r ' s s e e Mahmood ( 1 ) a n d M e l d r u m ( 4 ) , ( 1 3 ) . C a t e g o r i c a l

c o n s i d e r a t i o n s a r e i n Mahmood ( 1 ) - ( 4 )

,

See a l s o H e a t h e r l y ( 1 2 )

and Mahmood-Meldrum ( 1 ) .

J o h n (2).

d ) D I S T R I B U T I V E L Y GENERATED NEAR-RINGS WITH SS -F I N I T E N ECONDITIONS

6 . 1 4 THEOREM - (Ligh (3)).

N

Let

$. {Ol

be a d g n r . w i t h DCC on

monogenic N-subgroups. Then ( a ) N€Wl c->

N*

c o n t a i n s an e l e m e n t w h i c h i s n o d i v i s o r

o f tero.

( b ) b nEN*:(NEnl

and

nEN

i s invertible)



n i s no

zero divisor.

Proof. ( a ) "->"

a d i v i s o r o f zero. 3 k e I N : Nxk = NXk + l 3 e E N : x . x k = e-xktl. Hence

So a s s u m e t h a t 2 Now N x 4 x ?...

i s clear.

x-ex = 0

--

,.. . So

.

x

(9

0)

Therefore

This implies that ( x - e x ) x k = 0.

a n d we g e t

e x = x.

i s not

179

6d Finiteness conditions

Also, ( x e - x ) x = 0 a n d t h u s xe = x . S o \1 meN: (me-m)x = 0 w h e n c e me = m . Now t a k e some a r b i t r a r y n E N . Decompose x a s x = Coidi. Then x(en-n) = Cuidi(en-n) = = E u i ( d i n - d i n ) = 0 , i m p l y i n g t h a t en = n . (b) Let

i s c l e a r again. n ;P 0 be no z e r o d i v i s o r . T h e n N E ~ , by ( a ) . A s i n ( a ) , 3 kclN : Nnk = N n k + ’ = This i m p l i e s t h a t So 3 meN: n k Ink = ( 1 - m n ) n k = 0 , so 1 = m n . Also, (nm-1)n = 0 , s o n m = 1 and n i s i n v e r t i b l e . “->I’

... .

m a n k t 1 .

6.15 REMARK ( L i q h ( 1 3 ) ) . I f N i s a f i n i t e simple d g . near-ring then (N,+) i s a p e r f e c t ?roup ( i . e . N coincides with i t s commutator s u b g r o u p ) . See a l s o F e i g e l s t o c k ( 2 ) . There a r e s e v e r a l c o n n e c t i o n s between c h a i n c o n d i t i o n s , s o l v a b i l i t y of (N,+) a n d “weak d i s t r i b u t i v i t y ” ( s e e F r o h l i c h ( l ) , D e f . 4 . 3 . 1 ) . We s t a t e w i t h o u t p r o o f t h e f o l l o w i n g c o l l e c t i o n o f r e s u l t s ( s e e a l s o Beidleman ( 1 1 ) ) . 6.16 THEOREM L e t N be a d g n r . ( a ) (Frohlich ( 1 ) ) . I f (N,+) i s solvable then N i s weakly d i s t r i b u t i v e . If N2 = N, the converse also h o l d s . See a l s o Mason ( 1 ) . (b)

(Beidleman ( 4 ) ) . I f N i s f i n i t e and i f NrtzNq t h e n (l‘,+) i s s o l v a b l e i f f N r i s s o l v a b l e ( i . e . r has a normal s e q u e n c e ( 2 . 3 7 ) w i t h a b e l i a n q u o t i e n t s ) .

i s s o l v a b l e a n d N has ( c ) ( B e i d l e m a n ( 4 ) ) . I f (N,+) t h e D C C N t h e n e v e r y maximal l e f t i d e a l i s m o d u l a r a n d c o n t a i n s t h e commutator subgroup o f ( N , t ) . ’;J2(N) i s nilpotent and is a r i n a . Also, N has a 2(N 1 c e r t a i n kind o f ACC.

N/l

(d) (Ligh ( 3 ) ) . I f ( N , + ) i s s o l v a b l e such t h a t n o t a l l elements a r e d i v i s o r s of z e r o . Then t h e DCCL implies the ACCL.

D ISTR IBUTIVE LY GENE RATED NEAR-R INGS

86

180

e ) "FREE" DISTRIBUTIVELY G E N E R A T E D N E A R - R I N G S

Since the dgnr.'s

do n o t f o r m a v a r i e t y ,

f o r t h e existence o f " f r e e dgnr.

I s " .

t h e r e i s no g u a r a n t e e

Moreover,

t h i s concept

does n o t seem t o b e a p p r o p r i a t e f o r t h i s c l a s s o f n e a r - r i n g s . We a r e now g o i n g t o d e f i n e a s i m i l a r c o n c e p t . F i r s t o f a l l we n e e d a " r e f i n e d " v e r s i o n o f homomorphisms b e t w e e n d g n r . I s .

6.17 DEFINITION L e t

6.18

h: N

+

No

h(D)

5

D'.

be d q n r . ' ~ . A h o m o m o r p h i s m

(N',D')

(N,D),

(N,D)-(N',D')-homomorphism

i s c a l l e d an

N

EXAMPLE Each d g n r . - h o m o m o r p h i s m

N'

+

i s an

i f

(N,Nd)-

-(N',N,!,)-homomorphism. 6.19 PROPOSITION ( F r o h l i c h ( 2 ) ) .

and l e t

h:

(N,+)

-c

Let

(N',+)

a s e m i g r o u p homomorphism

be a g r o u p homomorphism and (D,.)

+

(D',.).

(N,D)-(N',D')-homomorphism.

Then h i s an

P r o o f . I t o n l y r e m a i n s t o show t h a t = h ( n ) h ( n' ). Let

n = Caidi,

n ' = Ca'.d!.

1

J

J

1

ioi(lujh(di)h(d;))

n,n'cN: Then,

J J

h(nnl) = h(lai(la!d.d!))

=

(N',D') b e d g .

(N,D).

1

J

J

= ( l o i h ( d i ) ) . ( l o j h ( d ! ) )J

J

1

=

using 6.4(d),

= loi(la;h(did!))

J

h(nn')

= =

J

= h(n)h(n').

We a r e now g o i n g t o d e f i n e s o m e t h i n g l i k e a " f r e e n e a r - r i n g dg. b y a g i v e n s e m i g r o u p

(D,.)".

We u s e a s l i g h t m o d i f i c a t i o n o f a m e t h o d d u e t o F r o h l i c h ( 4 ) and M e l d r u m ( 2 ) . C f . a l s o Zeamer ( 1 ) . 6.20 DEFINITION L e t

(D,.)

o f groups. Denote b y ( t h e s e t ) D.

be a s e m i g r o u p a n d v a v a r i e t y (FD,v,t)

t h e f r e e group i n v on

6e "Free dgnr.'s"

181

F D ,V c o n s i s t s o f a l l f i n i t e sums C a i d i s where e q u a l i t y i s determined b y v . If e.q. 2)= then " e q u a l i t y " i s "formal equal i t y " . D e f i n i n g ( C o i d i ) - ( C a ! d ! ) : = ~ o i ( ~ a ! d . d ! )y i e l d s

7,

J J

6.21 THEOREM (a)

1

J

J

1

J

Let Z'Y be a v a r i e t y of g r o u p s .

i s well-defined.

( b ) ( F D , v , t , * ) = : F i s a n r . , d g . by D , whose a d d i t i v e group belongs t o v . ( c ) F o r every d g n r . (N',D') w i t h (",+I E 79 ever!! semiqroup homomorphism D + D ' can u n i a u e l y be e x t e n d e d t o a (F,D)-(N',D')-homomorphism.

( N , D ) with ( N , + ) homomorphic image o f ( F , D ) .

( d ) Every d g n r .

E

? i sl a (F,D)-(N,D)-

P r o o f . ( a ) : h o l d s by t h e d e f i n i t i o n of e q u a l i t y v i a laws in V . By a r o u t i n e b u t somewhat i a s t y c a l c u l a t i c n o n e sees t h a t ( F D , V , + , * ) i s a n e a r - r i n q . By c o n s t r u c t i o n , F 'is f r e e o v e r D i n 'V, s o ( F , t ) E z ) a n d D g e n e r a t e s (bj:

(F,+). ( c ) : By d e f i n i t i o n , e v e r y map

f:D

+

D'

can uniquely

h:(F,t) * ( N , t ) . If f i s moreover a s e m i q r o u p homomorphism, h i s a n (F,D)-(N',D')-homomorphism by 6 . 1 9 .

b e e x t e n d e d t o a homomorphism

( d j : C o n s i d e r i n g t h e d i a g r a m ( 1 i s t h e i n c l u s i o n map) a n d rememberinq o r o u p t h e o r y ( o r making a r o u t i n e d i a g r a m a r g u m e n t ) gives the information t h a t h i s a g r o u p - e p i m o r p h i s m . Now h / D = i d D , whence h i s a ( F . 0 ) - ( N , D ) - e p i m o r p h i sm by 6 . 1 9 See a l s o John ( l ) , Mahmood ( 1 ) - ( 4 ) , Meldrum ( 1 3 ) a n d Rhabari ( 1 ) , ( 2 ) . R e p r e s e n t a t i o n s o f groups v i a f r e e d g n r . ' s a r e s t u d i e d i n Meldrum ( 4 ) a n d ( 1 3 ) .

182

§ 6 0 lSTR IBUTIVE LY GENE R A T E D NEAR- R INGS

f ) D-GROUPS A N D ( N , D ) - G R O U P S

L i k e n r . homomorphisms o f d g n r . ' ~ , t h e c o n c e p t o f N - q r o u p s c a n be " r e f i n e d " f o r a d g n r . ( N , D ) : we w a n t t h e e l e m e n t s o f D t o " d i s t r i b u t e over r " ( t h i s appeared already in 6 . 5 ) . 6 . 2 2 DEFINITION Let (N,D) be a d g n r . an ( N , D ) - g r o u p i f V y l , y 2 ~ r

v

. N r ~ Ni qs dED:

called

d(ylty2) = dyl+dy2.

6 . 2 3 DEFINITION L e t ( D , . ) be a s e m i g r o u p and r i s c a l l e d a D-group i f a m u l t i p l i c a t i o n

(I',+) a :

a group.

r

Dxr-

(d,Y) dY VdcD: d ( y l t y 2 ) = d y l t d y 2 . +

i s defined with

y1,y2€r

(N,D) i s d g . t h e n N r i s a n ( N , D ) - g r o u p 6.24 REMARK So i f i f f r i s a D-group ( w . r . t . t h e r e s t r i c t e d m u l t i p l i c a t i o n of N r ) . Now l e t

(D,.)

by D i n 2, ''

be a s e m i q r o u p a n d F D , z ) be t h e " f r e e n r . d q . a s I n 6 . 2 1 , w h e r e 0 i s some J a r i e t y i n ? .

6 . 2 5 T H E O R E M E v e r y D-group Proof.

If

ZaidicF

D ,V

rEl)

i s an

and

ycr,

(FD,D,D)-group. define

(Coidi)y: =

= Cui(diy). Again t h i s i s well d e f i n e d and c h e c k i n g the (FD,Q,D)g r o u p a x i o m s c r e a t e s no p r o b l e m .

A g a i n , l e t q be a v a r i e t y o f g r o u p s a n d of a l l We c o n s i d e r t h e c l a s s (N,D) ( N , D ) -groups. L e t Q be t h e f a m i l y o f o p e r a t i o n s

.d

(+So,-)

u (Wn)nEN

(2,0,1)

u (')nCN

(N,D) rE'U

a dqnr. which a r e

of type

'

L e t % be t h e c l a s s o f ( u n i v e r s a l ) a l g e b r a s o f t h i s t y p e .

183

6f D-groups and (N,D)-groups

L e t q b e t h e v a r i e t y determined by a l l laws which d e f i n e ,

r&,

t o be a g r o u p

(I',t,O,-)

EV

for

and by a l l laws

Then c l e a r l y 6.26

THEOREM

3=(N,Dl,v(g;

so t h e l a t t e r c l a s s i s a v a r i e t y .

F r o m u n i v e r s a l a l g e b r a we now g e t

6.27

COROLLARIES T h e r e e x i s t a l l f r e e ( N , D ) - n r o u p s ; unique up t o the

6.28

(N,D)-isomorphisms;

(N,D)-(N,D)-homomorphic

each

they are

(N,D)-group

imaqe o f a f r e e

is

(N,D)-qroup.

REMARKS ( a ) Meldrum (2) used these " f r e e and t h e f r e e variety

9

faithful

(N,D)-groups

by

(N,Dj-group

i n

Zf"

(N,D)

has a

( n o t even i n the f i n i t e case). (even n o t every f i n i t e dgnr.)

i n s u c h a way t h a t

E(r)

r

remain d i s t r i b u t i v e on

endomorphisms on

D

i n a s u i t a b l e non-abelian

c a n b e e m b e d d e d i n t o some

dsD

dg.

t o show t h a t n o t e v e r y d q n r .

Therefore n o t every dgnr. a l l

nr.'s

( = become

r).

O b s e r v e t h a t we k n o w f r o m 6 . 1 1 t h a t e v e r y f i n i t e d a n r . c a n b e e m b e d d e d i n t o some that all

deD

E(r), i f

remain d i s t r i b u t i v e .

Meldrum a l s o c o n s t r u c t e d i n ( 2 )

( n , D ) , (N,D) such t h a t of

(N,D)

with faithful (N,D) and

one does n o t i n s i s t

i s a

(N,D)

"nearest" dgnr.'s

( m , D ) - ( (N,D)-)qroups

(R,D)-(N,D)-homomorphic is a

imaqe

-

(N,D)-(N,D)-homomorphic

(N,D). Moreover he considered t h e "Dorroh-type" image o f

of an i d e n t i t y 1 t o a drlnr. adjoin 1 t o

D)

(cf.

See a l s o E l e l d r u m ( 7 1 ,

(N,D)

(Kertgsz), (10)-(13).

Th.

adjunction

(one has t o

3.13).

36 DISTRIBUTIVELY GENERATED NEAR-RINGS

184

( b ) F o r more i n f o r m a t i o n on

(N,D)-groups

see F r o h l i c h

(2). ( 4 ) . I n ( 4 ) . F r o h l i c h d e s c r i b e d f r e e sums a n d p r o d u c t s , o r t h o g o n a l sums, i n t h e case o f

f r e e bases and p r o j e c t i v i t y It turns out that the

(N,D)-groups.

s i t u a t i o n i s s i m i l a r t o t h e r i n g (-module)

case.

and ( N , D ) -

( c ) F r o h l i c h a l s o s t u d i e d c a t e g o r i e s o f N-

g r o u p s i n ( 5 ) and d e v e l o p e d a " n o n - a b e l i a n a l g e b r a " v i a these groups i n ( 6 )

-

homological

(8).

a \ S T R U C T U R E THEORY

We s t a r t w i t h a r e s u l t o n g e n e r a t o r s i n

6.29

N/I.

N and N

( N r ) b e fg.,

THEOREM ( ( G a s c h u t z ) ,

Lausch ( 4 ) ) .

N a dgnr. and I ( A )

be a f i n i t e i d e a l . Moreover,

N/I

(r/A)

Then

V

I;?

be t h e N-subgroup g e n e r a t e d b y

iEI1,

...,k )

t h e N-subgroup N

-Proof.

Let

3ei€Fi:

{el

,...,e,)

let

l,...,ek).

-

grznerates

(r).

As i n (Gaschiltz)

(where i t i s proved f s r groups;

t h i s p r o o f c a r r i e s o v e r t o groups w i t h o 3 e r a t o r s

-

see Lausch ( 3 ) ) . T h i s r e s u l t can be used t o p r o v e 6.31: 6.30

D E F I N I T I O N If N€ml

in

(N,*))

NEWl

I ( N ) : = {nENln i s i n v e r t i b l e

(N,.).

denotes t h e "group k e r n e l " o f

6 . 3 1 THEOREM ( L a u s c h ( 3 ) , Let

then

L a u s c h - N o b a u e r (l), Scott (1)).

be a f i n i t e dgnr. and l e t

homomorphism.

+)

h:N

+

m

be a nr.-

Then

+ ) T h e elements of I ( N )

a r e a l s o c a l l e d t h e " u n i t s " o f N.

185

6 g Structure theory

Proof.

(a) I f

iEI(N),

3

then

jEN:

= h(j)h(i) = h(l),

h(i)h(j)

( 0 ) Conversely,

if

t h e N-subgroup

h(N):

icI(h(N)) take

i j = j i = 1.

so

Hence

h(i)EI(h(N)).

Cil

then

generates

h ( n ) = Caih(di)ch(N) 1

and

3

= lukh(dk)

7.T = h ( 1 ) .

with

Then

11

= h(n).J.T

(iailuih(di)h(dk))i

= h(n).

1 1 1

S o b y 6 . 2 9 t h e r e i s some

h ( i ) = i and

with

is1

such t h a t t h e N-subgroup g e n e r a t e d by i e q u a l s N.-So t h e r e i s some

j E N

with

j i = 1.

Hence b y 1 . 1 1 3 ,

i i s invertible. 6.32

R E M A R K See L a u s c h ( 4 )

f o r some m o r e g e n e r a l v e r s i o n s o f

6.31. N e x t , we v i s i t p r i m i t i v e d q n r . ' ~ . w i t h D C C N and g e t 6.33

THEOREM ( L a r t o n ( 2 ) ) .

let

NcM(r)

be a f i n i t e d q .

non-

r i n g w i t h a l e f t i d e n t i t y . Then t h e f o l l o w i n g c o n d i t i o n s are equivalent: ( a ) N i s 1 - p r i m i t i v e on ( b ) N i s 2 - p r i m i t i v e on

r. r.

(c) N i s simple. ( d ) N = Mo(r) and m o r e o v e r

r

i s a finite,

non-abelian,

i n v a r i a n t l y simple group. Proof.

( a ) <=+

( b ) ->

(d):

4.6(b),

r

( b ) <->

( c ) f o l l o w s from 4.47(a).

N has an i d e n t i t y . By 4.60,

i s abelian,

i s a rina.

r . By M (r). GO

Assume t h a t N i s 2 - p r i m i t i v e o n N i s a b e l i a n b y 1.49,

Hence

r

i s non-abelian.

N

so by

i s an e n d o m o r p h i s m o f

r.

Since

all

dcNd,

whence

r

Nr

N

Since N i s f i n i t e ,

t h e same a p p l i e s t o r . r i s m o n o q e n i c , s o % 3 Y O E r : N / ( o : y o ) uN r by 3 . 4 ( e ) . So e v e r y

cannot c o n t a i n a n o n - t r i v i a l

If

6.6(c)

dsNd

i s N-simple,

it

subgroup i n v a r i a n t under

shows u p t o be i n v a r i a n t l y s i m p l e .

186

$ 6 DISTRIBUTIVELY GENERATED NEAR-RINGS

Now

i s f i n i t e and fixed-point-free, so i t

AutN(I')

c o n s i s t s e i t h e r of { i d } alone or contains a fixedp o i n t - f r e e a u t o m o r p h i s m o f p r i m e o r d e r . The p a p e r ( T h o m p s o n ) t e l l s us t h a t ( r , + ) i s n i l p o t e n t . B u t r i s invariantly simple a n d therefore abelian, a c o n t r a d i c t i o n . S o A u t N ( r ) = I i d l a n d M O(r) = Mo(r). c;

( d ) => ( b ) : I f I' by 4 . 5 2 ( b ) .

N = Mo(I')

t h e n F1 i s 2 - p r i m i t i v e o n

T h i s t h e o r e m h a s some i n t e r e s t i n o c o n c l u s i o n s ( s e e 7.46). We & o w c o l l e c t some r e s u l t s c o n c e r n i n g r a d i c a l s o f r e l a t e d d . g . n e a r - r i n g s . P r o o f s a n d more o n t h i s c a n be f o u n d i n K a a r l i ( 3 ) and ( 4 ) . 6 . 3 4 T H E O R E M ( K a a r l i ( 4 ) ) . L e t N be a d g n r , I a N and M SN N. (a) I f q&I i s quasiregular i n I then q is quasiregular i n N. 1 ( b ) J v ( M ) 3 J v ( N ) f l M f o r v = 0 and w = ~ . (c) If I , $ I 2 g . . . 4 I k a N and 1 4 1 , then

6.35 R E M A R K S ( a ) S u r p r i s i n g l y ( o r u n f o r t u n a t e l y ) , 6 . 6 ( a ) does not f o r c e

t h e v a r i o u s r a d i c a l s o f a d q n r . t o c o i n c i d e ( n o t even f o r f i n i t e d q n r . ' ~ ) . See s e v e r a l p a p e r s o f Laxton a n d Beidleman. Also, i s not necessary n i l i n t h i s c a s e . See a l s o S c o t t ( 1 1 ) .

22(N)

(b) For dgnr.'s

N E ? ~ ~ , Beidleman ( 8 ) d e f i n e d " s t r i c t l y

p r i m i t i v e " i d e a l s a s 2 - p r i m i t i v e maximal i d e a l s . The i n t e r s e c t i o n o f t h e s e o n e s c o n t a i n s C d 2 ( N ) and equals i n the c a s e o f D C C N ( t h i s f o l l o w s from 4.47(b)).

g2(N)

( c ) L a x t o n ( 3 ) c o n t a i n s a n e x a m p l e o f a f i n i t e d g n r . II w i t h

the property t h a t

g1,,2(N) i s

no i d e a l , w h i l e N h a s nilpotecr; l e f t ideals \ b u t o f course n o n i i p o t e n t i d e a l , .

187

6 g Structure theory

( d ) P e s k i n s ( 2 ) c o n t a i n s more i n f o r m a t i o n on t h e e N e ' s , w h e r e eEN i s some i d e m p o t e n t . (e) In (4), if

Tharmaratnam c a l l e s a dgnr.

EndN(N,+)

non-trivial instance.

l e f t ideals i s a division near-ring,

N - Mo(r).

for

N which i s not a

For a f i n i t e d i v i s i o n dgnr.

r i n g t h e r e i s some f i n i t e , with

N a d i v i s i o n dgnr.

A f i n i t e dgnr. without

AutN(N,+)u{6).

=

non-abelian

s i m p l e N-group

r

T h i s e s t a b l i s h e s a I - I - c o n n e c t i o n between

i s o m o r p h i s m c l a s s e s o f f i n i t e d i v i s i o n dg. ( f ) See T h a r m a r a t n a m ( 1 ) , ( 2 ) , ( 3 ) dgnr.'~": a topological nr. Beidleman-Cox ( 1 ) )

near-rings.

and ( 4 ) f o r " t o p o l o g i c a l N (def.

as u s u a l - s e e

i s c a l l e d a t o p o l o g i c a l dgnr.

if

Nd g e n e r a t e s N t o p o l o g i c a l l y .

If the topological nr.

N i s a dgnr.

then N i s a topo-

b u t t h e converse does n o t h o l d i n g e n e r a l .

l o g i c a l dgnr.,

Tharmaratnam a l s o d e s c r i b e d t o p o l o g i c a l

(N,D)-groups

and t h e s t r u c t u r e o f t o p o l o g i c a l d g n r . ' ~ , e s p e c i a l l y t h a t o f a 2 - p r i m i t i v e complete t o p o l o g i c a l dgnr.. ( 9 ) See L a x t o n ( 4 ) a n d L a x t o n - M a c h i n ( 1 )

f o r the behaviour

o f prime ideals i n d g n r . ' ~ . (h) Plotkin (1),(2)

transferes t h e concept o f a dgnr.

to

universal algebra. ( i ) See a l s o 9 7 c ) . ( j ) N i s c a l l e d a g e n e r a l i z e d dgnr. (gdg.nr.)

generates

(N,+).

nomial n r . ' s

D g n r . ' ~ , constant nr.'s

are of

by a l l g d g . n r . ' s

t h i s type.

3. E v e r y

is

i n a f i n i t e gdg.nr.. not a ring,

and " N

= M ( o ) ( r ) ' 'w i t h r

non-abelian

( k ) By 5 . 1 9 ( a ) a dgnr.

f i n i t e nr.

"I-primitive", finite,

are equivalent. and 6 . 3 4 ,

as a n i d e a l .

a n d many p o l y -

The v a r i e t y g e n e r a t e d

For f i n i t e gdg.nr.'s

and N o

i f Ndu Nc

N with i d e n t i t y

"2-primitive",

"simple"

i n v a r i a n t l y simple,

See P i l z - S o

n o t every N

c a n be embedded

€ a oc a n

(3). b e embedded i n

188

57

TRANSFORMATION NEAR-RINGS

T h i s c h a p t e r c o n t a i n s r e s u l t s on n e a r - r i n q s o f g r o u p mappings ( t h e "elements o f near-ring-theory''

o f 4.62)

r i n g s which a r e r e l a t e d t o these (97 d ) ) .

and o f n e a r -

We w i l l m a i n l y b e

concerned w i t h t h e i d e a l s t r u c t u r e o f t h e s e c l a s s e s o f n e a r rings

.

We s t a r t w i t h

fixed-point-free o f 2.50

where

M:(T)

i s shown t o f u l f i l l

i f f H h a s f i n i t e l y many o r b i t s o n

r.

We a l s o a n s w e r t h e q u e s t i o n , and

M!

r is

just H itself.

a l l conditions I n t h i s case,

M:(r)

i s s i m p l e and t h e f i n i t e t o p o l o g y on

Mi(T)

H i s some

group o f automorphisms of t h e ( a d d i t i v e l y

r.

w r i t t e n ) group

= (MH(r))o,

= MHU{61(l')

M:(T):

i s discrete.

M;

under which c o n d i t i o n s

(rl)

1

( r 2 ) are isomorphic, using semi-linear transformations 2 as i n r i n g t h e o r y . The a u t o m o r p h i s m g r o u p o f t h e M : ( r ) - q r o u p i n b ) we show t h a t f o r Mo(r)

Mo(I')

Turning t o

are equivalent:

a l l c o n d i t i o n s o f 2.50,

g e n e r a t i o n and f i n i t e n e s s o f

Mo(I')

a r e shown t o b e t h e

ones a r e a l l

(o:y)

(yer")

easy t o c h a r a c t e r i z e . (if

Ir(

2)

M(r)

r.

ACCL,

the followina

DCCL, f i n i t e

A l l minimal l e f t i d e a l s o f

(o:I*\{y?)

for

ycr*.

The m a x i m a l

a n d some o t h e r s , w h i c h a r e l e s s

C o n c e r n i n q i d e a l s we show t h a t are simple near-rinqs.

Mo(r)

and

There a r e no s u b n e a r -

Mo(r) a n d M ( r ) . I n c ) we s t u d y m a i n l y E ( r ) . E(r) i s 2 - p r i m i t i v e o n r i f f r i s i n v a r i a n t l y s i m p l e . I n t h i s case, E(r) = M o ( r ) . S i m i l a r A(r) a n d I ( r ) . E(r) h a s a l l results are obtained for

r i n g s s t r i c t l y between

c o n d i t i o n s o f 2.50 i f

r

i s t h e d i r e c t sum o f f i n i t e l y many

minimal f u l l y i n v a r i a n t subqroups. % i n v a r i a n t l y simple. Aut I ( r ) = r.

E(r)

i s simple i f f

r

i s

F i n a l l y we s t u d y n e a r - r i n q s o f p o l y n o m i a l s R[x] o r r[x] over a commutative r i n q R w i t h u n i t y o r a g r o u p r and t h e i r a s s o c i a t e d near-rings P(R), P ( r ) o f p o l y n o m i a l f u n c t i o n s . We show t h a t

7a Mfi(r)

iff R i s a finite field.

P ( R ) = f4(R)

i s simple i f f F i s i n f i n i t e . f u n c t i o n on F } ,

If

F =

If F i s f i n i t e but char

z2,

Z2,

but

and hence q u i t e w e l l - k n o w n ,

that

char F

2.

P(P)

i s simple i f f R i s a f i e l d

$; Z2

holds i f f

= M(T)

FLXJ

{ p ~ F [ x ] I p induces t h e zero

is a

e a c h i d e a l o f F[x]

Frx]

lrl>l.

F[x]

t h e r e a r e e x a c t l y 2 maximal i d e a l s .

F $. 2 ,

ring-ideal o f

-r [ x ]

If F i s a field,

If F i s finite,

c o n t a i n s e x a c t l y one maximal i d e a l :

iff

189

r

and

r[x]

B2

=

provided

or

r

i s simple i s a finite,

n o n - a b e l i a n s i m p l e g r o u p . We c o n t i n u e a n d c l o s e w i t h n r . ' s of

polynomials and polynomial f u n c t i o n s on R-groups.

Now we a r e g o i n g t o d e c o m p o s e

MH(r),

(r,+).

p o i n t - f r e e automorphism group o f I n c o n t r a s t t o 3.43,

w h e r e H i s some f i x e d -

we f i r s t d e c o m p o s e t h e i d e n t i t y a n d t h e n MH(r). B e f o r e d o i n q s o , we h a v e t o

get a decomposition o f f i x some n o t a t i o n .

7.1

approach i s i n Holcombe ( 4 ) .

A categorical

NOTATION T h r u u g h o u t t h i s s e c t i o n 7 a ) l e t

r

be a non-zero

group and H a f i x e d - p o i n t - f r e e automorphism qroup o f r . r = I o l u UBi b e a p a r t i t i o n o f r i n t o a d i s j o i n t ic1

Let

union o f orbits o f

r

under H.

Denote ( f o r i E I ) b y ei (4.28(a)) Of M H ~ { a( }r )

ei(Y) =

(SO

ei

t h e u n i q u e l y d e t e r m i n e d map with

Y

for

ycBi

o

for

y&Bi

i s l i k e the identity i n

M o r e o v e r , we a b b r e v i a t e s o m e t i m e s o r s i m p l y by M .

Bi

and o e l s e w h e r e ) .

MHJ(al

(r)

by

M;(r)

07

190

7.2

TRANSFORMATION NEAR-RINGS

As promised,

THEOREM ( B e t s c h ( 7 ) ) .

(a) IeiIiEI) ( b ) A l l Mei

let

M.

=:

M:(T)

i s a s e t o f orthogonal idempotents.

n

=

Li

=:

(0:B.) j+i J

a r e l e f t i d e a l s and R-

groups o f type 2 w h i c h a r e M-isomorphic t o

r

and f u l f i l l

f o r Li.

ZFI TI

(c) M

Li.

i E I

(d) If

1 Li

L: =

=

I*Li iE1

i e I

then

f i n i t e l y many o r b i t s o n

r;

i f f H has

L = M:(r)

i n t h i s case,

1 ei.

1 =

i E I

(e) Every non-zero i n v a r i a n t subnear-rinq S o f M contains L. Proof.

( a ) i s e s t a b l i s h e d b y an e a s y c o m p u t a t i o n .

4.28(aj,

Conversely i f

T h e map

[I ( o : B j ) . j+i 11 ( o : B .J) t t i e n j+i

Mei 5

whence m

E

f y : Mei

+

mei

+

r

m = me.EMei, 1

i s an M-epimorphisrn f o r

meiy

y€Bi.

Ker f Ker f

Y

= Mei n ( 0 : ~ ) . B u t

fl

=

(o:Bj)

=

IO),

(o:y)

and

fY

j E I

b e an M - i s o m o r p h i s m f r o m i s 2 - p r i m i t i v e on

to

L~

?r PM

r

Li

by 4.52(b),

= (o:Bi),

to

r.

so

i s unmasked t o Since

M:(r)

t h e same a p p l i e s

r.

( c ) i s s e t t l e d by t h e M-isomorphism s e n d i n q m i n t o (...,mei,...).

f:

M:(r)

+

Jl Li is1

191

7a M a ( r )

(d)

n

Li =

ieI

8

Li

h o l d s i f f I i s f i n i t e . Now a p p l y

ic1

2.30.

( e ) I f i e I a n d S as d e s c r i b e d , ( o : L i ) = ( o : r ) by 1 . 4 5 ( b ) . S u p p o s e t h a t Lin S = ( 0 1 . Then S L i c L i n S = = {Ol, s o S F ( o : L i ) = ( o : r ) = l o ) , a c o n t r a d i c t i o n . So L i n S { O l , whence L i n S = L i by ( b ) , s o a l l L j c S a n d t h e r e f o r e LsS.

+

R a m a k o t a i a h ( 7 ) showed t h a t t h e L i ' s a r e e x a c t l y a l l m i n i m a l l e f t i d e a l s o f M:(r). He a l s o c h a r a c t e r i z e d i n t h i s p a p e r a l l maximal l e f t i d e a l s ( a l s o i n terms o f t h e f i n i t e t o p o l o a y i n 4.26). The f o l l o w i n g r e s u l t g e n e r a l i z e s Theorem 5 . 7 o f B e t s c h ( 7 ) ( n o t a t i o n a s a b o v e ) . C f . Ramakotaiah ( 3 ) . 7.3

C O R O L L A R Y The f o l l o w i n q s t a t e m e n t s s r e e q u i v a l e n t :

(a) M = L. ( b ) MM f u l f i l l s a l l c o n d i t i o n s o f 2 . 5 0 . ( c ) H h a s f i n i t e l y many o r b i t s on r . Proof: apply 7.2. 7.4

C O R O L L A R Y I f M f u l f i l l s the c o n d i t i o n s o f 7 . 3 t h e n M has no n o n - t r i v i a l t w o - s i d e d i n v a r i a n t s u b n e a r - r i n g s . I n p a r t i c u l a r , M i s simple.

T h i s f o l l o w s from 7 . 2 ( e ) ( s i m p l i c i t y can a l s o be d e r i v e d from 4 . 4 6 ) . More on s i m p l i c i t y o f M:(r) c a n b e f o u n d i n Meldrum (12).

7.5

C O R O L L A R Y ( B e t s c h ( 7 ) ) . Let t h e n o n - r i n q N E f l o n % , be 2 - p r i m i t i v e on r . T h e n t h e f o l l o w i n a c o n d i t i o n s a r e equivalent:

( a ) NN i s " f i n i t e l y completely reducible" ( a l l conditions of 2.50 are v a l i d ) . and G h a s f i n i t e l y many o r b i t s on r . ( b ) N 1 M:(r) ( c ) N 2 ll;(r)

a n d t h e f i n i t e t o p o l o n y on

Mi(r)

i s discrete.

07

192

TRANSFORMATION NEAR-RINGS

Proof. 2 . 5 0 , 4.60, 7.3 a n d a.29. There i s an i n t i m a t e c o n n e c t i o n between t h e l a t t i c e s a l l H - i n v a r i a n t s u b g r o u p s o f r and f(M) = ( S S M I S M S S I " r i g h t - i n v a r i a n t su b n e a r - r i nqs " o f 11) We mention w i t h o u t p r o o f :

.

7 .6

yH(r) o f (the

T H E O R E M ( L a x t o n ( 2 ) , Be t sc h ( 7 ) , 9 8 ) . I f H has S E I N o r b i t s on r* t h e n t h e map f : g H ( r ) + Y(M) i s a l a t t i c e isomor A

phism w i t h Moreover,

+

(A:r)

+ q H ( r ) given

f-l: f(M)

[ ( j H ( r ) =l

\!f(M)I

S

by

S

+

sr.

2'.

Holcombe ( 6 ) s u g g e s t e d t h e f o l l o w i n g

7.7

DEFINITION Every c h o i c e B : = { b i l i ~ I ) o f r e p r e s e n t a t i v e s i s c a l l e d a H-base. dirnH(r): = I B I i s c a l l e d t h e H-dimension o f r . biEBi

T h i s comes from t h e e a s y - t o - p r o v e

7.8

( c f . 4.28)

PROPOSITION (Holcornbe ( 5 ) ) .

(a)

ycr*

( b ) Each map

3

3

icf B

+

r

hcH: y = h ( b i ) .

can be u n i q u e l y e x t e n d e d t o a map

MH(r)m Holcombe f o r m u l a t e d 7 . B ( b ) more g e n e r a l l y : "Every m a p B + I", where r ' i s a n o t h e r g r o u p o n which H o p e r a t e s ( r ' i s a n "H-group") c a n be e x t e n d e d t o a u n i q u e map r * f ' which commutes w i t h H " . So r i s i n a k i n d " f r e e " o n B . H o p e r a t e s on $ ( r ) i n a n a t u r a l way. From 7 . 8 ( b ) we g e t 7 .9

T H E O R E M (Holcombe ( 5 ) ) . I f

d i m H ( M H ( r ) ) = (S+l)'-l.

d i r n H ( r ) = SEIN t h e n

7a M;(r)

7.10 REMARK I n ( 7 ) , pp. 92-97,

Betsch studied t h e d i s t r i b u t i v e

0

D:

elements

193

= (MH(T))d

bl:(r)

of

and "monomial m a t r i c e s "

matrices over

D

which c o n t a i n i n each column

a t most one non-zero e n t r y

-

cf.

over D (i.e.

i f H h a s f i n i t e l y many o r b i t s o n

r.

isomorphic t o t h e monoid o f a l l

fcEnd(r)

with all

haH.

D i s

also Frohlich (3)).

shown t o b e embeddable i n t o t h e s e m i q r o u p

(End (M,+),o)

(D,.)

i s antiw h i c h commute

Deskins (2).

Cf.

M;

Now we c o n s i d e r t h e f o l l o w i n g p r o b l e m : w h e n a r e

(rl)

and

1

(r2) isomorphic ? 2 F o r r i n g s , t h i s p r o b l e m i s s o l v e d i n t h e f G l l o w i n o way ( s e e

M:

i s an homomorphism rings

(V1,V2

v e c t o r spaces o v e r t h e d i v i s i o n r i n o

r e s p e c t i v e l y ) t h e n h i s an isomorphism i f f t h e r e

D1,D2,

i s some 1 - 1 - s e m i - l i n e a r

!+EHomD ( V l , V l ) :

t: V 1

transformation

V2

+

such t h a t

h(4) = t 0 t - l .

1

We f o l l o w i n some w a y J a c o b s o n ' s d i s c u s s i o n a n d s t a r t w i t h

7.11 THEOREM ( H o l c o m b e (4), R a m a k o t a i a h ( 6 ) ) . (as usual) fixed-point-free

r

M:l(r)

then

(r)

= M;

H1EH2

If

are

groups o f automorphisns on H1 = H E .

<->

2 P r o o f . We o n l y h a v e t o s h o w "->". Suppose t h a t

H1=H2,

and c o n s i d e r t h e o r b i t s

ycr*

with respect t o Clearly

3

a n d t a k e some

H1,H2.

but

h2(y)€B2,

Then

fixed-point-free, 4.28(a)

0

B2 = H2y.

and

(since otherwise

hl

= h2

since

g u a r a n t e e s t h e e x i s t e n c e o f some

ml(h2(Y)) But

containino y

B1 = H1y

so

Take

H2

i s

a contradiction).

ml(y) = h 2 ( y ) '

with

Bl,B2

h2(y)BBl

~ ~ E H ~ Ch l (Hy ) ~ := h 2 ( y ) ,

h2cH1\H2.

and

6BB1:

ml(6) =

mlcM:l(r) Hence

0.

= 0-

= ml(h2(~))

= h2(ml(y))

m EM' ( r ) = M i ( r ) . T h u s H1 2 a c o n t r a d i c t i o n . H e n c e HI

= h2(h2(y))

h2(y) = H2.

0,

since

whence

y =

0,

$ 7 TRANSFORMATION NEAR-RINGS

194

MH1(r)

7.12 REMARK Observe t h a t

= MH2(r)

(r)

M:

<->

= M:2(r).

1 7 . 1 3 COROLLARY ( R a m a k o t a i a h ( 5 ) ) .

(r)

Aut

M;P) Proof.

H':

If

=

AUtMH(r)

(r)

then

H = H'

point-free

on

7.14 DEFINITION

M:2(r2)y

M:

ri

SEHom(rl,r2)

homomorphism

MH(r)= MH,(r).

b y 7.11.

( r l ) and 1 (i = ly2).

N e x t we c o n s i d e r

H I i s by 4.52

HEH'.

shown t o b e f i x e d - p o i n t - f r e e w i t h SO

= H.

if

3

w h e r e Hi

are fixed-

i s called a semi-linear H1*

s:

W

H2

yl€.T1

S(hl(Y1)) = S(hl)(S(Y1)).

If

S

+

tl

6,

hlEH1:

s i s uniquely

d e t e r m i n e d and c a l l ed t h e isomorphism associated w i t h

-S .

w i l l a l s o speak a b o u t t h e s e m i - l i n e a r monomorphism

( Y ).

We

(S,S).

7 . 1 5 T H E O R E M ( R a m a k o t a i a h (5), f o r t h e f i n i t e - d i m e n s i o n a l

(7.7)

case a l s o B e t s c h ( 7 ) and Holcombe ( 5 ) ) . homomorphism

0

(rl)

f : PIH

+

tl;

A near-rino i s a n i s o m o r p h i s m <=>

(r2)

1 2 <-> t h e r e i s some s e m i - l i n e a r i s o m o r ! J h i s m mcMil(r,): f(m) = Som0S-l.

w

Proof.

We a b b r e v i a t e

M:

(ri)

by

Mi

S:r,1

(iE{1,2)),

+

r2

with

and

i keep t h i s n o t a t i o n f o r i. <-:

If

(S,s)

t h e n t h e map

m o r p h i sm 1 i s an isomorphism:

i s a semilinear i f : M1 m

+

M2

+

SmS-l

T o s e e t h i s we f i r s t s h o k t h a t f maps

M1

Clearly

and

f(m)cMo(r2).

So t a k e

y2cr2

into

+

r2

M2.

h2cH2.

7a M g ( r )

3

3

ylcrl

hlOm

hlEHl:

= h2(S(m(yl)))

= y2 A s(hl)

S(yl)

So

= mohl.

195

S(hl(m(Y1)))

= s(hl)(S(m(Y1)))

= h2(SomoS-1(S(yl)))

On t h e o t h e r h a n d , w e

Also,

= h2.

=

= h2(SomoS-l(y2)).

can compute

S(hl(m(yl)))

i n a

second way: S(hl(m(Y1)

1)

= S ( m ( h 1 (Y1) 1) = ( S 0 m o S - l ) ( S ( h l ( Y l ) )

h2o(SomoS-l)

s o we g e t

y2cr2,

(SomoS-l)oh2;

=

=

(S0moS-l)( h 2 ( Y 2 ) ) .

= (somos-l)(s(hl)(s(Yl)))

T h i s h o l d s f o r each

1

SomoS-1EM2.

hence

I t i s e a s y t o show t h a t f i s an i s o m o r p h i s m . ->:

Assume now t h a t

(a)

M1

f : M1

M2

+

i s an isomorphism.

can be c o n s i d e r e d t o b e 2 - p r i m i t i v e on

M1Xr2

since

+

+

(“19Y2)

+

f ( m 1 )Y 2

( b ) We s h o w t h a t t h e r e i s a n i s o m o r p h i s m -1 mlEM1: f(ml) = SomloS

M1

By 7 . 2 ( b ) ,

say).

rl

So b y ( a )

= ml(S(yl))

and

W

rl r2 W

s(ml(vl))

mlEM1:

whence

= f(ml)(S(yl)),

r2

By ( a ) ,

4.56(a)

a r e M1-isomorphic

Ylcrl

( c ) Now we c l a i m t h a t

hl

r2.

and o n

-c

Soml

(by S , =

= f(ml)oS

= Somlo~-l.

f(ml)

Clearly

r1

contains a minimal l e f t ideal.

i s 2 - p r i m i t i v e on

assures t h a t

or

S:

.

with

M1

r2,

does t h e r e q u i r e d j o b .

r2

S0hloS-’cEnd(

commutes w i t h a l l

( b ) and 7.13

hltzH1:

SohloS-1cH2.

r2). I f f ( m l ) = r n 2 we g e t f r o m

mlcM1.

m20(Snh,oS-’)

=

f(ml)o(SohloS-’)

=

= S a m , ~ S - ’ ~ S ~ h l o S - ’= S 0 r n l ~ h l ~ S - l = SohlomloS - 1 = ( S O ~ ~ O S - ’ Henc’e ) ~ ~ ~S o. h l 0 S - ’

( d ) N e x t we o b s e r v e t h a t isomorphism, procedures.

s : H1 hl

EAutM

+

+

2

(r)

H2

=

= ti2.

i s an

SohloS-’

a f a c t which can be seen b y t h e usual

196

§ 7 TRANSFORMATION NEAR-RINGS

( e ) F i n a l l y , we h a v e t o c h e c k t h e s e m i - l i n e a r i t y c o n d i t i o n 7.14

some

hlEH1

for

t a k e some

(S,s):

.

= AutMl(r)

= (Soh1oS-l) (S(Y1) 1

Then

S(hl(yl))

= s ( h 1 ) (S(Y1)

and

yl~rl

=

1.

The p r o o f i s now c o m p l e t e . From t h a t we c a n d e d u c e i n t e r e s t i n g r e s u l t s a b o u t t h e automorphism o f n e a r - r i n g s o f t h e t y p e 7.16

COROLLARY ( R a m a k o t a i a h ( 5 ) ) . then S on

fEAut

r

M

with

<=>

M:(T):

M = Mi(r)

If

and

fEEnd(M)

t h e r e e x i s t s a s e m i - l i n e a r automorphism

f ( m ) = SomoS’l

for all

mcM.

M 2 = M.

M1

T h i s f o l l o w s f r o m 7.15 b y s p e c i a l i z i n q

7.17 THEOREM ( R a m a k o t a i a h ( 5 ) ) . L e t G b e t h e p r o u p o f s e m i l i n e a r a u t o m o r p h i s m s o n r a n d G ‘ : = Aut M:(l‘). % Then G / G n H G’,

-

Proof. Define some

a:

as f o l l o w s :

G’

-*

M:(r)

fEAut

(by 7.16).

G

with

if

there i s

SEG, f(m)

mEMi(r):

= SornoS-’

Observe t h a t t h i s f i s unique. P u t

a ( S ) : = f.

F i r s t we p r o v e t h a t a i s a homomorphism. take

S,TEG.

Then f o r a l l

a(ST) = :

g,

a ( S ) =:

mEM:(r)

q(m)

fl,

= (ST)m(ST)-l

= S T ~ T - ~ S= - s~ f 2 ( m ) S - l = f 1( f 2( m 1 )

Hence

To do t h i s , a ( T ) =:

g = flf2# i m p l y i n g t h a t

f2.

=

.

a(ST) = a ( S ) a ( T ) .

Now a i s a n e p i m o r p h i s m : i f f E G ’ then there i s some S E G w i t h b mcM:fr): SornoS-’ = f f m ) . Thus a(S) = f.

F i n a l l y we c o m p u t e K e r a . I f SEKer a t h e n b m E M i ( r ) : m = i d ( m ) = SmS”. So

SEAut

$(r)

(I-)

Conversely,each T h i s shows t h a t

= H

(7.13).

element o f

6/Gn

%

G n H G’.

Hence is i n

SEGnH.

Key a.

7b M ( r ) and M,(r)

197

U is C l a y ( 1 4 ) d e t e r m i n e d the g r o u p U o f u n i t s o f ME(r): isomorphic t o the wreath p r o d u c t o f G w i t h t h e symmetric group u n d e r G . He a l s o p o i n t e d on t h e i n d e x s e t I o f t h e o r b i t s o f o u t t h e i n t i m a t e c o n n e c t i o n b e t w e e n U and t h e g e n e r a l l i n e a r g r o u p s i n l i n e a r a l g e b r a . He a l s o d e f i n e d a " d e t e r m i n a n t

f u n c t i o n " on U. S e e a l s o p . 3 7 6 . I n ( 3 ) , R a m a k o t a i a h showed t h a t M i ( T ) i s a r i n g i f f dimensional vector space over t h e skew-field H .

r

is a 1-

M:(r) f o r H s E n d ( r ) i s s t u d i e d i n 99 h ) . Don't f o r g e t t o read t h i s c h a n t e r a s we1 1 !

b)

M,(r)

AND

M(T)

T h e r e a r e a l o t o f t h i n g s w h i c h we c a n g e t by s p e c i a l i z i n g H = { i d ) i n t h e p r e v i o u s s e c t i o n . By 1 . 1 3 i t i s j u s t i f i e d t o consider primarily Mo(r). We s t a r t by c o n s i d e r i n g l e f t i d e a l s i n M o ( r ) . C f . Holcombe ( 4 ) . 7.18 C O R O L L A R Y ( H e a t h e r l y ( l ) , ( 4 ) , Ramakotaiah ( 7 ) ) .

(a) If for then

dcr,

e

{e616Er*)

6

:r

+

r

cf. a l s o Frohlich ( 3 ) ,

with

{

ed(y) =

d

if

y = 6

o if yS.6 i s a s e t of orthogonal idempotents.

( b ) A l l Mo(r)e6 =: L6 = (o:r\C6) a r e l e f t i d e a l s and M,(r)-groups o f t y p e 2 ( h e n c e minimal M o ( r ) - s u b g r o u p s ) g e n e r a t e d by e d a n d M o ( r ) - i s o m o r p h i c t o r.

If

then

iff

r

is finite.

( e ) Every non-zero i n v a r i a n t s u b n e a r - r i n a S o f c o n t a i n s L.

Proof: 7.2.

Mo(r)

§ 7 TRANSFORMATION NEAR-RINGS

198

Hence Mo(r) i s a 2 - p r i m i t i v e n r . o n T w i t h i d e n t i t y a n d a minimal l e f t i d e a l ( s e e 5 4 d 3 ) ) . 7 . 3 and 7 . 1 8 g i v e 7.19 C O R O L L A R Y ( H e a t h e r l y ( 3 ) , M . Johnson ( 6 ) ) . The f o l l o w i n g are equivalent: (a)

Mo(r)

1'

=

L

= L.

6E r *

(b) Mo(r)

has D C C L .

( c ) Mo(r) h a s A C C L . (d)

Mo(r)

i s completely r e d u c i b l e i n t o f i n i t e l y many

minimal l e f t i d e a l s .

( e ) Mo(r) (f)

Mo(r)

(9)

r

has o n l y f . g .

l e f t ideals.

i s finite.

is finite.

Mo(r)

C l e a r l y (by 7 . 4 )

i s s i m p l e i n t h i s c a s e . However, we

w i l l extend t h i s r e s u l t t o t h e a r b i t r a r y c a s e ( 7 . 3 0 ) . B u t f i r s t we examine t h e l e f t i d e a l s more c l o s e l y ,

7 . 2 0 T H E O R E M ( H e a t h e r l y ( 3 ) ) L e t L be a l e f t i d e a l of (a)

V

( b ) If

A:

= I 6 ~ r l L 6=

Proof. ( a ) I f

b

or

YEr: L y = I01

Ly

+

I01

rl

Ly = -/=

d

then

mcMo(r): m R E L , whence f o r t i o r i Ly = r .

Mo(r).

r. then

3

EEL:

1L

6 ~ E A Ey

5

L.

4 0.

ImLyJmcMo(r)) =

But

r

and a

( b ) I t s u f f i c e s t o show t h a t i f L6 4 I o l t h e n L6%L. This t r S v i a l l y holds f o r l r l r 2 s i n c e then l M o ( ~ ) 1 ~ 2 .So assume t h a t 11-123.

Suppose t h a t k 6 ~ L 6 ( ~ E A ) . Denote E , ( 6 ) = : 8 . Choose some I1cL w i t h t ( 6 ) = 6 = E 6 ( 6 ) . This i s p o s s i b l e by ( a ) . e y = e Take m.nEMo(r) with m(y) = 0

YS;O

199

7b M ( r ) and M,(r)

(so

=t 6

Y

m = ee

d h e -

n(y)

1: =

Then

-

of 7 . 1 8 ( a ) ) ,

and

b u t for

n(6) = 0,

L(Y)>.

and for

m(n+l)-mnEL

y =/= 6

we q e t

t(v)

= m(n(y)+k(v))-m(n(v)) 0-0 = 0 L6(y), w h i l e T(S) = m ( n ( a ) t ! L ( & ) ) - m ( n ( 6 ) ) = m ( e ) - m ( o ) =

e

=

=

So

e,(s).

-

l6 = R E L

a n d we a r e t h r o u g h .

7 . 2 1 COROLLARIES ( B l a c k e t t ( I ) ,

( a ) The L 6 ' s of M o ( r ) .

(6Er*)

H e a t h e r l y ( 3 ) , PI. J o h n s o n ( 1 ) , ( 3 ) ) .

a r e e x a c t l y a l l minimal l e f t i d e a l s

( b ) Every l e f t i d e a l of Mo(r) which i s c o n t a i n e d i n L 1 L i s i s o m o r p h i c t o a d i r e c t sum o f s u i t a b l e 6Er*

L * k ( c ) L c a n n o t be a n o n - t r i v i a l d i r e c t summand o f Proof. -

Wo(r).

( a ) i s a c onse que nc e of 7 . 2 0 .

( b ) f o l l o w s from ( a ) and 2 . 5 5 . ( 6 )

If

L i L ' = Mo(r)

L' dL Mo(r) i s suc h t h a t

L ' $. {Ol t h e n ( b y 7 . 2 q ) t r a d i c t i on.

3

6Er*:

Lg

5 L n L ' ,

and

a con-

S i n c e we have been v e r y s u c c e s s f u l i n d e t e r m i n i n g a l l minimal l e f t i d e a l s o f Mo(r), we t u r n t o maximal l e f t i d e a l s . ble r e a d i l y g e t some of them: 7 . 2 2 E X A M P L E ( H e a t h e r l y ( l ) , ( 3 ) ) . For e v e r y

i s a maximal Mo(r)-subgroup i d e a l ) o f Mo(r).

-Proof.

By 3 . 4 ( e ) a n d 7 . 2 we qe? N / ( o : v )

where

N =

y~r*,

(o:y)

( h e n c e a l s o a maximal l e f t

Mo(r). Now a p p l y 3 . 4 ( h ) .

T

=N

Lyy

07

200

TRANSFORMATION NEAR-RINGS

B u t woe: 7.23

PROPOSITION ( H e a t h e r l y ( 3 ) ) .

If

r

i s i n f i n i t e then L

i s c o n t a i n e d i n a maximal l e f t i d e a l o f Mo(r), b u t n o t i n a n y ( o : y ) (yEr*). (as i n 7.18(d))

-

P r o o f . The f i r s t s t a t e m e n t i s s e t t l e d b y 1 . 5 3 ( a )

Mo(r)cgl Lyc(o:y)

since

a n d b y 7 . 1 8 ( d ) . Now i f L ~ ( o : y ) t h e n a n d e y ( y ) = 0 , w h e n c e y = 0.

So t h e r e a r e o t h e r m a x i m a l l e f t i d e a l s b e s i d e t h e

(o:y)'s,

which i m p l i e s more t r o u b l e f o r us. But, f o r t u n a t e l y ,

M.

Johnson

h a s s o l v e d t h i s p r o b l e m . See a l s o R a m a k o t a i a h ( 7 ) a n d ( 8 ) .

7 . 2 4 NOTATION F o r

mEMo(r)

call

CyErlm(y)

= 0 1 = : Z,

(the

"zero s e t o f m"). We s t a t e w i t h o u t p r o o f 7.25

PROPOSITION ( M . J o h n s o n ( 3 ) , ( 6 ) ) . i n Mo(r). T h e n

( a ) EEL <=> (b) 7.26

(o:ZE)eL.

3

al,L2cL

NOTATION L e t

= IL

L e t L be a l e f t i d e a l

E E L : ZEln Z E 2 = Z E .

$(r)

b e f o r t h e moment

=

x

=

AE Mo(r)l \ & E L : ZL i s i n f i n i t e ] .

Now we a r e i n a p o s i t i o n t o c h a r a c t e r i z e t h e m a x i m a l l e f t ideals o f 7.27

Mo(r).

THEOREt4 ( ? 4 . J o h n s o n ( 6 ) ) . L e t L b e a ? e f t i d e a l o f M o ( r ) . L i s m a x i m a l <=> ( 3 y E r + : L = ( 0 : ~ ) )v ( L i s m a x i m a l i n

t(r)). P r o o f . =>: that

Suppose t h a t yEr: L ( 0 : ~ ) . Assume m o r e o v e r E E L : Z1, i s f i n i t e . F o r YE:T c o n s i d e r e, of

3

7.18(a).

S i n c e L i s no

(o:y),

all

eyEL

b y 7.20(b).

7b

M(r)and M,(r)

201

Assume t h a t l z t l = n E N . We c l a i m t h a t 1 kcL: z k ( 0 1 a n d p r o v e t h i s by i n d u c t i o n on n. ( a ) T h i s i s t r i v i a l f o r n = 1. ( b ) Suppose t h a t n > l and t h e s t a t e m e n t h o l d s f o r n-1. Now t + e E L and Zll+e = Z,\{y1, so Y Y IZllte 1 = n-1. Y S o L = Mo(r) by 7 . 2 5 ( a ) s i n c e ( o : Z k ) = ( o : o ) = LEL(r) = Mo(r)cL. T h i s i s a c o n t r a d i c t i o n . Hence a n d s i n c e L i s a maximal l e f t i d e a l , L i s maximal i n $.(r).

=

I n v i e w o f 7 . 2 2 i t s u f f i c e s t o c o n s i d e r niaximal e l e m e n t s o f $. L e t K qll Mo(r) p r o p e r l y c o n t a i n L . T h e n 3 kcK: l Z k l = n e N . A g a i n we use i n d u c t i o n t o show t h e e x i s t e n c e o f some klcK w i t h = {ol, zkl n = 1 is ti-ivial again. <-:

t a k e some y e Z k , y =/= 0. T h e n e E L = Y Y = (0:n{y1) by 7 . 1 8 ( b ) . Lt(o:ll{y})c& SO L t ( o : I " \ { y ) ) = L s i n c e L i s maximal i n $ . Hence e E L and so k+e E K w i t h IZkte I = n - 1 a s b e f o r e . Y Y Y Mo(r) a n d L i s shown t o be a maximal l e f t Hence K i d e a l o f Mo(r). If

Since

g(r)

n>l,

i s empty i f

7.28 COROLLARY Let

r

r

is finite

b e f i n i t e . Then

( a ) The m i n i m a l l e f t i d e a l s o f L,IS.

we g e t f r o m

Mo(r)

are exactly the

( Y E P ) .

( b ) The maximal l e f t i d e a l s o f

Mo(r) a r e e x a c t l y t h e

( 0 : ~ ) ~ s( y E r * ) .

( c ) The l e f t i d e a l s o f

Mo(r)

a r e exactly the

(o:A)'s

(Acr). I n S c o t t ( 1 4 ) i t i s shown t h a t 7 . 2 8 ( c ) a l s o h o l d s f o r f . g . l e f t i d e a l s i n a n y Mo(r). These l e f t i d e a l s a r e even generat,ed by a single element.

87

202

TRANSFORMATION NEAR-RINGS

We c a n g e n e r a l i z e 7 . 2 8 ( c )

7 . 2 9 THEOREM ( M .

e a s i l y t o t h e i n f i n i t e case:

Johnson ( 3 ) ) .

L

Every

Mo(r)

Ak

i s t h e sum o f

annihilator l e f t ideals. T h i s i s a consequence o f 7.25,

L =

for

1

(o:Z,).

EEL

Now we e x t e n d o u r r e s u l t t h a t

MG(r) i s s i m p l e i f

i s finite

(7.19). 7 . 3 0 THEOREM ( B e r m a n - S i l v e r m a n

(Z),

Meldrum ( 1 3 ) ) .

r.

Mo(r) i s s i m p l e f o r e v e r y g r o u p

r

P r o o f . We may a s s u m e t h a t

I

-4

{Ol.

If

I 9 Mo(r),

i s i n f i n i t e . Take

rnEMo(r),

call

I{m(y)lyErll

=:

rk(m)

t h e r a n k o f m. ( a ) F i r s t we r e m a r k t h a t f o r e a c h maximal s e t

Aer

yEr*

there i s a

with respect t o the property that

( A t v ) = kf. T h i s f o l l o w s b y Z o r n ' s lemma.

A

n

( b ) The s e t A o f ( a ) s a t i s f i e s

IAl =

Irl.

To s e e

consider A "(Aty). A u ( A t y ) = T, t h e r e i s n o t h i n n t o p r o v e . 6 ~ ri s n o t i n A u ( A + y ) , consider A ' : = Ay{bI. Since A i s maximal, A ' n ( A ' t y ) 4 8. But this,

If If

A ' n (A'+Y) = ( A uI61) n ((Aty) u I6+yI) = A n{6+y1 b y t h e a l q e b r a o f s e t s and t h e assumptions. So 3 ~ E A : a = 6 t y , w h e n c e ~ E A - Y . Thus

r

= IA-yl

lrl

= A cr(Aty)v(A-y). and t h e f a c t t h a t

Since

r

\ A [ = IAtyl =

i s i n f i n i t e we q e t

= 1AI.

( c ) Now we s h o w t h a t I c o n t a i n s some i w i t h rk(i) =

\rl.

Take

ycr*

and A as i n ( a ) and ( b ) .

7b M ( r ) and M,(r)

Since I f

b

{O}, t h e r e i s some y l & r w i t h

i: = f o ( i d + h ) - f o i d s 1 ,

Then

acA\{'o):

that

rk(i) =

y

:=

j(y,) f 0.

since h = j"gcI = -a.

i(a) = f(a+h(a))-f(a)

r k ( i ) 2 IA\{oIl

Hence

203

= IAl =

Irl,

thus proving

Irl.

( d ) Now we p r o v e t h e t h e o r e m . L e t m be a r b i t r a r y

Since

cMO(r).

r k ( r n ) I Il'l

= rk(i)

(ia s i n ( c ) l t h e r e i s some i n j e c t i v e map

m(r)

f:

with y

*

+

i(r).

For each

i ( y ' ) = f(m(y)).

y&r

b y 9. We may a s s u m e t h a t

y'

choose one

y l ~ r

Denote t h i s correrpondence f(o)

5:

o

and

i ( o ) = 0. D e f i n e t h e ma:,

r(6):

m(y) =

r as follous: if

1

y&r: 6 = f(m(y))

+o

otherwise

Hence r o i o g = m a n d mE1. T h i s s h o w s t h a t I = Mo(r) a n d t h e p r o o f i s f i n i s h e d .

87

TRANSFORMATION NEAR-RINGS

O f c o u r s e one i s l e d

t o the question whether

204

M(r)

i s simple

o r n o t . We t r y t h e s i m p l e s t e x a m p l e :

7 . 3 1 EXAMPLE

ll(n2)

has one n o n - t r i v i a l

M(r)

But t h i s i s t h e o n l y case t h a t

ideal,

namely

Mc(r)

i s n o t simple:

7 . 3 2 THEOREM ( B e r m a n - S i l v e r m a n ( 2 ) ) . T h e r e a r e no p r o p e r s u b n e a r - r i n g s P r o o f . L e t N be a s u b n e a r - r i n q o f L e t n be

rn

3

eMM0(r).

Mo(r)

between M(r)

with

If y i s arbitrary

d e n o t e s t h e map w h i c h i s c o n s t a n t

Y

rnoEMo(I')

and

m

Y

with

m,(n(o))

= -nAtn'EN

showing t h a t

N

2

(where

= y.

Er

m n =:

But

M c ( r ) . Hence N

0

C

=

M(r).

N=M,(T).

= y

n' = nAtn'

and

and i f then n'EN,

as i n 1 . 1 3 ) ,

Mc(r).

7 . 3 3 THEOREM ( B e r m a n - S i l v e r m a n (2), M e l d r u m ( 1 3 ) ) . M(r) i s s i m p l e i f f \ r \ =+ 2.

M(r). Then Io: = InYo(r) ( 2 . 1 8 ) i s an i d e a l i n Mo(r). H e n c e I,: = I01 o r I. = M o ( r ) . I f I. = {Ol then IFMC(r). Examininq 1 . 2 7 13) shows t h a t I = I 0 1 i f Irl =/= 2. I f I, = Mo(r) t h e n I = M o ( r ) o r I = M ( r ) by 7 . 3 2 . S i n c e M o ( r ) M c ( r ) ~ M c ( r ) , Mo(r) $I M ( r ) . It r e m a i n s I = M(r).

P r o o f . L e t I b e an i d e a l o f

Cf.

A d l e r ( 1 ) and B l a c k e t t ( 1 ) .

O t h e r s u b s t r u c t u r e s have a l s o

been c o n s i d e r e d . We m e n t i o n w i t h o u t p r o o f :

7 . 3 4 T H E O R E M ( B e r m a n - S i l v e r m a n ( 2 ) ) . The o n l y t w o - s i d e d i n v a r i a n t s u b n e a r - r i n q s o f M(r) a r e M c ( r ) = I f E M ( r ) l r k ( f ) = 1 1 , {fcM(I')lrk(f)
M,(T).

7b M ( r ) and M,(r)

205

7.35 R E M A R K S e e H e a t h e r l y ( 3 ) f o r a d i s c u s s i o n o f t h e r i q h t ideals o f

Mo(r).

7.36 COROLLARY By 7.6, t h e l a t t i c e s o f a l l s u b q r o u p s o f a f i n i t e group

r

and o f a l l r i g h t i n v a r i a n t subnear-rinqs

Aut

M O W

Mo(r)

a l s o H e a t h e r l y (4)).

are isomorphic (cf.

7.37 COROLLARY

o f

(r)

= {id).

P r o o f . 7.13. T h e n e x t c o r o l l a r y g i v e s o n e h a l f o f 1.99 o n c e a g a i n i n a different version.

7.38 COROLLARY

Mo(r) a n d M o ( A ) a r e i s o m o r p h i c ( s a y b y f ) i f f t h e r e i s some i s o m o r p h i s m S : r + A w i t h

'd

mEMo(r):

Proof.

f(m) =

SomoS

-1

.

7.15.

We now t u r n t o t h e a u t o m o r p h i s m s o f

Mo(r).

7.39 THEOREM ( R a m a k o t a i a h (5)). A h o m o m o r p h i s m i s a n a u t o m o r p h i s m <->

v

( r ,+) w i t h Moreover, Aut

mE:Mo(

7.38 o r

7.16

Proof.

(r,+)

f :

M0(r)

+

Mo(r)

t h e r e i s some a u t o m o r p h i s m S o f

I-) : f ( m ) = SornoS-l,

- Aut 'L

(MO(r),+,.). H = lid)).

(for

See a l s o N o b a u e r ( 1 2 ) .

7.40 R E M A R K S ( a ) See C l a r k ( 1 ) f o r e x a m p l e s o f

M(r)

r

with

= Zp,

r = Z

r = I p r . P9' ( b ) See B l a c k e t t ( 4 ) , ( 5 ) a n d ( 6 ) f o r e x a m p l e s o f s i m p l e s u b n e a r - r i n q s o f M(IR) a n d M(b) a n d f o r " d e n s e " subnear-rings of

M(IR)

.

(c) Beidleman (11) proved f o r a f i n i t e oroup n ilpotent (solvable, supersolvable) applies t o (!.lo(r ) .+)

.

r

that

i f f t h e same

r

i s

87

206

TRANSFORMATION NEAR-RINGS

( d ) M a l o n e - M c Q u a r r i e (1) showed t h a t i f

r

i s torsion

(Mo(r),t)

without elements of order 2 then

i s

(uniquely) halvable. (e)

M.

J o h n r o n ( 3 ) c h a r a c t e r i z e d t h e c a s e s when l e f t i d e a l s

and normal non-abelian

M.

Johnson showed t h a t i f

r

b!e

i s a finite, and 6 . 6 ( a ) ) .

i s f i n i t e then l e f t ideals

Mo(r)

coincide i n

M,(r)-subqroups

iff

not a b e l i a n .

i s

F o r i n f i n i t e groups

3

r

if

i n v a r i a n t l y s i m p l e group (6.33

and normal

r

Mo(r).

coincide i n

Mo(r)-subgroups

know a l r e a d y t h a t t h i s happens e.4.

[r:

yEr:

C(y)]

=

o f y. Or, equivalently: group

r

r t h i s happens i f f Irl, w h e r e C(y) i s

the centralizer

t h i s c o i n c i d i n a happens f o r

i f f every normal

Mo(r)-subgroup

some

Mo(r)

of

Mo(r)

i s t h e sum o f a n n i h i l a t o r l e f t i d e a l s o f

(7.29). (f)

Mo(r), r

f i n i t e , has an i n v o l u t i o n i f f r i s n e i t h e r an elementary a b e l i a n 2-orouo n o r isomorphic t o Z3 ( S c o t t ( 1 3 ) ) .

E(r),

c)

A(r)

I(r)

AND

I n t h i s s e c t i o n we w i l l e x a m i n e s i m i l a r i t e m s f o r t h e d g n r . ' s

E(r), A(r) morphisms, for

M(r) a n d

radicals,

g e n e r a t e d by a l l e n d o m o r p h i s m s

I(r),

and

i n n e r automorphisms,

Mo(r)

(one-

automorphisms,

and two-sided

etc.).

r e s u l t s f o r t h e case t h a t

r

r,

respectively) of

As i n

ideals,

(auto-

a s we d i d simplicity,

b ) we w i l l g e t t h e b e s t

i s finite.

A l o t o f f u r t h e r i n f o r m a t i o n on t h e s e n e a r - r i n g s c a n b e f o u n d i n Meldrum ( 1 3 ) , ch.

10 and 1 1 .

See H o l c o m b e ( 3 )

for a categorical

approach. F i r s t we d e f i n e o u r o b j e c t s i n q u e s t i o n .

7.41 DEFINITION L e t

A(r)

(I(r))

r

be a group. S i m i l a r t o

E(r)

we d e f i n e

as t h e n e a r - r i n g s a d d i t i v e l y g e n e r a t e d b y

a l l automorphisms ( i n n e r automorphisms) o f

r.

207

7.42 REMARKS (a) Let

End(r),

Aut(T)

and I n n ( r )

groups o f a l l endomorphisms

E ( r ) , A(r)

and

I(T)

are d g n r . ' ~ .

A(r)

(c) S i m i l a r t o 6.4(d), f i n i t e sums

inner

r.

automorphisms) o f (b)

denote t h e semi-

(automorphisms,

Iakak

(I(r)) consists o f a l l ak = +_I and a l l a k E A u t ( r )

with

k (a,EInn(r)).

(d) I(r) 6.6(c) 7.43

5

A(r) c E(r)

t

Mo(r).

g i v e s us

PROPOSITION If .-

r

i s abelian then

E(r), A ( T )

and

I(r)

are rings. H e n c e we w i l l b e i n t e r e s t e d i n n o n - a b e l i a n 7.44

PROPOSITION The

qroups

E ( r ) - (A(r)-, I ( r ) - ) s u b g r o u p s a r e

exactly the f u l l y invariant (characteristic, subgroups o f

r. normal)

r.

P r o o f . 6.24. This implies 7.45

C O R O L LARY

(a)

E(r) i s 2 - p r i m i t i v e o n r

(b) A(T)

s im p l e

(c) I(r)

i s 2-primitive on

. i s 2-primitive on

<=+

r r

i s invariantly simple.

r

r

<==>

r

i s simple.

Now we c h a r a c t e r i z e i n w h i c h c a s e s

<==>

i s characteristically

E ( r ) , A(r)

and

I(T)

c o i n c i d e w i t h Mo(r) a n d g e t a s a g e n e r a l i z a t i o n o f r e s u l t s o f J o h n s o n (I), F r o h l i c h ( 3 ) and Maxson ( 1 4 )

208

7.46 THEOREM

87

TRANSFORMATION NEAR-RINGS

r

be a n o n - a b e l i a n , f i n i t e qroup.

Let

(a) E(r) = Mo(r) c-> (b) A(r) =

Mo(r)

r

i s c h a r a c t e r i s t i c a l l y simple.

<->

(c) I(r) = Mo(r) <=>

is invarlantly simple.

r

i s simple.

T h i s r e s u l t follows f r o m 7.45 and 6.33. O b s e r v e t h a t i f r i s invariantly s i m p l e then E(r) is a non-rinq: E ( r ) is 2-primitive o n r and can be no rinq by 4.8. T h e s a m e applies to A(r) and I(T). The l a s t theorem can be expressed a s a "purely group-theoretic" result:

7.47 THEOREM Let r be a f i n i t e n o n - a b e l i a n aroup. Then e a c h m a p r + r which f i x e s o can be expressed as a f i n i t e s u m C a i d i w i t h di€End(r) (Aut(T), Inn(r)) <=> r is invariantly s i m p l e (characteristically s i m p l e , s i m p l e , respectively). Ramakotaiah (9) c o n t a i n s information a b o u t the infinite case.

7.48 COROLLARY L e t (a) If

r

r

be as 7.47.

is characteristically s i m p l e then

A(r)

=

E(r)

=

= Mo(r).

(b) I f

r

i s s i m p l e then

I ( r ) = A(r) =

E(r)

=

Mo(r).

Thus E(T), A(r) and I ( r ) can b e viewed to b e characterized by 7 b ) if r is a s i n t h e c a s e s o f 7.46. The next s t e p is to c o n s i d e r the c a s e t h a t r i s a non-abelian group w h i c h is t h e d i r e c t s u m o f f u l l y i n v a r i a n t (characteristic, normal) subgroups. W i t h o u t takinq t h e hands o u t o f t h e trouser pockets w e d e d u c e from 7.44 t h e following result: 7.49 COROLLARY L e t r b e the direct s u m o f minimal f u l l y invariant (characteristic, normal) subgroups. Then r is a c o m p l e t e l y reducible (2.48) E(r)- ( A ( I - ) - , I ( r ) - ) group.

E(r)

Now w e w i l l t r y t o decompose with

(...)

itself.

Ne s t a r t

a n d f i x o u r n o t a t i o n , Our d i s c u s s i o n f o l l o w s p a r t l y

E(r)

and g e n e r a l i z e s Johnson ( 1 ) .

r

7 . 5 0 NOTATION L e t groups

Oi

b e t h e d i r e c t sum o f f u l l y i n v a r i a n t s u b (don’t forget 7.44!).

(iEI)

c a n u n i q u e l y b e decomposed as

Then e a c h

1 c$~

y

iEI

with

( n o t e t h a t t h i s sum i s a c t u a l l y f i n i t e ) . I f mEE(r) a n d ieI t h e n “(i) r

Now we i m p r o v e P r o p o s i t i o n 5 . 7

7.52

PROPOSITION

Proof.

(a)

t(iE1:

If

Qi€0i

+ r

o f Johnson (1).

E(i)(r)

9 E(r).

m(i)cE(i)(r)y

decompose rn as

consider

mfE(r)

with a l l = ii‘kek

eli)EEnd(r)

yfr

by 7.511b).

Usino 7,51(c)

ekeEnd

and

r.

we o b t a i n

A11

87

210

TRANSFORMATION NEAR-RINGS

= lukeii),

whence

m(i)EE(r).

This shows that

E(i)(r)

t

k

E(r).

(b) Applyinq 7.51(c) again shows that E(i)(r) s E(r). r ( c ) Now let n = 1 ujfj€E(r) (all fjEEnd r ) and j=l take some m(i)cE(i)(r). Then ntm(if-n = ulflt.. .turfrtm(i)-orfr-. But if

1 I $ ~= oit$ i€ 1

y =

(urfr+m(i) -urfr)(y)

where

5

$ =

. .-ulfl. then

k i +k

= u r f r ( + i t ~ ) + m ( i ) ( + i + 9 ) - u r f r ( 6 i t $ )=

= u f (+i)tm(i)(~i)-urfr(6i) = (urfr+m( 1 ) -urfr)(’)(y),

r r since fr($)

4k commutes with m(i)(@i)EOi. k,i This shows that urfrtm(’)-urfrEE(’)(r). E

Proceeding i n this way yields n+m(i)-neE(i)(I‘) and E ( i ) ( r ) is normal a n d by (b) and 6.6(a) we know that

E(i)(r)

At

E(r).

n =

(d) Filially consider again rn

Pick

(i).

(i)

m(’)n

=

= k1 ;.

eki)fj(y) (

and

Then by 6.4(d),

Jlujek(i)f

j’

Now if

= eii)(fj(bi)tfj($))

= (eii)fj)(i)(y),

lujfj

J

whence

e[’)f

a s before,

y =

= eii)(fj(bi))

j

E

=

E ( i ) ( ~ ) and hence

rn(i)ncE(i)(r).

E ( i ) ( r ) has no chance any more t o escape from being a n ideal.

So

The following result is implicit i n Johnson (1).

7 . 5 3 T H E O R E M L e t r b e f i n i t e , n o n - a b e l i a n a n d t h e d i r e c t sum o f f i n i t e l y many m i n i m a l f u l l y i n v a r i a n t s u b q r o u p s Q i (icI). Then, as near-rinqs,

Proof.

v

If

mcE(r),

y =

cQisr:

1 m"), f o r 2.30 implies t h a t iE1 m ( y ) = m(CQi) = c m ( + i ) = rm(i)(pri) = m =

i = Em ( y ) .

(r)n(

If

f

j

E(')(r'))

t h e n f o r each

Qi€Qi

i

c o n s e q u e n t l y rn ( j ) = 6 a n d t h e sum i s d i r e c t . The r e s t f o l l o w s f r o m 2 . 3 0 , 2 . 2 8 , 7 . 5 1 ( d ) , 7.43 and 7.46(a).

m(')(Qi)

= 0;

Using 4 . 4 6 and 7 . 5 3 one g e t s

E ( r ) i s f i n i t e l y completely 7.54 COROLLARY I f r i s a s i n 7.53, r e d u c i b l e a s a n e a r - r i n g a s well as a n E ( r ) - g r o u p ( 2 . 5 0 ) . E ( r ) i s s i m p l e i f f 111 = 1 , i . e . i f f r i s i n v a r i a n t l y simple.

S o t h e s t r u c t u r e o f ( l e f t ) i d e a l s o f E(r) seems t o b e c l e a r f o r a group r as i n 7 . 5 3 ( s e e 2 . 5 0 and 2 . 5 5 ) . 7 . 5 5 C O R O L L A R Y L e t t h e f i n i t e n o n - a b e l i a n q r o u p r be t h e d i r e c t s u m o f m i n i m a l f u l l y i n v a r i a n t s u b g r o u p s O1,. ,Qk. L e t be a b e l i a n a n d Q t + l , . . . , @ k n o t . Then

..

k

t

E(r) 2 Q

i=1

(where

End(Qi)

a'

8

j=t+l

'o(Qj)

d e n o t e s t h e e n d o m o r p h i s m r i n n on

Qi).

P r o o f . 7 . 5 3 , 7 . 4 3 and 7 . 4 6 ( a ) ( o b s e r v e t h a t t h e minimal f u l l y i n v a r i a n t subqroups Q i are i n v a r i a n t l y simple).

87

212

TRANSFORMATION NEAR-RINGS

A(r)

It i s harder t o g e t s i m i l a r r e s u l t s f o r since

oEAut(T)

does n o t i m p l y

automorphism i n general. 7.56

r

T H E O M Let

( jE J).

a(i)cE(i)(r)

E(T) 4

characteristic)

II

E(mj)

II

=

jsJ

J

n

A(T) 4

A(Pj)

jsJ Proof.

t o be an

b e t h e d i r e c t sum o f n o n - a b e l i a n

Then =

I(r),

A t l e a s t we g e t t h e f o l l o w i n g

f u l l y i n v a r i a n t (minimal

9.

and

n

jcJ

Mo(mj),

minimal

subqrougs Mo($.) and J

respectively.

jcJ h:

An e m b e d d i n g map c a n b e

E(r) m

+

and l i k e w i s e f o r t h e o t h e r two cases.

ll E ( Q j ) jsJ ( . . . ,m / ,...) j'

Observe t h a t t h e r e s t r i c t i o n o f an endomorphism i s a n e n d o m o r p h i s m a n d t h a t o f an a u t o m o r ? h i s m .

E

For

( 8 ri)

see Fong ( 2 ) and Fong-Meldrum ( 1 ) .

iEI 7 . 5 7 COROLLARY I f

r

i s as i n 7.53 t h e n

So t h e s i t u a t i o n i s q u i t e c l e a r i f

r

i s t h e d i r e c t sum o f

f i n i t e l y many m i n i m a l f u l l y i n v a r i a n t s u b g r o u p s .

I f t h i s i s n o t t h e c a s e , l i f e i s much m o r e c o m p l i c a t e d ( e . q . if

r

i s t h e dihedral group w i t h 8 elements).

We m e n t i o n o n l y t w o r e s u l t s w i t h o u t p r o o f s

i n t h e f i n i t e case:

(but with hints)

7.58 --__

T H E O R E M ( J o h n s o n ( l ) , ( 2 ) ) . Let

r

be a n o n - a b e l i a n

finite

group. Then

( a ) E ( r ) i s s i m p l e <=> r i s i n v a r i a n t l y s i m p l e . (we know <-; c o n v e r s e l y , i f A i s a p r o p e r f u l l y i n v a r i z n t subgroup of ideal o f

r

(o:A)

then

i s a proper

E(r)).

...

( b ) q 2 ( E ( r ) ) = Cdl(E(r)) = =p(E(I')) ( b u t n o t n e c e s s a r i l y = {O)). (gZ(E(r)) t u r n s o u t t o b e n i l p o t e n t . Now a p p l y 5 . 6 1 ( b ) ) ,

F o r more o n t h a t s e e Lyons ( 5 ) , ( 6 ) , Lyons-Meldrum ( I ) , Meldrum ( 6 ) , ( 7 ) ,( 1 3 ) .

We c o n t i n u e w i t h a r e s u l t d u e t o F r o h l i c h ( 3 ) . 7.59 T H E O R E M (Ramakotaiah ( 5 ) ) . Let a b e l i a n g r o u p . Then each

h cAut(I(r))

Proof.

r

be a f i n i t e s i m p l e non-

Aut(T) (as groups) and i s o f t h e f o r m m + ama - 1 , w h e r e a E A u t ( r ) . Aut(I(T))

7 . 3 9 and 7 . 4 6 ( c ) .

I f we a r e g o i n g t o l e a v e f i n i t e g r o u p s r i t becomes p r e t t y d a r k . I n ( 3 ) , F r o h l i c h mentioned t h a t one can s e e ( b y comparing t h e c a r d i n a l numbers) t h a t i f r i s a n i n f i n i t e s i m p l e group then I n t r o d u c i n g a s u i t a b l e t o p o l o g y i n Mo(r) g i v e s t h e r e s u l t t h a t M o ( T ) i s t h e c o m p l e t i o n o f E ( T ) . More c a n be f o u n d i n Ramakotaiah ( 9 ) .

E(T) < M o ( T ) .

We c o n c l u d e t h i s s e c t i o n w i t h some r e m a r k s . 7 . 6 0 REMARKS ( a ) An e x c e l l e n t s u r v e y on s p e c i a l E ( T ) ' s can be f o u n d i n Meldrum ( 1 3 ) . m,ncE(r)

(A(r), I ( r ) )

(A(r), I ( r ) )

is a rinq.

r

be h a l v a b l e . I f f o r a l l T-7-;T = 0 t h e n E(i-)

( b ) (Malone-McQuarrie ( 1 ) ) . Let

mtn n m

87

214

TRANSFORMATION NEAR-RINGS

( c ) (Flaxson ( 1 4 ) ) . I f r i s abelian i f f

E(r) i s a r i n q a n d r i s f g . t h e n i s monoqenic. ( T h i s c a n w r s o E ( r ) can b e a r i n q a l t h o u q h

r a c t u a l l y happen; i s n o t a b e l i a n ! - Compare 7 . 4 6 ) . If r i s a finite p-group and E ( T ) i s a semisimple ring then r i s abel i an. S e e a l s o Lyons ( 5 ) , Malone ( 3 ) , ( 1 0 ) , ( 1 2 ) ,

McQuarrie ( 5 ) .

( d ) (Chandy ( 2 ) ) . C a l l a group r a n L-qroup i f f I(r) is a r i n g . Cf. S c o t t ( 9 ) . Then r i s an L - g r o u p <=> a l l c o n j u g a t e d e l e m e n t s commute <==> t h e c e n t r a l i z e r o f e a c h Y E r i s a normal subgroup. I f r i s an L-qroup a n d n i l p o t e n t o f c l a s s 53 t h e n ( I ( r ) , t ) i s n i l p o t e n t o f c l a s s 1. ( e ) (Chandy ( 3 ) ) .

Iir)

i s a commutative r i n ? i f f

r

is

n i l p o t e n t of c l a s s 2 . ( f ) (BeidIerr,an ( 1 1 ) ) . I f

r

i s f i n i t e and s o l v a b l e ( n i l -

p o t e n t ) t h e n t h e same a p p l i e s t o ( g ) ( F a i n ( l ) , Lyons ( 3 ) , ( 4 ) ,

(E(r),t).

Lyons-Malone ( 2 ) , Johnson ( 1 ) ) .

If 0, i s the dihedral group with 2n elements ( n o d d ) t h e n E ( D n ) = A ( D n ) = I ( D n ) . S e e a more d e t a i l e d d e s c r i p t i o n o f t h e s e d g n r . ' s i n Meldrum ( 1 3 ) . L y o n s - M a l o n e ( 2 ) c o n t a i n s a l s o a d e s c r i p t i o n o f a l l d g n r . ' s d e f i n a b l e on

D n . S e e Meldrum ( 1 3 ) f o r i n f o r m a t i o n s on E ( Q , w h e r e 0, i s t h e i n f i n i t e d i h e d r a l cjroup. E ( T ) f o r f r e e T ' s a r e s t u d i e d i n Zeamer (3),(4),(5) a n d L o c k h a r t ( 4 ) . ( h ) S e e Lyons ( 4 ) , f o r a p r o o f o f t h e f a c t t h a t i f b i s a f u l l y i n v a r i a n t a b e l i a n d i r e c t summand t h e n E ( A ) embeds i n E ( T ) a s a d i r e c t summand. (i)

I(r) h a s t h e

DCCL i f f

E({y}) i s studied

r

i s f i n i t e (Scott ( l o ) ) .

in S c o t t ( 1 2 ) .

( j ) A d e n s i t y r e s u l t f o r E(T)

i s in Kaarli

(5).

( k ) C a l l e c E n d ( r ) normal i f a c I n n ( r ) : ea = a e , Let N(I') be t h e n r . q e n e r a t e d ( a d d i t i v e l y ) by a l l normal e n d o m o r p h i s m s o f r . By a r o u t i n e c h e c k we g e t

21 5

7d Polynomial near-rings

( l ) , P l o t k i n (3), 6 . 1 . 3 . 2 ) :

t h e f o l l o w i n q r e s u l t (Heerema

N(r)

i s a rino.

(1) (B.H.

Neumann ( 2 ) ) .

morphism o f

A(r)

of

r

L e t a be a f i x e d - p o i n t - f r e e

A,

o f o r d e r 3 and

q e n e r a t e d b y a.

auto-

the subnear-rino

A,

Then

i s a rinq,

too.

( m ) S e e S c o t t (1) f o r much i n f o r m a t i o n a b o u t t h e i n v e r t i b l e elements i n c e r t a i n subnear-rinos o f

E(T).

I(T)

r

proved t h a t i f

r

(n) ( S c o t t (8)). L e t p o i n t - f r e e and

(0)

N

h a s t h e DCCL t h e n

be f i n i t e ,

the nr.

(r,t)

See a l s o

and 6 . 3 5 ( d ) .

dJ-

POLYIJOI4IAL

fixed-

a

generated by a.

i s nilpotent iff 7 d)6)

aeAut(r),

He a l s o

i s finite!

Then

(N,t)

i s nilpotent.

NEAR-RINGS

1, ) POLYNOMIALS A N D POLYNOMIAL F U N C T I O N S

O.K., y o u a r e r i g h t : p o l y n o m i a l n e a r - r i n o s o n some a l o e b r a A a r e no t r a n s f o r m a t i o n n e a r - r i n o s o n A i n o e n e r a l .

But they are

so c l o s e t o t h i s s u b j e c t and necessary t o t r e a t n e a r - r i n o s o f polynomial

f u n c t i o n s t h a t we i n c l u d e t h e m i n t o 5 7 .

P o l y n o m i a l s c a n be d e f i n e d o v e r any a l q e b r a i c s t r u c t u r e A and any s e t X ( w i t h Vcontaininq A. Roughly spoken,

A n X = $)

o f "indeterminates"

i n any v a r i e t y

A(X,'V).

Denote t h e s e t o f t h e s e p o l y n o m i a l s by

A(X,V)

i s t h e " s e t o f a l l words i n

A u X

where e q u a l i t y i s d e f i n e d i n accordance w i t h t h e laws d e f i n i n o

v".

I n fact,

A(X,V)

i s d e f i n e d as a " s u i t a b l e " f a c t o r

o f t h e word algebra over algebra o f

V ,

AuX

the "polynomial

characterization o f

A(X,?J)

i n u ,

Another possible

i s the followinn:

f r e e u n i o n o f A and t h e f r e e a l n e b r a a detailed exposition.

hence becomino a l s o an

alqebra".

See L a u s c h - N o b a u e r ( 1 ) ( w h o l e b o o k ,

alaebra

F

A(X,D)

i s the

over X i n v .

p a r t i c u l a r l y pp. 12/13) f o r

216

§ 7 TRANSFORMATION NEAR-RINGS

I n some i n s t a n c e s f o r t h e s e p o l y n o m i a l s ( = e q u i v a l e n c e c l a s s e s

Au X )

o f words o v e r

N o b a u e r (l), p p .

t h e r e e x i s t "normal

22 f f . ) , e . p .

forms"

(see Lausch-

f o r t h e case o f commutative

r i n g s w i t h i d e n t i t y and f o r groups. We w i l l r e s t r i c t o u r c o n s i d e r a t i o n s t o t h e c a s e o f a s i n q l e "variable"

X = Ex1

(9) a n d

groups identity

and t o two v a r i e t i e s :

t h e variety??.

t h e one o f a l l

o f a l l commutative rincls w i t h

1.

T h e n we h a v e t h e f o l l o w i n q p o l y n o m i a l a l o e b r a s ( i n n o r m a l f o r m s ) (see Lausch-Nobauer

7.61 N O T A T I O N I f (a)

RE%

I

R[x]:

(I), 1.8.11 a n d 1.9.11):

r

and

=

(r,+)Eq

' 1 aix i I n E I N o , ai€R, i= O

p

a,

0 ) u{01.

1+n 2 x c . . . + ~ ~ - ~ + n ~ x + y ~ ,/ Yr i cE rl, N ~

( b ) ~ [ x J : = {yo+nlx+y n i E P

then

tj

and

tEE1,

...,r-11:

yt

$. 01.

Our i n t e r e s t i n p o l y n o m i a l s stems f r o m t h e e a s y - t o - e s t a b l i s h

7.62 PROPOSITION U n d e r a d d i t i o n " + " a n d s u b s t i t u t i o n p o l y n o m i a l s ( d e f i n i t i o n as u s u a l ,

77)

p.

(R[x]

,+,o)

and

(r[x]

cf.

,+,o)

by

R[x]

and

T h r o u g h o u t t h i s s e c t i o n R w i l l mean a n e l e m e n t o f

7.63 D E F I N I T I O N (a)

If

p~R[x]

then

-p :

R r

+

R

+

p o r =:

i s called p(r)

t h e polynomial f u n c t i o n induced by p. P(f?): = f T l p ~ R [ x ] ) . ( b ) S i m i l a r l y we d e f i n e Trivial,

b u t good t o n o t e i s

5; f o r

pEr[X)

(l),

are near-rinas.

We w i l l c o n t i n u e t o d e n o t e t h e s e n e a r - r i n r l s r[x].

of

"0"

Lausch-Nobauer

and

p(r).

'41.

21 7

7d Polynomial near-rings

7.64

PROPOSITION - If

-

r

is abelian then every polynomial function

5: r

+

r

Y

-c

Yo+nY

PROPOSITION T h e c o r r e s p o n d e n c e

h: p

homomorphism. Hence

P(r) a r e s u b n e a r - r i n q s

pEP(r)

i s o f the form

nEINo.

7.65

of

M(R)

Proof: 7.66

P(R)

and

with

yO€r and

i s a near-rinq

+

(M(r), r e s p e c t i v e l y ) .

straightforward.

REMARKS

(a)

P(k)

and

P(r)

are called the near-rinns o f polynomial

(r,

functions on R

respectively).

(b) Throughout t h i s section,

h w i l l have t h e meaning o f

7.65. Take

h i s n o t n e c e s s a r i l y a n i s o m o r p h i s m R[x] R = : Zp a n d q: = x . ( x - 1 ) . x-(p-1)).

Then

p

+

P(R):

. . . a (

$. 0 ,

but

= b ( z e r o f u n c t i o n on ZP).

h(p)

K e r h 9 R[x]

w i l l p l a y a decisive r61e. S i m i l a r c o n s i d e r a t i o n s a p p l y t o g r o u p s ( i fe . 9 . r = Z3, q = 3 x b e h a v e s a s a b o v e ) . Anyhow,

Hence one c a n n o t i d e n t i f y p o l y n o m i a l s and p o l y n o m i a l ( a s one i s used t o do o v e r I R ) i n q e n e r a l .

functions But: 7 . 6 7 REMARK I f (R,+)

i s t o r s i o n - f r e e ( ( A c z e l ) ) or i f R i s an i n f i n i t e

f i e l d (use t h e theory o f l i n e a r equations and t h e Van-derM o n d e - d e t e r m i n a n t ) t h e n h i s an i s o m o r p h i s m L a t e r o n , we s h a l l

V[x]

also consider

( V a v e c t o r space)

n o w we r e m a i n a t

R[x]

NIX]

(N

R[x]

* P(R)

a near-rinq),

and so on t o g e t more n e a r - r i n q s . and

r[xl,

a l r e a d y be seen t h e r e and d i s c u s s seems t o b e m o r e f a m i l i a r t h a n

s i n c e m o s t phenomena c a n RTx]

r[xl.

f i r s t since

REX]

But

87

218

2.)

TRANSFORMATION NEAR-RINGS

R[x]

We s t a r t w i t h some e l e m e n t a r y , y e t f u n d a m e n t a l as o f t e n

R[x]

i s an a b e l i a n n e a r - r i n q w i t h i d e n t i t y x.

' aix i

p =

I f we w r i t e

1

,

=o

p,q~R[x]:

p = 0

(1.61(a))

See a l s o Mc O u a r r i e ( I ) .

PROPOSITION ( L a u s c h - N o b a u e r (a)

if

an = 0

we a l l o w

t o a v o i d s e p a r a t i n q cases.

7.68

Let,

([O] : = 0).

be t h e deqree o f p

[p]

Observe t h a t

properties.

[poq]

5

(1). p . 134).

[p].[qJ

(with equality iff

R is

an i n t e g r a l domain). (b)

If

p~R[x]

i s invertible (w.r.t.

If R i s a field,

t h e conve:se

then

0 )

( c ) I f R i s an i n t e g r a l domain t h e n each > 0

fp]

= 1.

[p]

also holds. p~R[x]

with

i s r i g h t cancellable.

'

1 aiq i

poq =

= a n b m x n m t summands o f l o w e r d e g r e e .

i= O

(b) follows from (a).

( c ) If

f o p = qop

= 0,

[f-q] and so

f-q

q o p = fop

7.69 DEFINITION C a l l

[f]

implies

so

p~R[x]

= [p]

then

or

( f - q ) o p = 0.

i s a constant c. qop+c,

whence

(a)

[p]cP

(b)

If p1

[p]

f = q+c

p = fog

= [p].

7 . 7 0 PROPOSITION ( L a u s c h - N o b a u e r ( 1 ) ) . domain and

Hence

c = 0.

indecomposable i f [g]

By ( a ) ,

L e t R b e an i n t e q r a l

p~R[x].

->

p i s indecomposable.

> 1

then t h e r e e x i s t indecomposable polynomials

,. . . , p s ~ R [ x ]

with

p = p10p20.. .ops.

7d Polynomial near-rings

Prosf. (a) i s t r i v i a l . ( b ) follows by i n d u c t i o n on

21 9

usinq 7,68(a).

[p!,

7 . 7 1 DEFINITION ( a ) L e t p be c a l l e d normed i f

a n = 1.

(b) p i s called linear i f

= 1.

[p]

Now we can s h a r p e n 7 . 7 O ( b ) a n d s t a t e w i t h o u t p r o o f : 7 . 7 2 T H E O R E M ( L a u s c h - N o b a u e r ( 1 ) ) . I f F! i s a f i e l d o f c h a r a c t e -

p~R[x]

r i s t i c 0 and i f

has

[p]

1

then

~ ? o p ~ o . . . o pw ~ i t h II l i n e a r , (a) p indecomposable and normed.

(b)

If

p = moqlo

of p then and q j ' s

...Oqt

S E N and

pl,

...,p,

i s a n o t h e r such "prime decomposition"

m = II, s = t a n d t h e d e g r e e s o f t h e p i ' s a r e t h e same ( u p t o o r d e r ) .

( c ) There e x i s t o n l y f i n i t e l y many d e c o m p o s i t i o n s o f

p of

t h e form ( a ) . T h i s u n i q u e n e s s can even be s t r e n q t h e n e d by t h e deep theorem 2.46 o f Lausch-Nobauer ( 1 ) . 7.73 R E M A R K See Graves-Malone ( 2 ) f o r a n o t h e r r e s u l t of "prime f a c t o r decompositions" in "Euclidean near-domains'' ( s e e 9 . 6 0 and 9 . 6 7 ( c ) ) .

7 . 7 4 DEFINITION C a l l

REQ p o l y n o m i a l l y c o m p l e t e ( N o b a u e r ( 6 ) )

o r 1 - p o l y n o m i a l l y complete (Lausch-Nobauer ( 1 ) ) if P ( 2 ) = M ( R ) , i . e . i f e a c h map R R i s a ?olynomial -f

function.

87

220

TRANSFORMATION NEAR-RINGS

O f c o u r s e , Z , I, IR a n d (c a r e n o t p o l y n o m i a l l y c o m p l e t e . O n t h e o t h e r h a n d , t h e f a c t t h a t q, i s p o l y n o m i a l l y c o m p l e t e i s of high v a l u e i n mathematical l o g i c . P e r h a p s we now f e e l w h i c h R m i g h t be p o l y n o m i a l l y c o m p l e t e and prove the f o l l o w i n g i n t e r e s t i n o r e s u l t , w h i c h w i l l a l s o prove useful i n the sequel. 7 . 7 5 T H E O R E M ( N o b a u e r (6), L a u s c h - N o b a u e r

R i s p o l y n o m i a l l y c o m p l e t e <=>

(1)). R i s f i n i t e and simple

(hence a f i n i t e f i e l d ) . P r o o f . ->: I f R i s i n f i n i t e , [ P ( R ) I S IR[x]I = IRI < so R c a n n o t b e p o l y n o m i a l l y < IRIIRI = IM(R)I, complete. Let I b e a n o n - t r i v i a l i d e a l o f R . T a k e some nonzero iEI. I f ~ E P ( R ) then p ( i ) - p ( O ) c I , since I SIR. Let f c M ( R ) f u l f i l l f ( i ) - f ( O ) e I . Then f t P ( R ) and a g a i n R i s n o t polynomially complete. T h i s f o l l o w s from L a g r a n y e ' s i n t e r p o l a t i o n theorem ( o r f r o m t h e f a c t t h a t P ( R ) i s 2 - p r i m i t i v e

<-:

o n R).

4 . )I D E A L

T H E O R Y IN

R[xl

Rtx]

c a n a l s o be c o n s i d e r e d a s a r i n g ( R l x ] , + , * ) , w h e r e 'I 'I means o r d i n a r y m u l t i p l i c a t i o n . T h e r e i s some c o n n e c t i o n b e t w e e n t h e r i n q - a n d t h e n e a r - r i n q i d e a l s o f RLx]: O f course,

7.76 P R O P O S I T I O N ( S o ( 1 ) ) . I f p,q,rER[x] then r divides ( W . r . t . " . " ) p o ( q + r ) - p o q . I n p a r t i c u l a r , r i p o r f o r each p € R O [ x ] .

n Proof. Let

p =

1 aixi .

i=O

po(q+r)-poq = o f r.

Then

n

1 ai(q+rli - 1 aiqi

i =O

i =O

is a multiple

7d Polynomial near-rings

7.77

(1)). Every i d e a l o f

COROLLARY ( L a u s c h - N o b a u e r

i s a l e f t i d e a l of See 7.94

(REX] ,+,*)

(R[x],+,o).

f o r a ( p a r t i a l ) converse.

N o t e t h a t i f we w r i t e version

22 1

(R[x],t,o).

we a l w a y s mean t h e n e a r - r i n c l R[x] I t ' s a l s o h i g h t i m e t o f i x one more

notation.

7 . 7 8 NOTATION 1 a i x i laiER (a) ( R [ X ] ) ~ = { ' i=l = : R,[x].

(b)

n

(Rbl),

= {

1 aixi

A nElN

JaoERl = {PI

i= g

1 = {plp(o) = 01 =

rP]

01

=:

Rc[x].

( O f course, Rc[x] c a n b e i d e n t i f i e d w i t h R, we w i l l d o s o i n t h e s e q u e l . ) ( c ) The e l e m e n t s o f zero-symmetric

7.79

Ro[x]

(b) (c)

rER

If

Proof.

E

V

XEL

are called the

(constant) polynomials.

PROPOSITION ( C l a y - D o i ( a ) Ro[x]0L

(Rc[x])

and

( Z ) , So ( 1 ) ) . L e t

L slP.R r x ] .

Then

L.

EL: r L E L . then

R,[x]

F

L.

(a) f o l l o w s from 1.34(a).

( b ) rP. = ( r x ) o L c L

by (a).

(c) a l s o f o l l o w s from (a). 7.80

REMARK Observe t h a t

R

i s an

i n t h e n a t u r a l way ( p r : =

R[x]-

-p ( r )

if

and an p~R[x]

P(P)-group

and

rER).

We w i l l m i l k t h i s o b s e r v a t i o n i n t h e n e x t n u m b e r . Now we t a k e a g l a n c e t o t w o i n t e r e s t i n a i d e a l s o f

R[x]:

S 7 TRANSFORMATION NEAR-RINGS

222

-7.81

E X A M P L E S (Nobauer ( l ) , Lausch-Nobauer ( 1 ) ) . Let Then

( a ) ( J ) : = J[x]

J 4 R.

9 R[x].

( b ) < J > : = ( L J : R ) ~ C ~ I4 R C X ] .

show u p t o be a l s o These examples of i d e a l s i n ( R [ x ] , t , o ) ideals in (R[x],+,*). This l e a d s t o t h e followinrl i s c a l l e d a f u l l i d e a l (Nobauer) o r 7.82 D E F I N I TION I 9 REX] T - 0 - i d ea l (Menger) o r c o m p o s i t i o n i d e a l ( A d l e r ) i f a l s o

I 4 (R[x],+,*). 7 . 8 3 R E M A R K Thus f u l l i d e a l s a r e e x a c t l y t h e " i d e a l s " o f t h e c o m p o s i t i o n r i n g ( T O - A l p e b r a ) ( s e e 1 . 1 1 7 ) (R[x] , t , * , o ) . a r e much b e t t e r e x p l o r e d In f a c t , t h e s e f u l l i d e a l s of R[x] t h a n t h e " o r d i n a r y " i d e a l s o f R[x]. Not e v e r y i d e a l o f R[x] i s a f u l l i d e a l , b u t t h i s holds in s p e c i a l , important cases (see 7.93). I f e . g . R happens t o be a f i e l d t h e n a l l f u l l i d e a l s a r e p r i n c i p a l ( a well-known r e s u l t o f r i n q t h e o r y ) a n d so q u i t e e a s y t o o v e r l o o k . I t i s n o t k n o w n u n d e r which c o n d i t i o n s each i d e a l o f R[x] i s " p r i n c i p a l " ( = g e n e r a t e d by one s i n g l e polynomial), or a t l e a s t f . g . , The i d e a l s of 7 . 8 1 show u p t o be q u i t e i m p o r t a n t o n e s : 7.84 T H E O R E M (Nobauer ( l ) , Lausch-Nobauer ( l ) , Hule ( 1 ) ) . L e t I be a f u l l i d e a l o f R t x ) . Then t h e r e e x i s t s e x a c t l y one i d e a l J of R w i t h ( J ) ' I I " < J > (namely J = I n R ) . J : = I nR. Then c l e a r l y ( J ) = J [ x ] e I . Also, i f iE1 a n d r E R t h e n i ( r ) = i o r E I n R = J , whence I c < J > . I f J ' 9 R a l s o f u l f i l l s (J')sI'<J'> then ( J ' ) c < J > , whence J ' e < J > a n d s o J ' c J . S i m i l a r l y , J e J ' and hence J = J ' .

P r o o f . Let

223

7d Polynomial near-rings

7 . 8 5 DEFINITION- L e t I , J be a s i n 7 . 8 4 . Then J i s c a l l e d t h e " e n c l o s i n q i d e a l " o f I ( J i s n o t i d e a l i n R[x] in general ! ) .

F o r (much) more i n f o r m a t i o n c o n c e r n i n g t h e s e e n c l o s i n g i d e a l s s e e t h e c o m p r e h e n s i v e book L a u s c h - N o b a u e r ( 1 ) . C f . a l s o Mlitr (1).

We w i l l g e t more p o w e r f u l r e s u l t s when

?i

i s a s s u p e d t o be

a field:

5.) F r x l Throughout t h i s number, l e t F d e n o t e a commutative f i e l d . 7 . 8 6 PROPOSITION ( C l a y - D o i ( Z ) , S t r a u s ( 1 ) ) . w i t h L n F =/= I 0 1 ( c f . 7 . 7 8 ( b ) ) . (a) F

E

(b) If

Let

L AII F[x]

L. IF1 > 2

then

L = F[x].

P r o o f . ( a ) L e t II f 0 be c L n F . By 7 . 7 9 ( b ) , f . E - l . 9 . = fEL.

Then t a k e some f e F .

By ( a ) , fEL ( b ) I f c h a r F $. 2 , t a k e f : = 2 - ' . 2 2 2 and a l s o f E L . Hence x o ( x + f ) - x O X E L , so x + f 2E L , w h e n c e X E L . Use ( a ) a n d a p p l y 7 . 7 9 ( c ) t o get L = I f char F = 2 , and I F / > 2 , then i n p a r t i c u l a r 3 3 2 char F 3 . Hence x o ( x + l ) - x O X = x + x + ~ E L , s o x2+xEL. T a k e some f e n { O , l I a n d d e n o t e f - l * x 3 by p . Then po(x+f)-pox = x 2+fx+f2E L . 2 Since x +XEL a n d FcL, ( f - l ) x e L , s o by 7 . 7 9 ( b ) a n d t h e f a c t t h a t F i s a f i e l d we g e t xeL a n d a q a i n F[x] = L .

FIX].

87

224

TRANSFORMATION NEAR-RINGS

7 . 8 7 R E M A R K B r e n n e r ( 1 ) h a s shown t h a t 7 . 8 6 ( b ) d o e s n o t h o l d f o r F = B 2 . See 7 . 9 8 ( b ) .

7 . 8 8 PROPOSITION

(Clay-Doi i s an

(a) (F,t)

(2)). a n d P ( F ) -group o f type 2.

F[x]-group

( b ) I f h ( 7 . 6 5 ) i s a n isomorphism ( c f . 7 . 6 6 ( b ) ) t h e n Fcx] i s 2 - p r i m i t i v e on ( F , t ) = F. (c) P(F)

i s a l w a y s 2 - p r i m i t i v e on F .

Proof. I f f , f ' E F , n a m e l y p o = ft:-'xo:

then 3 p o ~ F 0 [ x ] : p o ( f ) = f ' , The r e s t i s e q u a l l y o b v i o u s .

I f t h e r e a d e r i s s t i l l i n t e r e s t e d , he i s c o r d i a l l y i n v i t e d t o

a nearly complete trip t o the ideals of F i r s t we s e t t l e t h e q u e s t i o n , f o r w h i c h F be s i m p l e .

F[x] a n d P(F). F[x] happens t o

Fcx]

7.89 T H E O R E M ( S t r a u s ( 1 ) ) . Let F be i n f i n i t e . T h e n simple.

is

Proof. I f I 9 Fcx], I 9 {Ol, t a k e Some i c 1 , i 0. By 7 . 6 7 , i 6 , s o 3 f s F : i ( f ) = i o f 0. Hence i o f s I n F a n d 7 . 8 6 ( b ) i m p l i e s I = F[x].

+

+

S o t h e i n f i n i t e c a s e i s s e t t l e d a n d we t u r n t o f i n i t e f i e l d s . T o d o s o , we f i r s t d e t e r m i n e a l l f u l l i d e a l s ( 7 . 9 0 ) o f F cx ] a n d then we w i l l s e e ( 7 . 9 3 ) t h a t , i f c h a r F 2, a l l ideals o f FLx] are full ideals. 7.90 T H E O R E M (Menger ( 2 ) , Milgram ( l ) , Lausch-Nobauer

(l),

Straus (1)). Let F be a f i n i t e f i e l d and I a n i d e a l o f ( F [ x ] , t , * ) . Since (F[x],+,*) i s a P I D , I i s some p r i n c i p a l i d e a l ( p ) o f t h i s r i n g ( F [ x ] , + . . ) . Then t h e f o l l o w i n a c o n d i t i o n s are equivalent:

(a) I = (p) (b)

SEF[X]:

is a full ideal. p/po~.

225

7d Polynomial near-rings

( c ) There e x i s t

ml,.

and

kEN,

. .,mkf

Wo

~ ( x q

with

ml

Proof.

'k

,..., ( x

-XI

where

'2

nl

- ~ ) ~ ' ( )( xnq

I = ((xq

(then

l s n I < n2< . . . c n k

with

'1 p = 1.c.m.

...,

nl,

m2 -x)

IF1 = q

... n ( ( x q k'

)n

m - x ) ')).

(Straus). ( a ) -> ( b ) i s t r i v i a l . L e t C be the algebraic closure o f t h e

(c):

( b ) ->

f i e l d F.

-

p ( c ) = 0.

L e t c be i n C w i t h

m u l t i p l i c i t y of multiplicity

L e t m be t h e

t h e r o o t c . Then c i s a r o o t o f

>m

o f each

pos

p has a z e r o o f m u l t i p l i c i t y

(sEFLx1).

Hence

a t each element

2m

o f F(c) ( f i e l d extension o f F by adjunction o f c). A p p l y i n g t h e t h e o r y o f f i n i t e f i e l d s we s e e t h a t p : i s d i v i s i b l e by ( X ~ " - X ) ~ ,w h e r e n : = [F(c):F! dsF(c):

(x-d)"/p,

It

But

hence

TI

n

dcF(c)

(x-d)m

d i v i d e s p.

(x-d)m = (xq - x ) ~ .

dEF( c ) S t a r t i n g w i t h a r o o t c w i t h maximal

[F(c):F!

we

arrive successively a t (c). ( c ) =>

( a ) : By 7.77,

a r i g h t ideal.

Let s

i t s u f f i c e s t o show t h a t

R[x].

F o r m,n

induces the zero function.

But xq-x

E

degree zero f u n c t i o n i n P ( F ) , I f we d o t h i s f o r n = niy

E

IN

(p) i s ( x q n - X ) ~ sO

i s the lowestn whence ( x q - x ) l ( x q - x I m .

m = m .1 ( 1 2 i 2 k ) we g e t o u r

result. 7.91 REMARK The r e p r e s e n t a t i o n i n 7 . 9 0 ( c ) see Lausch-Nobauer

( l )Ch. ,

i s moreover unique:

111, 7 . 2 1 .

7 . 9 2 DEFINITION A p o l y n o m i a l p as i n 7.99 (Milgrarn (1)).

i s called saturated

87

226

TRANSFORMATION NEAR-RINGS

7 . 9 3 T H E O R E M ( S t r a u s ( 1 ) ) . Let F b e a f i n i t e f i e l d . T h e n : E v e r y i d e a l o f F[x] i s a f u l l i d e a l <*> c h a r F 4 2 .

P r o o f . ->: A s s u m e t h a t c h a r F = 2 . Isle show t h e e x i s t e n c e o f a n i d e a l I of FIX? which i s n o t a f u l l i d e a l : Let

n IF1 = : q

I:

Consider

I

4

and

F[x2]:

= {

1 a2ix2iIncINo,

i= O

a2i~F).

( ~ ~ t x ) ~ . F ~ x ~ ! t ( x ~ t x ) ~ . F [ x ] .

=

(F[x] , + , * )

since

Let p : = ( x q + x ) ' 1 2 (Fcx] , + , o ) :

(xqtx)2cI,

= x Z q + x 2 E F[x2].

but

~ . ( x ~ t x ) ~ $ I .

We s h o w t h a t

( a ) C l e a r l y , ( I , + ) i s a normal subgroup o f (F[x],+). i t suffices t o 2 c o n s i d e r u : = x o ( r + p t + p s ) - x n o r w i t h r , s E F[x] and t E F[x2]. I f n i s e v e n , u = ( r + p t + p 2 s ) ' - r n = z pntntp2nsn E I . I f n i s odd t h e n u = r " - l ( p t + p 2s ) + v w i t h v E ( p2 ) . So u = p ( t r n - ' ) + ( p r "Istv) E I , s i n c e ( b ) I n o r d e r t o show t h a t I SRF[x]

n

n-1

i s even.

( c ) I n p r o v i n g I a F [ x ] we h a v e t o s h o w t h a t f o r a l l r t E F[x 2 ] a n d a l l s , r E F [ x ] we g e t w : = ( p t + p 2 s ) o r E I . Now w = ( p o r ) ( t o r ) + ( p 2 0 r ) ( s o r ) . S i n c e c h a r F = 2 , t o r E F [ x 2 ] a n d p o r E F [ x 21. Now p ( p o r b y t h e s a m e a r g u m e n t a s i n the p r o o f ( c ) = > ( a ) of 7.90 a n d p 2 1 p 2 0 r as w e l l . Hence ( p 2 0 r ) ( s o r ) E p2F[x], and (por)(t.r) ( 3 :

Let

we n e t

E

pF[x2]F[x2j

c pF[x2].

This proves

W &

I.

I S F l x J . Ther, F o r a l l iEI and a l l p E =:x: i-p = -1( x 2 O ( i + p ) - x * o p - ( x 2 o ( i t 0 1 - x 2 0 0 ) E I . 2

7 . 9 4 R E M A R K T h e p r o o f o f " 7 . 9 3 <==" a l s o s h o w s : I f F is f i n i t e with char F 2 then the left ideals Of (F[x],+,o) are e x a c t l y t h e i d e a l s o f (F[x],+,*) ( c f . 7.77).

+

7d Polynomial near-rings

227

7 . 9 5 R E M A R K S t r a u s ( I ) a l s o showed t h a t i f F i s f i n i t e w i t h c h a r F = 2 b u t I F 1 > 2 a n d i f I 4 F[x] then J o f ( F [ x ] ,+,*) which a n d I c o n t a i n s an i d e a l F[x2]*I e I

i s g e n e r a t e d by ( il , i 2 ~ I ) .

.-.

i i L = i - i l i E I l ; J contains a l l

il-i2

7 . 9 6 C O R O L L A R Y I f F i s f i n i t e t h e n F[x] does not f u l f i l l 2 t h e D C C ( b e c a u s e o f ( x ) ( ~x )z. . . n o t e v e n t h e D C C L ) F o r more i n f o r m a t i o n s e e S o ( 1 ) . 7 . 9 7 C O R O L L A R Y I f F i s a f i n i t e f i e l d o f c h a r a c t e r i s t i c =/= 2 f u l f i l l s t h e A C C , b u t n o t t h e D C C on i d e a l s . then F[i] Hence F[x] c a n n o t be c o m p l e t e l y r e d u c i b l e ( 2 . 5 0 ) .

So i t remains t o c o n s i d e r f i n i t e f i e l d s F with c h a r F = 2 a n d t h e r e i n p a r t i c u l a r F = Z2. A s u s u a l i n a l g e b r a , c h a r a c t e r i s t i c 2 c a u s e s a l o t o f t r o u b l e . The i d e a l s t r u c t u r e o f Z,[x] i s much more c o m p l i c a t e d t h a n t h a t o f F I X ] i n 7.89. o r 7.90/7.93.

We g e t s a t i s f a c t o r y r e s u l t s c o n c e r n i n g t h e i d e a l s o f F c x ] w i t h c h a r F = 2 o n l y i n t h e c a s e o f maximal i d e a l s :

7.98

THEOREM (Clay-Doi

(2).

( a ) If F is i n f i n i t e , F

Brenner ( 1 ) ) .

I01

i s t h e o n l y maximal i d e a l o f

1x1 '

P2 t h e n Ker h = (PEF[x]IP = 61 i s t h e u n i q u e maximal i d e a l o f F c x ] .

(b) If F i s f i n i t e , b u t

( c ) Z,[X] V:

=

T: =

h a s e x a c t l y two maximal i d e a l s :

{ ~ E F [ x ] ( Pi s

c o n s t a n t 1 and 3 t h e ( n e a r - r i n g ) i d e a l g e n e r a t e d by x

.

57

228

TRANSFORMATION NEAR-RINGS

P r o o f . C o n s i d e r h o f 7.65 and t h e diaqram ( o b s e r v e 1 . 3 0 ) . ( a ) i s s e t t l e d by 7 . 8 9 . S o we may a s s u m e t h a t F i s f i n i t e . By 7 . 7 5 , P(F) = M ( F ) . So i f P(F) IF1 2, P(F) i s s i m p l e and Ker h Ker h i s m a x i m a l . If IF/ = 2 t h e n Mc(F) i s a maximal i d e a l (01I01 i n M ( F ) = P ( F ) by 7 . 3 1 . Hence i t s pre-image under h (= V ) i s maximal i n F[x]. The f a c t s t h a t T i s a n o t h e r maximal i d e a l i n Z,[x] and t h e u n i q u e n e s s s t a t e m e n t s i n ( b ) a n d ( c ) i n v o l v e some t e c h n i c a l r e a s o n i n q a n d we o v e r q o t h e p r o o f s . T h a t o f ( b ) i s i n Clay-Doi ( 2 ) , w h i l e t h a t o f ( c ) can b e found i n Brenner ( 1 ) . F

CXI

+

I\

1

7 . 9 9 C O R O L L A R Y ( N o b a u e r ( 6 ) , Hule ( l ) , C l a y - D o i Nobauer ( 1 )) . i s s i m p l e <-> F i s i n f i n i t e . F[x]

(2),

Lausch-

Theorem 7 . 9 8 h a s some a p p l i c a t i o n s . Ve m e n t i o n 7.100 C O R O L L A R Y (Clay-Doi Proof. If

(2)).

P E P , (pZ)[x]

Z [ X ] / ( ~ ~[xl ~ 2 ZP[x],

n[x] 9 z[x]

c o n t a i n s maximal i d e a l s . ( 7 . 8 1 ( a ) ) , and

which c o n t a i n s a t l e a s t

o n e maximal i d e a l by 7 . 9 8 . gives the statement.

An a p p l i c a t i o n o f 1 . 3 0

A l l maximal i d e a l s o r a l l f u l l i d e a l s o f Z [x] a r e n o t known ( C l a y - D o i ( Z ) , L a u s c h - N o b a u e r ( I ) , p . 1 3 1 ) . One a l s o m i g h t r a i s e t h e q u e s t i o n , which P ( R ) h a p p e n t o be s i m p l e . We a r e happy t o h a v e a f u l l a n s w e r : 7 . 1 0 1 T H E O R E M ( N o b a u e r (6)). P ( R ) i s s i m p l e <-> R i s a c o m m u t a t i v e f i e l d w i t h IR1,2.

229

7d Polynomial near-rings

( a ) L e t I be a n o n - t r i v i a l

P r o o f . ->:

h b e a g a i n as i n 7.65. consequently

h(I[x])

i d e a l o f R.

1 1 x 3
By 7 . 8 1 ( a ) ,

9 P(R).

= P(1)

Considerinn t h e constant polynomial

9 {el

P(I) Since a

and

P(R)

P(I)

+

Let

functions yields

a contradiction.

P(R),

i s assumed t o b e s i m p l e ,

we a r r i v e a t

nonsense.

(b)

P ( Z 2 ) = M(Z2)

<=-:

L e t R be a f i e l d .

b y 7.75

P(F?)

i s n o t s i m p l e b y 7.31. If R i s finite

= M(R)

=/=

i s s i m p l e (7.33).

i n f i n i t e t h e n b y 7.67

P(R)

3

Z2

then

If R is

which i s simple

R[x]

b y 7.99. We g e t o n l y p a r t i a l r e s u l t s o n t h e r a d i c a l s o f 7.102

F[x] :

REMARKS

( a ) L e t F be i n f i n i t e . Then =

{Ol.

)62(F[x])

T h i s h o l d s by 7 . 8 8 ( b ) .

(b) For any i n t e q r a l domain R, f o r by 7.68(a)

(c) Clay-Doi

(Z),

R[x]

Cf.

V(R[x])

=

... 7.123

=?(F[x]) (a).

= '$'(R[x])

=

=

{Ol,

has no n i l p o t e n t elements

0.

t 4 l i t z ( 1 ) and ( 3 ) d e t e r m i n e d r a d i c a l s o f

some

F[x]'s,

which do n o t always c o i n c i d e w i t h o u r

a2(N),

. . . ,p(N).

B u t one can a e t immediate r e s u l t s

on p o l y n o m i a l n e a r - r i n q s o v e r f i n i t e f i e l d s F : ( b y 7.88

l z ( F [ x ] ) C Ker h

char F 2 n(F[x])= I 0 1 ( t h i s f o l l o w s

...

( a ) ) and i f

then 21,2F([x]) = = f r o m 7 . 9 4 ) . See a l s o 7 . 1 2 3 .

( d ) The s i t u a t i o n i n t h e g e n e r a l c a s e does n o t seem t o be c l e a r . A n y h o w , we o b s e r v e t w o s t r a n q e p h e n o m e n a : ( 1 ) The n e a r - r i n a - r a d i c a l s m i a h t d i f f e r s u b s t a n t i a l l y f r o m t h e r i n g - r a d i c a l s o f R[x] ( t h e l a t t e r ones a r e always = I01 i f - R i s a f i e l d ) .

( 2 ) The s m a l l e r F, t h e more c o m p l i c a t e d i s t h e structure o f

F [x]

.

( e ) S e e So ( 1 ) f o r a d e t a i l e d s t u d y o f t h e i d e a l s t r u c t u r e o f

R[x]

and

P(R).

87

230

TRANSFORMATION NEAR-RINGS

( f ) E v e r y f i n i t e f i e l d c a n be o b t a i n e d by f o r m i n g t h e r i n g (Zp[x] ,+,.) f o r some p r i m e p , c h o o s i n g a n i r r e d u c i b l e p o l y n o m i a l p a n d f o r m i n g Z p [ x ' ] / ( p ) . One m i g h t wonder which n e a r - f i e l d s c a n be o b t a i n e d a s ( H p [ x ] , + , o ) / I f o r some m a x i m a l i d e a l I . The a n s w e r i s v e r y s u r p r i s i n g : none a t a l l ! I n ( 2 ) , Y.S. So has shown: I f R i s a c o m m u t a t i v e r i n g w i t h i d e n t i t y and if F : = (R[xl , + , " ) / I i s a near-field then F i s a f i e l d and moreover a f a c t o r f i e l d o f R ! I n t h i s paper i t i s a l s o shown t h a t e v e r y n e a r - f i e l d c o n t a i n e d i n

R[x]

i s a f i e l d . Also, every planar near-ring

8 . 8 5 ) c o n t a i n e d i n some R[x]

(see

is a field.

7 . 1 0 3 P R O P O S I T ION (Lausch-PJobauer ( I ) ) . Every normal subrlroup D of

r[x]

is a l e f t ideal of

Proof. S i m i l a r t o 7.77:

then w i t h

p

if

r[x]. dcD 9 (r[x],+)

and

p,q~I'[x]

= yotnlx+yl+n2x+ ...+yr-l+ nrxtyr

we q e t

Po(d+q)-Poq = Yo+n~(d+q)+Y1+".+Yr-~+nr(d+q)+Y,- y r - n r q - ...-y ,ED.

The d e f i n i t i o n 7.74 o f polynomial completeness i s c a r r i e d o v e r i n t h e o b v i o u s w a y . S i m i l a r t o 7 . 7 5 we ? e t t o r[x]

7 . 1 0 4 T H E O R E M ( L a u s c h - N o b a u e r ( I ) ) . The p o l y n o m i a l l y c o m p l e t e groups a r e e x a c t l y

Z+

and a l l f i n i t e , n o n - a b e l i a n

simple groups. ( a ) As i n 7 . 7 5 , a p o l y n o m i a l l y c o m p l e t e n r o u p i s shown t o b e f i n i t e a n d s i m p l e . I f r i s a b e l i a n and lr[i3 then r i s s o m e EP ( P E P , ~ 2 3 ) .T a k e f E V ( Z ) w i t h f ( 0 ) = 0 , P f(1) = 1 a n d f ( 2 ) = 0.

Proof.

r

231

7d Polynomial near-rings

f

+ S,

and

-

-

(7.64). q has t h e form q : y * yotny f ( 0 ) a n d q ( l ) = f ( 1 ) we o e t y o = 0 B u t t h e n q ( 2 ) = 2 4 f ( 2 ) , whence

If ScPjr], From q ( 0 ) and n = 1 .

r

i s n o t polynomially complete.

( b ) C o n v e r s e l y , a2 i s e a s i l y shown t o be p o l y n o m i a l l y c o m p l e t e , w h i l e f o r f i n i t e n o n - a b e l i a n s i m p l e clroups each p 6 : y 6ty-6 (6Er) i s i n P Cr]. Consequently I ( r ) E P[r]. B u t I ( r ) = M(r) f o r t h e s e o r o u p s ('7.46). +

7 . 1 0 5 R E M A R K T h i s r e s u l t i s t r a n s f e r r e d t o Q - g r o u p s by L a u s c h ( 2 ) . See p a r t 7 . ) of t h i s p a r a a r a p h . The f o l l o w i n g r e s u l t i s e a s y t o p r o v e .

A g a i n , r c a n be c o n s i d e r e d a s an r [ x ] and a n (cf. 7.80). B u t i n c o n t r a s t t o 7 . 8 8 we g e t

P[r]-group

i s a s r[x]and P r r l - o r o u p n o t o f t y p e 2 i n g e n e r a l . ( C o n s i d e r an a b e l i a n , b u t n o t s i m p l e n r o u p a n d you have a c o u n t e r e x a m p l e . )

7.107 REMARK

B u t the theory of e n c l o s i n o i d e a l s s t i l l works: 7 . 1 0 8 EXAMPLES Let

A 4

rcy.

(a) (A)

( = t h e i d e a l of

(b) :

= (A:r)

4

Then

r[x]

g e n e r a t e d by A ) sl rrx].

r[x].

See Lausch-Nobauer ( 1 ) o r Hule (1) f o r a C o m p l e t e l y s i m i l a r t o 7 . 8 4 we g e t

d e s c r i p t i o n of ( A ) .

87

232

TRANSFORMATION NEAR-RINGS

I sl

7.109 T H E O R E M (Lausch-Nobauer ( 1 ) ) . For each A 9

i s e x a c t l y one

r

with

r[x]

there

(A)cIt.

7 . 1 1 0 DEFINITION A g a i n , A i s c a l l e d t h e e n c l o s i n q i d e a l

of I .

C o n c e r n i n g s i m p l i c i t y we g e t s u r p r i s i n g l y

r[x]

7.111 THEOREM ( H u l e ( 1 ) ) .

Irl

i s never simple (unless

= 1).

Proof. Let

r

h a v e a t l e a s t 2 e l e m e n t s and t a k e some n o n -

-

zero YET. S u p p o s e t h a t r[x] i s simple. C o n s i d e r t h e n e a r - r i n o epimorphism 9:

r [XI

yo+nlx+.,.+n xty Clearly

g

+

r

6.

r

+

z2

(nl+ ...+ n r ) * l

Hence Ker g

=

a n d g i s an

{ij}

i s o m o r p h i s m . C o n s e q u e n t l y IrLx] i s q u i t e hard t o f u l f i l l s i n c e

1

=

1Z21 = 2

r[x]

which

i s infinite.

7 . 1 1 2 T H E O R E M ( M e l d r u m - P i l z - S o ( 1 ) ) . The o n l y i d e m p o t e n t s i n r[x] a r e x a n d t h e c o n s t a n t p o l y n o m i a l s y i n r r x ] .

F o r t h e proof s e e t h e paper mentioned in 7 . 1 1 2 . 7 . 1 1 3 C O R O L L A R I E S L e t r be a g r o u p . ( a ) r [ x ] has e x a c t l y l r l + I idernpotents. ( b ) If e i s idempotent in ro[x] then e = 0 o r e = x . ( c ) I f { O l t N < r [ x ] and N h a s an i d e n t i t y e t h e n e = x 7 . 1 1 4 T H E O R E M All s t r i c t l y maximal l e f t i d e a l s o f

c

r[x]

are:

( a ) Lp:= { y i + z i x / y i EI', z i E Z Z ~ z z i E ~ 3Z f o r some 1 prime p . ( b ) L A : = ( A : y ) , w h e r e A i s a maximal normal s u b g r o u p o f r n o t c o n t a i n i n g t h e d e r i v e d g r o u p [ I ' , r ] a n d Y E r\A

or y

= 0.

( c ) L b : = ( B : o ) , w h e r e B i s a maximal normal s u b g r o u p o f r containing [r,r]

233

7d Polynomial near-rings

1 i: y i + z i x

( d ) L(@,p):=

i

p r i m e p a n d @ E Hom

I

t-T'[x]

(r,

@ ( C y.) 5 1 z . 1

1

(mod

n))

for a

B ). P

The p r o o f c a n be f o u n d i n M e l d r u m - P i l z - S o

(2).

The i n t e r s e c t i o n

o f a l l these f o u r c o l l e c t i o n s o f s t r i c t l y maximal l e f t i d e a l s gives the "d-radical

o f 17[x-]. F o r a g r o u p

G , l e t U ( G ) be t h e

i n t e r s e c t i o n o f a l l maximal normal subgroups o f G ( a ( G ) o f G).

as the Baer-radical

Note t h a t t h i s a p p l i e s t o a l l s o l v a b l e groups, 7.116

COROLLARY

Concerning

v =

Jv( I -

2

1

Z[x]

i s known

T h e n i t i s s h o w n i n t h e same p a p e r :

=

f o r instance.

10) f o r each v .

we g e t ( a g a i n i n

t h e same p a p e r )

7 . ) POLYNOMIALS O V E R R - G R O U P S We now s e e , p.

how t h e g e n e r a l i z a t i o n s m e n t i o n e d a t t h e e n d o f

2 1 7 c a n come t r u e .

We s t a r t b y f i x i n g a v a r i e t y P o f R -

g r o u p s and, a c c o r d i n g t o t h e l i n e s on p.

216,

we r e s t r i c t o u r

c o n s i d e r a t i a n s t o t h e case X = { x l . T h i s i s t h e reason f o r w r i t i n g

7 . 1 1 8 NOTATION Av[x] over A i n

v.

:=

A({x},D)

i s the polynomial algebra

P(A) i s the s e t o f a l l polynomial functions

f r o m A i n t o A. Even i f ? p i s a " w e l l - b e h a v e d " "normal

f o r m s " f o r t h e elements i n Av

the variety o f a l l rings, RR[x]

variety,

[x].

For instance,

v: 2 2 +z x +r5x +...I k 3

i f ?? i s

we g e t f o r R E

x+xr +Tr.xr ri&R,zi~Z} 2 ~ a n d t h e s i t u a t i o n i s e v e n much w o r s e i f b i s t h e v a r i e t y o f =

{ ro+zlx+r

there might not exist

1

57

234

TRANSFORMATION NEAR-RINGS

n e a r - r i n g s , f o r example. On t h e o t h e r h a n d , f o r t h e v a r i e t y & o f a b e l i a n g r o u p s a n d A E J# we g e t A Jk [ x i = { a + z x l a L A , Z E Z l . I t s h o u l d be n o t e d , h o w e v e r , t h a t A [ x ] d o e s n o t d e p e n d on t h e variety A i s taken from. For i n s t a n c e , f o r the ( a b e l i a n ) g r o u p Z we g e t i n t h e v a r i e t y $ o f g r o u p s . "@[XI = {ZOfZlX+Z f Z X+...+Z x+zn+,I zi r 3 2 3 n ~

z

while

Zd[x]

[XI

but

=

I z o + z l x ~ z o , z lI Z l ;

=

Cx-rz +z x I z o , z , 0

1

F_

Z l holds in both cases.

We s t a t e w i t h o u t t h e t h e s i m p l e p r o o f 7.119 -

PROPOSITION I f zp t h e n A [x] and

v

i s a v a r i e t y o f ! L - g r o u p s a n d A c 3P ( A ) a r e n e a r - r i n g s w.r.t. + and c

.

The c o r r e s p o n d e n c e p+P i s a n e a r - r i n g e p i m o r p h i s m f r o m

AD;^]

onto

.

P(A)

As b e f o r e , we d e n o t e t h e z e r o - s y m m e t r i c p a r t s o f A].[' and P ( A ) by A:[x]and P o ( A ) , r e s p e c t i v l y . I f A , B ~ v a n dh i s an epimorphism form A t o B then h e x t e n d s u n i q u e l y t o a ( n e a r r i n g ) epimorphism h * : Av[x] + B3[ x ] w i t h h * i A = h and h ( x ) = x . I f h i s a n i s m o r p h i s m , h * i s an i s m o r p h i s m , t o o ( s e e ( L a u s c h - N o b a u e r ( 1 1 , p . 1 5 ) . There e x i s t , h o w e v e r , n e a r - r i n g homomorphisms h b e t w e e n p o l y n o m i a l n e a r - r i n g s w i t h h(x)

x:

7.120 EXAMPLE ( S o ( 1 ) ) . Let N = Z Z p [x] ( i n t h e v a r i e t y o f c o m m u t a t i v e r i n g s w i t h i d e n t i t y ) . Tlien n n h l ( a o + a l x +. . . +a n x ) = p ( a o + a 1 x t . . . t a n x ) n and h 2 ( a o + a X + . . . + a n x ) = p ( a , + . . .a n ) x + p a o 1

d e f i n e two d i f f e r e p t q e a r - r i n g hornornorohisfis f r o m N t o

7d Polynomial near-rings

7.121 -__

235

D E F I N I T I O N L e t A b e a n y !:-group. i s c a l l e d _. c om p a t i b l e i f f o r each i d e a l

fFM(A)

we g e t a l : a 2

(mod I ) - - f ( a

) - f ( a 2 )

(mod I ) .

1 s e t o f a l l compatible functions from A t o A.

I of A

C(A)

i s the

Hence each i d e a l of

A i s an i d e a l o f t h e C ( A ) - g r o u p ( A , + ) .

In particular f ( i ) E

I

each f

E

I o f A,

f o r each i d e a l

each i c I and

Co(A).

-~ 7 . 1 2 2 PROPOSITION F o r e a c h c ? - g r o u p A ,

C(A)

i s a near-ring with

P(A) 5 C(A) 5 M(A). -~ Proof.

C(A) 5 M(A)

i s e a s i l y seen.

functions are compatible, { i d A } " Mc(A) Near-ring algebra

i d A and t h e c o n s t a n t

so i s each p c P(A),

generates P(A).

theory not only receives contributions

(cf.

1.60),

e.g.

since

from universal

b u t a l s o pays something back.

We g i v e

such an a p p l i c a t i o n and t h e n numerous a p p l i c a t i o n s o f t h i s application.

r

I f y o i s a n e l e m e n t o f some c l r o u p generated by ko o f the form ~ =

+

c o n s i s t s o f a l l f i n i t e sunis o f e l e m e n t s

~ w~i t h-

{ p ( y ) I p cPo(T)}.

then t h e normal subgroup

y I ,

i.e,

To = { f ( y o ) l f c

=

This motivates the following r e s u l t which

describes generated i d e a l s i n 0-groups 7.123 -

I(r)}

completely.

THEOREM L e t A b e a n ( 2 - g r o u p a n d a € A .

Then t h e i d e a l

(a) generated by a i s given by

(a)

{Tp(a)

=

1 T E Po(A)I.

If B r A then the ideal

(6)

=

(B)

generated bv B i s q i v e n by

C (b).

b EB

Proof.

I t s u f f i c e s t o show t h e p a r t f o r ( a ) .

L e t N:= whence

Po(A).

Since ( a ) i A,(a)<

N

A,hence

( a l l

N

I p ( a ) l p E P o ( A ) l = Na c _ ( a ) .

Conversely,

we w i l l

o f A containing a,

s h o w t h a t t h i s Na i s a n i d e a l f r o n i w h i c h we g e t ( a ) S N a .

57

236

TRANSFORMATION NEAR-RINGS

( i ) Because o f i d A ( i i ) Clearly,

E

N,

a = i d ( a ) E Na.

Na i s a s u b g r o u p o f ( A , + ) .

b + c -b ) ( a ) , where ( i i i ) b b E A\dp(a) E Na:b+p(a)-b = (b i s the constant polynomial function w i t h value b Now b + F - b-

E

P(A) and (b + p -b)(O)

b + p - b N~ a n d

so

b+p(a)-b

E

Na.

b+p(O)-b

=

=

b-b = 0,

H e n c e Na i s n o r m a l

i n (A,+). ( i v ) L e t w b e an n - a r y o p e r a t i o n on A a n d bl

,... ,bn

b:

=

w(

E

A, F 1 ( a )

bl+Fl ( a )

,... , p-n ( a )

,. . . , b n + F n

(a)

c Na. C o n s i d e r

1-

cu(bl,.

. . ,bn).

L e t q be t h e polynomial w

(bl+pl

and

,.. . , L n + p n ) -

One p o s s i b l e a p p l i c a t i o n ( c f . 7.124

w (bl

q(O)=O, t h e r e f o r e

,. . . , b n ) .

b = :(a)

2.52

(b)!)

T h e n :(a)

=

b

EN^.

i s given in

THEOREM L e t t h e a - g r o u p A be a s u b d i r e c t p r o d u c t o f

s i m p l e n - g r o u p s A i ( i ~I ) a n d s u p p o s e A h a s t h e D C C I . T h e n A i s a f i n i t e d i r e c t sum o f t h e s i m p l e a - g r o u p s A ( j

E

J s I ) .

-~ Proof.

Since A i s a s u b d i r e c t p r o d u c t o f t h e Ails

j

there

there i s a (by the DCCI f i n i t e ) family o f ideals K. J n ( s a y w i t h j E {I , n ) 1 a n d fl K . = C O I . S u p D o s e t h a t j=lJ

,. . .

n i s minimal w i t h t h i s property. each

K.

i s a maximal

Since A/K

Aj, j

i d e a l . We s h o w t h a t A =

J

n

@

j=1

A.. J

,..., O,aj,O, . . . , 0 ) b u t a 4 Ai t h e n a + 0

I t s u f f i c e s t o show t h a t e a c h ( 0

whenever a . J

E

A.. J

If a

E

fl

At

tEJ t+j

and ( a ) = { p ( a ) l n E P o ( A ) I by 7.123.

Hence t h e r e i s

some p. E P ( A . ) w i t h a j = p . ( a ) E A . c A . J 1J O J Again by 7.123, a = pj(a) E At. Hence a l l t - t h j t+j components ( w i t h t t j ) o f a a r e z e r o , whence

n

(0,. . . , O,aj,O

,..., 0)

=

a

E

A.

E A

7d Polynomial near-rings 7.125 -

T H E O R E M Let

2'

237

be a v a r i e t y o f R - g r o u p s ,

i d e a l o f A and a E A \ M . T h e n ( M : a ) = Cp E A [ x ] ( p ( a ) = p o a E M ) ( s e e 3.92) l e f t i d e a l of AV[x].

v

A E V , M a maximal

i s a s t r i c t l y rraxirnal

V

P r o o f . L e t N : = A [ X I . A i s a n N - g r o u p , so ( M : a ) S p , N by 1 . 4 2 . S i n c e a 4 M , x $ ( M : a ) , whence ( M : a ) { N. Now s u p p o s e t h a t U 4N N i s s t r i c t l y between 0

( M : a ) a n d N . T a k e u E U \ ( 1 l : a ) . T h e n u o a & M . Now u - u o a E ( M : a ) c U , whence uoa E U . Hence M c U A A . The i d e a l ( U n A ) g e n e r a t e d by U A A e q u a l s A . S i n c e U5 N , AGU by 7 . 1 1 9 . H e n c e x - a E ( M : ~ ) E U , a n d NO V a E U i m p l i e s x E U . S i n c e A " Ex) g e n e r a t e s A [ X I , IT U = A [XI, a contradiction. T h i s e n a b l e s us t o c o m p u t e a n u p p e r b o u n d f o r t h e ( n e a r - r i n g ) 7J r a d i c a l s o f A [x] and P ( A ) .

v IV(A

7 . 1 2 6 T H E O R E M F o r w E { 0 , 1 / 2 , 1 , 2 1 we g e t [ x ] )c_ (R(A):A) where X ( A ) i s t h e i n t e r s e c t i o n o f a l l maximals i d e a l s o f A ( t h e " r a d i c a l of A " ) .

Proof. jv(Av[x] = (

7.127

n

If 22(Av[X]

M:A)

=

15

n

(M:a)

=

M max. aEA

(~(A):A).

M max.

EXAMPLES L e t be t h e v a r i e t y o f c o m m u t a t i v e r i n q s w i t h identity. ( a ) L e t RET be s e m i s i m p l e w i t h ( R , + ) t o r s i o n f r e e .

Then ~,(Rv[x]),(O:R) = {O}, h e n c e ~ ( R v[ X ] ) = t o } . V I n p a r t i c u l a r , we r e p r o v e d 7 . 1 0 2 ( a ) . 7 ( b ) If R i s a f i n i t e f i e l d o f order >2 than &(R[x]) = = (0:R).

zr

( c ) By 7 . 9 8 ( c ) , . d 2 ( Z 2 [ x ] ) c ( 0 : Z l 2 ) . ( d ) See a l s o 7.115 - 7.117. I n P i l z - S o ( 1 ) i t i s shown t h a t i f F? i s a f i e l d w i t h c h a r R f 2

21,2(R*[x])

then a l s o proved.

= {O}.

I n t h i s paper, the following r e s u l t i s

,$ 7 TRANSFORMATION NEAR-RINGS

238

7 . 1 2 8 T H E O R E M L e t R be a r i n g w i t h i d e n t i t y . Then P o ( R ) i s a r i n g i f f R i s a B o o l e a n r i n g . I n t h i s c a s e we g e t f o r all v

E

a,(

{0,1/2,1 y2}:

7

where i s t h e Jacobson r a d i c a l P ( R ) ) = ? ( P o ( R))+8.( R ) o f r i n g t h e o r y and Z ( R ) i s t h e i n t e r s e c t i o n o f a l l maximal submodules of t h e P o ( R )

- module R .

The s t a t e m e n t c o n c e r n i n g J v ( P ( R ) ) w i l l f o l l o w f r o m 9 . 7 7 . We rema i n a t P ( R a n d c i t e a r e s u l t o f ( K e l l e r - O l s o n ) . 7.129

= P =

I P ( Z n )I

$ -1 ) I

B(k)lP(Z

IP(ZD)l

=

In t h e r e , B ( k ) i s the smallest t E M with

pp.

b y r e p e a t e d appl i c a t i o n of 7.129.

There a r e numerous n e a r - r i n g s 7.130 _~

for k ? 2 and

between P ( A ) a n d M ( A ) .

DEFINITION L e t A be a n R - g r o u p a n d n E N .

(a) LnP(A):= (b) LP(A):=

if

fi

E

M(A)l

LnP(A).

'd

-

T s A , I T l < n , v d P E P(A):flT = P / ~ ? The e l e m e n t s i n L P ( A ) a r e c a l l e d

n EW

l o c a l polynomial f u n c t i o n s . Hence l o c a l p o l y n o m i a l f u n c t i o n s c a n be i n t e r p o l a t e d b y p o l y n o m i a l f u n c t i o n s , a n d we a r e b a c k t o t h e t o p i c s t r e a t e d o n p a g e s 1 3 3 / 1 3 4 and 2 1 9 / 2 2 0 . F i r s t we s t a t e 7 . 1 3 1 P R O P O S I T I O N L e t A be an R - g r o u p . Then L P ( A ) and e a c h L n P ( A ) are near-rings with P ( A ) < L P ( A ) . ... i L n P ( A ) i . 5 L ~ P ( A ) IL ~ P ( A )= C ( A ) Z L ~ P ( A ) = M ( A ) . Proof.

I t i s easy t o see t h a t L P ( A ) ,

L n P ( A ) and C ( A )

a r e subnear-rings of M ( A ) with P ( A ) s L P ( A ) s L n P cL,(A) = M ( A ) . Let n - 2 and a : b in A .

..

7d Polynomial near-rings

239

T h e n t h e r e i s some p c P ( A ) w i t h f ( a ) = p ( a ) a n d f(a) = p(b).

Hence f ( a ) : f ( b )

shown t h a t L n P ( A ) c C ( A ) . L2P(A)2C(A).

by 7.122

L e t f E C ( A ) a n d s u p p o s e t h a t a,b F A .

Let x zy iff x - y i

(b-a).

S i n c e a :b

we h a v e

f ( a ) z f ( b ) , whence f ( b ) - f ( a ) E ( b - a ) . i s some p EP,(A)

with p(b-a)

q:=

c P ( A ) f u l f i l l s :(a)

=

a n d we h a v e

I t r e m a i n s t o show t h a t

p9(=)+f(a)

=

By 7 . 1 2 3

f(b)-f(a).

f ( a ) and q ( b ) = p ( b - a ) + f ( a ) = f ( b ) .

there

Now

p(O)+f(a) =

=

Hence

f E L2P(A).

L o o k i n g b a c k t o 7.75 7.132

we ( r e - ) d e f i n e :

DEFINITION A i s c a l l e d ( a ) e o l y n o m i a l l y complete i f P(A)

=

M(A).

( b ) a f f i n e complete i f P(A) = C(A). ( c ) l o c a l l y p o l y n o m i a l l y complete i f LP(A) = M(A).

( d ) l o c a l l y a f f i n e complete i f LP(A) = C(A). Obviously, simple.

( l o c a l l y ) p o l y n o m i a l l y c o m p l e t e a l g e b r a s m u s t be

We r e m a r k t h a t o u r d e f i n i t i o n s d i f f e r - s l i g h t l y f r o m t h e

ones i n Lausch-NGbauer

( l ) , s i n c e we a r e o n l y c o n c e r n e d w i t h

p o l y n o m i a l f u n c t i o n s i n one v a r i a b l e .

In (II),

Nobauer c h a r a c t e r i z e d c o m p a t i b l e f u n c t i o n s on t h e

r i n g s Z a n d B n . We m e n t i o n w i t h o u t p r o o f . 7.133

THEOREM C ( Z ) = t f : Z + Z I f ( x ) C

7.134

~

ZE , A ( i ) = 1 . c . m .

1 ~ ~ A ( i ) ( ~ j w~h e) r ei , i= O

, . . . ,i .

EXAMPLE ( N o b a u e r ( 1 1 ) ) . f : Z + Z , b u t n o t a polynomial

7.135

o f 1,2

=

x

-t

1 4 2 -2( x + x ) i s c o m p a t i b l e ,

function.

EXAMPLES ( a ) A commutative r i n g R w i t h i d e n t i t y i s p o l y n o m i a l l y complete i f f R i s a f i n i t e f i e l d (7.75).

87

240

TRANSFORMATION NEAR-RINGS

( b ) By L a g r a n g e ' s

theorem,

R i s locally polynomially

complete.

Z4, Z8 a r e p o l y n o m i a l l y c o m p l e t e ,

( c ) The r i n g s Z 2 ,

a f f i n e complete, n o t a f f i n e complete,

respectively

(see P i l z - so ( I ) ) . ( d ) By 7 . 1 0 4 ,

Z 2 and f i n i t e s i m p l e n o n - a b e l i a n

groups

are polynomially complete. ( e ) From 4.66

( a ) we k n o w t h a t L 3 P ( A )

LP(A) = M ( A ) ;

= M(A)

implies

hence A i s l o c a l l y p o l y n o m i a l l y complete

i n t h i s case. ( f ) A near-ring N i s polynomially complete i f f N i s f i n i t e and s i m p l e and i f N has e i t h e r n o n - a b e l i a n

addition

o r a b e l i a n a d d i t i o n w i t h a m u l t i p l i c a t i o n depending on b o t h arguments ( I s t i n g e r

-

Kaiser ( I ) ) .

We s h a l l i m p r o v e t h e s e r e s u l t s c o n s i d e r a b l y . d e f i n e a c o n c e p t due t o S.D.

F o r t h a t , we

Scott, which i s r e l a t e d t o

7.121

(see 7.140).

7.136

DEFINITION L e t N be a n e a r - r i n g and r an N-group

r

i s c a l l e d compatible i f f o r a l l y E T and n E N t h e r e

i s some m~

N w i t h n(y+li')-ny

N i s c a l l e d compatible

N-group

r

=

m6 f o r a l l 6 ~ " .

i f N has a f a i t h f u l

compatible

(we e x p r e s s t h i s b y s a y i n g t h a t N i s c o m p a t i b l e

on r). T h i s c o n d i t i o n means t h a t N a d m i t s a l l h o r i z o n t a l a n d " m a n y " v e r t i c a l t r a n s l a t i o n s (see P i l z ( 6 ) ) : P

T'

I

.'

/'

24 1

7d Polynomial near-rings

w i t h n(6) = n(y.6)

and m ( 6 )

( T h i s p i c t u r e shows a n r .

-_ 7.137

7.138

EXAMPLES M(i-1,

group

r),

(9.69;

r

N

=

E(&)-n(y) = n(y+6)-n(y). M(T)

\

~ ~ ( r ) , ~ ~ ~( r )~ ,

Mdiff(R)

(see 1.4

r.)

which i s compatible on

(a)),

~ ( f o ~r a ~t o p o(l o g ir c a l )

P(R) a n d M a ( r )

abelian) are compatible.

PROPOSITION An N - g r o u p

r

i s compatible i f f i t i s

c o m p a t i b l e as an N o - g r o u p .

r

If

Proof.

and y all

6

i s a c o m p a t i b l e N-group then f o r a l l n

No

E

~ t hr e r e i s some r n c N w i t h n ( y + S ) - n y = m 6 E

r.

D e c o m p o s i n g m i n t o m=mo+nic

1.13 g i v e s moR+mc&

=

6

E

r

0

and

according t o

mo6+mco o n t h e r i g h t s i d e .

C h o o s i n q 6 = o y i e l d s o = m o+mco. for all

for

Hence n ( y + 6 ) - n y

i s a c o m p a t i b l e No-group.

=

rno4

The c o n -

v e r s e i s even e a s i e r and o m i t t e d . 7.139

COROLLARY I f N i s c o m p a t i b l e o n on

7.140

r

t h e n No i s c o m p a t i b l e

r.

PROPOSITION E v e r y n e a r - r i n g N b e t w e e n P o ( A ) a n d C ( A ) ( i n p a r t i c u l a r , e a c h member o f t h e c h a i n i n 7 . 1 3 1 )

i s

c o m p a t i b l e on ( A , + ) . -___ Proof.

f

N a c t s on A i n t h e o b v i o u s and f a i t h f u l way. E

N and a

E

A.

T h e n 4 : = f o ( a-+ i d ) - f o a-,

f u n c t i o n which i s c o n s t a n t l y =a,

Let

where a i s the

i s i n N and

g ( b ) = f ( a + b ) - f ( a ) f o r a l l b E A. W i t h o u t p r o o f we m e n t i o n a r e s u l t o n c o m p a t i b l e N - q r o u p s . 7.141

12, b e c o m p a t i b l e on THEOREM ( L y o n s - S c o t t ( 1 ) ) L e t N I f N h a s t h e DCCL t h e n r i s " n i l p o t e n t b y f i n i t e " , Nr.

i.e.

r

has a n i l p o t e n t normal subgroup A such t h a t

is finite.

r/A

87

242

TRANSFORMATION NEAR-RINGS

P r i m i t i v e compatible near-rings are studied i n Scott (17), i f t h e n e a r - r i n g NEX,

i t i s shown t h a t

Nr w i t h

t i b l e on

ACCL t h e n e i t h e r N

sparse i n a c e r t a i n topology

S.I).

Mo(r)

is finite or N i s

Nr).

( a r i s i n g from "zero sets" i n

Scott mentions i n p r i v a t e conmunications t h a t i n t h i s

r is

second case ( i f else

2

where

i s 2 - p r i m i t i v e a n d compa-

r

infinite)

i s divisible (cf.

We w i l l

r

e i t h e r has p r i m e exponent o r

9.19O(c)).

r e t u r n t o c o m p a t i b l e N - g r o u p s i n $9 9 ) .

N be

7 . 1 4 2 THEOREM - Let

R-group. Proof.

as i n 7.140

and A ( n o n - z e r o )

Let

B be a n o n - z e r o s u b g r o u p o f (A,+)

n o b c B f o r a l l n r N o a n d b~ B . o p e r a t i o n i n A and al Then ai

pi(b)

=

p . , q . E Po(") 1

, . . . ,a n

and bi

= qi(b)

by 7.123.

1

~

(

Let

&A,bl

iii

such t h a t

be an n - a r y

, . . . ,b n

f o r some b

EB. B and

E

L e t p be t h e zero-symmetric

p o l y n o m i a l f u n c t i o n z+(w(pl+ql

-

simple

T h e n N i s p r i m i t i v e on ( A , + ) .

,.. . , p n + q n )

-

,,... p p n ) ) ( z ) . T h e n w ( a , + b l ,... , a n + b n ) . . . ,a n ) = p ( b ) c B. H e n c e B i s a n i d e a l

- w(al,

o f A,

w h e n c e B = A.

S.D. S c o t t e n a b l e s u s t o d r a w i m p o r t a n t

A powerful r e s u l t o f

We w i l l m e n t i o n t h i s r e s u l t i n 9 . 1 7 0

conclusions. 7.143

THEOREM L e t A be a s i m p l e R - g r o u p .

dense i n M ( A )

P roof.

Then P ( A ) i s e i t h e r

(then A i s l o c a l l y polynomially complete) ( i n which case A i s a vector

o r P ( A ) i s dense i n M a f f ( A ) s p a c e o v e r Hom

(4).

Po ( A )

By 7 . 1 4 2

(A,A)

and F o ( A )

and 7.139,

compatible on(A,+).

i s a ring.

P o ( A ) i s p r i m i t i v e and By T h e o r e m 9 . 1 7 0 ,

Po(A) must

e i t h e r be a p r i m i t i v e r i n g o r dense i n Mo(A). 4.52

gives the result.

Now

I n order t o extend t h i s r e s u l t

t o some n o n - s i m p l e R - g r o u p s we n e e d m o r e i n f o r m a t i o n , 7.144

P R O P O S I T I O N L e t A be a s u b d i r e c t p r o d u c t o f R - g r o u p s

( i E I ) . Then f o r e v e r y f c C ( A ) t h e r e a r e u n i q u e Ai f i € C ( A i ) w i t h f ( . . . ,a i , . . . ) = ( . . . , f i ( a i ) , . . . ) for all

( . . . ,ai , . . . )

E

A.

If f

E

i E I.( F o r t h e p r o o f s e e e . g .

P(A) then fi P i l z (25)).

E

P(Ai)

for all

7d Polynomial near-rings

7.1_ 45 _ THEOREM _ _ _ ~L e t A b e a s

i n 7.144.

o f A i n P(A) then P(A)/Ji

=

243

I f Ji

denote the a n n i h i l a t o r

i s i s o m o r p h i c t o a sub-

P(Ai)

d i r e c t product o f the P(Ai)'s. P-r o o f .

We a s s i g n t o e v e r y f E P ( A )

fi E P(Ai)

i n 7.144.

f r o m P(A)

i n t o P(Ai)

with kernel

= z e r o map}

= Ji.@ i s

i f p i € P(Ai)

t h e n pi

) with aik) EA.

i d

(.

Ai

. . ,ai

(k)

k e r @= {p!pi

an endomorphism: i s a w o r d p . = w ( a i( 1 )

$

=6=

,. .. ,ai ( n ) ,

We r e p l a c e a i k ) b y some

,. . . ) = : d ! k ) A a n d fL\(xi ( 1 ) ,... , Z j n ) , i d A ) E E

1

Then p : =

the uniquely determined

T h i s g i v e s a homomorphism

i d

by idA. Ai P ( A ) a n d @ ( p ) = pi.

Now t h e h o m o m o r p h i s m t h e o r e m d o e s t h e r e s t o f t h e

job,

n Ji

together w i t h t h e remark t h a t

iEI T h i s shows t h a t e a c h P ( A i )

i s a homomorphic

=lo1 .

image o f P(A) i f

A i s a s u b d i r e c t product o f t h e Ails. The n e x t r e s u l t f o l l o w s 7.146

and 7.145.

COROLLARY L e t A b e a s u b d i r e c t p r o d u c t o f t h e s i m p l e Q-groups P(Ai)

Ai

NOW 7.124

( i

E

I).

Then P ( A ) i s s e m i s i m p l e a n d e a c h

i s dense i n M(Ai)

i n M - (Ai) atf

7.147

from 7.143

( i f Po(Ai)

( i f Po(Ai)

i s not a ring) or

i s a ring).

g i v e s us

THEOREM L e t A b e a s u b d i r e c t p r o d u c t o f s i m p l e i 2 - g r o u p s

Ai

( i

F;

I ) s u c h t h a t P o ( A ) h a s t h e DCCL.

Then P ( A ) i s t h e

d i r e c t sum o f f i n i t e l y many o f t h e P ( A i ) ' s .

7.148

e i t h e r equal

t o M(Ai)

( w i t h d i m Ai

finite).

( w i t h Ai

COROLLARY L e t A b e a s i n 7 . 1 4 7 Po(Ai)'s

are rings (c.f.

finite)

Each P(Ai)

i s

o r t o Maff(Ai)

such t h a t none o f t h e

7.128!).

Then P ( A ) i s f i n i t e

and 2-semisimple. Finally,

we c l o s e w i t h some r e m a r k a b l e e m b e d d i n g t h e o r e m s .

p r o o f s c a n b e f o u n d i n M e l d r u n i - P i l z - So ( 1 ) .

The

5 7 TRANSFORMATION NEAR-RINGS

244

7.149 -

T H E O REM ( a ) For every n e a r - r i n g N t h e r e i s a v a r i e t y v o f R-groups a n d some A

EV

with NGA

v-

[XI.

( b ) T h e r e e x i s t d . g . n . r ' s w h i c h c a n n o t b e embedded i n some r [ x ] (1' i n t h e v a r i e t y i o f g r o u p s ) .

s

( c ) E v e r y f i n i t e n e a r - r i n g c a n b e e m b e d d e d i n some P ( 1 ' ) for a finite,

simple non-abelian

group

r.

( d ) F o r e v e r y g r o u p T t h e r e i s some g r o u p A w i t h P ( A ) (A c a n b e c h o s e n a s ( T4[ x ]

r 4LX]G

,+)I.

(e) Not every abstract a f f i n e near-ring (see 9.71) b e e m b e d d e d i n some A a b e l i a n groups.

J(

[ X Iw,h e r e A

can

i s t h e v a r i e t y of

But every near-ring N i s abstract

a f f i n e i f f N i s i s o m o r p h i c t o some A

x

[XI, where

i s a v a r i e t y o f ( r i n g - ) modules. ( f ) W i t h a s i m i l a r i d e a as f o r ( a ) (namely b y a d d i n g u n a r y o p e r a t i o n s ) , one can f i n d f o r each c o m p a t i b l e nr.

N some R - g r o u p A w i t h M

and 7.137,

=

P(A).

Hence, b y 1.86

e v e r y n r . c a n be embedded i n a c o m p a t i b l e

one, e v e n i n a P ( A ) - t y p e one ( S . D .

Scott,

private

communication).

F o r many p u r p o s e s i t w o u l d b e v e r y v a l u a b l e t o h a v e a b e t t e r knowledae of t h e i d e a l o f a l l p o l y n o m i a l s which i n d u c e t h e zero function.

This i s j u s t the kernel o f the (near-ring)-epi-

rnorphism w h i c h a s s i g n s t o e a c h p o l y n o m i a l As we h a v e s e e n i n t h i s c h a p t e r ,

i t s polynomial function.

t h i s k e r n e l d e c i d e s i f one can

i d e n t i f y polynomials and polynomial f u n c t i o n s .

I t a l s o has se-

veral connections with the r a d i c a l s o f polynomial near-rings. I n Meldrum-Pilz

( 1 ) these questions are f u r t h e r investigated,

but they are f a r from being solved.

8 ) CONCLUDING R E M A R K S We c l o s e t h i s s e c t i o n w i t h some r e m a r k s c o n c e r n i n g q u e s t i o n s r e l a t e d t o p o l y n o m i a l a n d p o l y n o m i a l - 1 ike n e a r - r i n g s .

7d Polynomial near-rings 7.150

245

R E F~ I A R_ KS _

R,S& R[x] 2 S[x] i m p l i e s t h a t R 2: S ( t h i s f o l l o w s from 7 . 1 1 9 ) . He a l s o r e m a r k e d t h a t e a c h s u b n e a r - r i n g o f M ( F ) ( F a f i e l d ) which c o n t a i n s a l l c o n s t a n t f u n c t i o n s i s automatically simple. P ( R ) i s d i r e c t l y decomposable i f f this a p p l i e s t o R .

( a ) Nobauer ( 6 ) remarked t h a t f o r

( b ) If C i s a c o m p o s i t i o n r i n q a n d U i s a m a p C * C t h e n D i s c a l l e d a d e r i v a t i o n ( M e n g e r (3). MUller ( I ) , Lausch-Nobauer ( l ) , Nobauer ( 9 ) ) i f f o r a l l a,bcC: (1) D(atb) = D(a)tD(b)

("sum r u l e " )

( 2 ) D ( a . b ) = D(a).bta.D(b) ( 3 ) D(aob) = ( D ( a ) o b ) . D ( b )

("product r u l e " ) ("chain rule")

C l e a r l y t h e z e r o e n d o m o r p h i s m on C i s a ( t r i v i a l ) d e r i v a t i o n . R[x] has a l s o a n o n - t r i v i a l d e r i v a t i o n , namely the u s u a l o n e : D: p * p ' , All o n R[x] a r e v i v e n by D r : p rep', where r E R i s i d e m p o t e n t -+

.

(La.usch-Nobauer ( 1 )) N o b a u e r ( 6 ) showed t h a t t h e c o m p o s i t i o n r i n q M ( R ) h a s no d e r i v a t i o n s e x c e p t t h e t r i v i a l o n e . I f R i s a f i n i t e f i e l d , t h e same a p p l i e s t o P ( R ) (by 7 . 7 5 ) . I f R i s a n i n f i n i t e i n t e g r a l domain t h e n Y i i l l e r ( 1 ) showed e . 9 . t h a t i f ( R , t ) i s t o r s i o n - f r e e , t h e suma n d t h e c h a i n r u l e imply t h e p r o d u c t r u l e . ?!uller

stLdied a l s o " d e r i v a t i o n s " i n near-rings as well a s " i n t e r v a t i o n s " ( s e e ( 8 ) ) . C f . a l s o SeppXla ( 1 ) . ( c ) I n v e r t i b l e elements ( w . r . t . 0 ) are s t u d i e d i n LauschN o b a u e r ( 1 ) a n d Suvak ( ( l ) , ( 2 ) ) . T h o s e PER[X] s u c h t h a t 5;; i s b i j e c t i v e ( = i n v e r t i b l e ) a r e c a l l e d p e r m u t a t i o n p o l y n o m i a l s , were c o n s i d e r e d by many a u t h o r s a n d a r e p r e s e n t e d e x t e n s i v e l y i n Lausch-Nobauer ( 1 ) .

( d ) C l e a r l y R[x] and P ( R ) a r e i n g e n e r a l non-commutative n e a r - r i n q s . T h o s e p o l y n o m i a l s w h i c h commute w i t h a c e r t a i n f a m i l y o f o t h e r s w e r e s t u d i e d e . g . by Kautschi t s c h ( 1 ) and Lausch-Nobauer ( 1 ) . C a l l C c_ F[x! (F a f i e l d ) a P-chain ("permutable

37

246

TRANSFORMATION NEAR-RINGS

chain") i f V c E C : jcI > O , v k E N j c E C : [c] = k a n d c , o c 2 = c 2 c c , f o r a l l c 1 , c 2 c C. E x a m p l e s : ( 1 ) The P - c h a i n o f p o w e r s { x , x 2 , x 3 , . . . I . -~

( 2 ) The P - c h a i n o f t e b y s h e v p o l y n o m i a l s I t l , t 2 , t 3 , . . I (where t n i s d e f i n e d via cos n$ = tnocos $ over F = Q a n d t h e n t r a n s f e r r e d t o F(x] for an a r b i t r a r y f i e l d F:

.

t.1 = x t 2 = - 1 + 2 x 2 -, t3 = -3xt4x 3 t q = 1 - 82~t 8 x 4 Also,

tnotm =

trim.)

I f 11 i s a l i n e a r p o l y n o m i a l and C i s a P - c h a i n t h e n C,: = ~ L o c ~ L - ~ l c t z Ci ls a P - c h a i n , t o o , c a l l e d a conjuqate P-chain. One c a n s e e ( t h e p r o o f s a r e n o t t o o e a s y - s e e L a u s c h N o b a u e r ( l ) , p . 156 - 1 5 9 ) : (a) I f

c i s a P-chain then C c o n t a i n s t o each e x a c t l y one c w i t h [c] = k .

kEIN

( 6 ) All P-chains o v e r a f i e l d F a r e conjucrates of

e i t h e r t h e P-chain o f powers o r o f t h e P-chain o f F e b y s h e v polytiomi a 1 s . ( e ) L a u s c h - N o b a u e r ( l ) , c h . 5 , c o n t a i n s more i n f o r m a t i o n on r[x] and P ( r ) For example, t h e c l a s s e s E k ( r ) o f a l l k - p l a c e f u n c t i o n s c r e n e r a t e d by a l l " k - p l a c e e n d o m o r p h i s m s on r " a r e c o n s i d e r e d ( " k dimensional composition qroups"). T h e s e a r e more e x a m p l e s o f d g n r . ' s , a n d r e s u l t s s i m i l a r t o o u r 6.33 and 7 . 4 6 a r e o b t a i n e d .

.

Fo[x] ( F a f i e l d ) . This i s a near-ring w i t h identity, b u t w i t h o u t divisors of zero. Fo[x] i s a l s o n o t r e c r u l a r $ 9 f ) ) . The

( f ) Heatherly ( 7 ) considered

7d Polynomial near-rings

ideals

Ik:

247

ktn = {

1

aix

InEIN

So

Fo[x]

i=k

descendinq chain.

aiEF1

0'

form a s t r i c t l y

does n o t f u l f i l l t h e

7.97).

DCC o n i d e a l s ( c f .

( 9 ) Nobauer ( 6 ) a l s o c o n s i d e r s t h e n e a r - r i n q s R(x) and R ( x ) o f a l l " r a t i o n a l p o l y n o m i a l s " and " r a t i o n a l

polynomial

functions".

7.75).

a f i n i t e f i e l d (cf.

R is it

decomposable i f f

( h ) T h e n e a r - r i n q s Ro[[x]] RE%

over

-

R(x)

i s directly

( c f . Remark ( a ) ) . of a l l formal

K a u t s c h i t s c h (1) - ( 8 ) and o t h e r s .

...,gn)

component o f

If

f E g:

II + II

. . ,X,,I~)~, defined

M : = ( ! I 0[cx,,.

i n t h i s s e t a composition

degree,

power s e r i e s

were considered by F r o h l i c h ( 9 ) , Cartan ( l ) ,

Frohlich ( 9 ) studied = fi(gl,

iff R i s

R(x) = M(R)

Again

(where

by

"0"

fi

denotes t h e i - t h

fEM.

all

c=>

fi

and

qi

t h e n one can c e f i n e i n

i n t h a t way t h a t

(M/:,t,o)

h a v e t h e same M/E

an a d d i t i o n

i s a near-ring

o f number-theoretic relevance. C a r t a n ' s r e s u l t was a l r e a d y m e n t i o n e d i n 1 . 1 2 . Graves-Malone

(3) looked a t the subnear-rina

N s a t i s f i e s t h e r i g h t Ore c o n d i t l o n ( 1 . 6 4 ) and i s integral.

( i )H e l l e r ( 1 ) d e f i n e d g e n e r a l i z e d p o l y n o m i a l s p i n

a

composition r i n g R by t h e p r o p e r t y t h a t f o r a l l f E R there are n €No =

c +clf+...+c,f 0

and c o n s t a n t cO, n

.

. . . ,c n €

R

C

with pof =

There e x i s t composition r i n g s i n which

every element i s a oeneralized polynomial, b u t n o t a polynomial. Anyhow,

t h i s s e c t i o n seems t o b e a w i d e f i e l d f o r f u r t h e r

research.

248

5 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

T h i s c h a p t e r b r i n g s u p t w o i n p o r t a n t c l a s s e s o f n e a r - r i n n s . Isle s t a r t w i t h p e r h a p s t h e most i m p o r t a n t c l a s s , t h e n e a r - f i e l d s . A t h o r o u q h t r e a t m e n t w o u l d r e q u i r e n e a r l y a whole b o o k . B u t t h e r e a r e s e v e r a l e x c e l l e n t p r e s e n t a t i o n s o f p a r t s of t h i s t h e o r y ( e . 4 . Karzel ( I ) , Kerby ( 7 ) a n d I r ! a h l i n r l ( 6 ) ) s o t h a t we d a r e t o n i v e t h e t h e o r y D a r t l y b ! i t h o v t p r o o c s . r i r s t we c h a r a c t e r i z e t h o s e n r . ' s which happen t o be r i f . ' s . A f t e r showinq t h s t t h e a d d i t i v e a r o u p o f a n f . i s a b e l i a n w e rrive a s u p e r - s o n i c t r i p t h r o u c r h t h e r e l a t i o n s between n e a r - f i e l d s and geornet.1.y ( i n c i d e c e q r o u p s , c o o r d i n a t i s a t i o n o f p l a n e s , planar near-fields).

I n b ) we d e a l w i t h p a n a r n e a r - r i n q s . ; h e i r s t r u c t u r e i s 0) arc. d e f i n e d explored ( 8 . 9 0 , 8 . 9 6 , "blocks" aN+b ( a a n d i t i s shown t h a t a p l a n a r f i n i t e n e a r - r i n o t o n e t h e r w i t h i t s b l o c k s forms a t a c t i c a l c o n f i c u r a t i o n ( N , B ) . ? h e c a s e i s a balanced incomplete block desian i s when ( N & ) c h a r a c t e r i z e d i n 8.115 a n d s e v e r a l consequences a r e deduced. The a u t h o r t h a n k s Dr. G . Betsch f o r l e a v i n o h i n u n p u b l i s h e d lecture notes concernin? t h t s paragraph.

249

8a Near-fields

NEAR-FIELDS

al

1.) CONDITIO!4S T O B E A N-EAR-FIELD

We s t a r t w i t h ( c f . 1 . 1 5 )

8.1

PROPOSITION ---I f N i s a nf. then e i t h e r i s z e r o - s y mine t r i c

N

.

nC€Nc,

Proof. I f ---

+ 0,

nC

%

=

Mc(Z,)

-1 1 = nc n c = n C ,

then

Gr N

when c e

lEWc.

So nEN*: n = In = 1 , r e s t i s obvious.

hence

N = {0,1?.

The

8. 2--CONVENTION -

In a l l of o u r s u b s e q u e n t d i s c u s s i o n of nearMc(Z2) o f o r d e r 2 . f i e l d s we w i l l e x c l u d e t h i s s i l l y n f . ( c f . Malone ( 2 ) ) . Evidently, every near-field i s simple. We now c h a r a c t e r i z e t h o s e n e a r - r i n y s w h i c h a r e n e a r - f i e 1 d s : 8.3

THEOREM (Lioh

2 ) , Maxson ( l ) , B e i d l e m a n E q u i v a l e n t a r e f o r N&no:

1). F a i n ( 1 ) ) .

(a) N is a near-field. ( b ) lid $:

{o}

and

nEN*:

Nn = N .

( c ) N h a s a l e f t i d e n t i t y a n d NF1 i s Pi-simple. ( d ) N h a s a l e f t i d e n t i t y a n d N i s 2 - p r i m i t i v e on ( e ) N h a s a l e f t i d e n t i t y a n d N i s 1 - p r i m i t i v e on P r o o f . ( a ) -> ( b ) i s c l e a r .

(a): a,bcN* 3 a ' , b ' c N * : Thus a'(ab) = (a'a)b = b'b = a N i s integral. ( b ) ->

+

".

".

b'b = a A a'a = b'. 0, s o a b 0 and

250

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

T a k e some dEFI:. 3 eEN: ed = d . S o ( d e - d ) d = d e d - d d = 0 . From a b o v e , we n e t

de = d

N o w l e t n be Then

EN*. d ( e n - n ) = den-dn

b

Finally,

nEN*

This shows,that a near-field.

( c ) <->

( a ) ->

8.4

3

n'EN":

(N",.) ( d ) <->

4

R E -M- A R K ( L i g h ( 2 ) ) .

=

0,

whence

en = n .

n ' n = e. i s a oroup and

(e)

(N,t,-)

is

a r e obvious (observe 4.6)

Of c o u r s e , e . 0 .

( c ) i n 8.3 can b?

nEN" 3 n ' c N * : n ' n $. 0 r e p l a c e d by ( c ) ' : "Nd ==! { o } , a n d N N i s N - s i m p l e . " ( F o r ( c ) ' => ( b ) ==> ( c ) => ( d ) =o

=>

(c)'!)

W i t h o u t p r o o f we m e n t i o n t h e f o l l o w i n n r e s u l t s o f Liobi ( 2 ) a n d (1):

8.5

THEOREM Let N I 0 1 be a d g n r . . N i s a s k e w - f i e l d <=> nEN" 33 n ' E N : n n ' n <->

8.6

t/

Nn

nEN":

=

=

n <=>

N.

C O R O L L A R Y A f i n i t e i n t e q r a l d g n r . i s a commutative f i e l d .

+

A dgnr. N i s N-simple.

{Ol

with l e f t i d e n t i t y i s a f i e l d i f f i t

n $: 1 ,

8.7

T H E O R E M FIET)onn, i s a n f . <-> e v e r y R E N , ( i n Eeidlenzn's sense - see 3 . 3 7 c ) ) .

8.8

R E M A R K See A n d r e ( 3 ) f o r a development of a t h e o r y o f

i s qr.

1 i n e a r a1 g e b r a o v e r n e a r - f i e 1 d s " a n d " n e a r - vec t o r - s p a c e s " ( c f . a l s o B e i d l e m a n ( 1 ) ) . S e e a l s o Grijger (l), P e l l e g r i n i ( 1 ) a n d Rado ( 1 ) a s well as 7 . 1 0 2 ( f ) . A very good survey on t h e a p p l i c a t i o n s o f n e a r - f i e l d s i s K a r z e l - K i s t ( 1 ) . "

8a Near-fields

251

2 . ) T H E A D D I T I V E G R O U P OF A N E A R - F I E L D

L e t t h e c h a r a c t e r i s t i c c h a r N of a n e a r - f i e l d N be d e f i n e d a s usual - (\{ahling ( 6 ) d e f i n e s char N : = c h a r Nd but this g i v e s t h e same ( s e e 8.23)).

-

Then one s e e s a s f o r f i e l d s :

8.9

P R O P O S I T I O N L e t N be a n f . a n d i n ( N , + ) . Then

o(1)

be t h e o r d e r o f 1

( a ) If

~ ( l )i s f i n i t e t h e n c h a r II = o ( 1 ) . ( b ) If o(1) i s i n f i n i t e then char N = 0 . ( c ) c h a r N i s e i t h e r 0 o r a prime.

For the foilowinq r e s u l t , c f . a n d apply 1 . 5 . 8 . 1 0 P R O P O S I.___ T I O N ( K a r z e l ( l ) , Maxson ( I ) , Linh-Neal Let N be a n f . . T h e n

(a)

tl

nc:N: ( n 2

(b)

v

n,n'EN: n ( - n ' )

=

(1)).

I <=> n c { 1 , - 1 ] ) , (-n)n'

=

-nn'.

P r o o f . ( a ) : "<=" i s c l e a r ; s o l e t n L = 1, b u t n 4 1. I f char N = 2 then (observe t h a t ( N , t ) i s abelian 2 in t h i s case) (nt1)n = n t n = ( l t n ) . l ; now n+l 0, whence n = 1 , a c o n t r a d i c t i o n . Now l e t c h a r N be 6 2 , a n d 1+1 = : 2 (i 0). Then 2 ( - 1 ) = ( l t l ) ( - 1 ) = - 1 - 1 = - ( l + l ) = (-1)2. ( q - 1 = ( 2 . ( - I ) ) - L ( - 1 ) - 1 . 2 - 1 = -2-1. so

+

Observe t h a t ( 2 - 1 ( - 1 ) t 1 ) ( - 2 ) = 1 - 2 = - 1 , whence ( 2 - ' ( - 1 ) + 1 ) = 2 - l . L e t m : = 2 - 1( n - 1 ) t I . T h e n rn-n = Z - l . ( n - l ) . n + n = 2 ( n 2 - n j t n = 2 - l ( l - n ) + n - 1 t 1 = 2 - ' ( - 1 ) ( n - I ) t ( n - 1 ) + 1 = ( 2 -1 ( - 1 ) t 1 ) ( n - 1 ) t 1 = = (2-')(n-1)+1

n

+

1

n = -1.

qives

= rn = m . 1 .

m = 0,

so

n-1

=

.?.(-I) = -1-1,

wberlce

48 NEAR-FIELDS AND PLANAR NEAR-RINGS

252

( b ) I t s u f f i c e s t o a s s u m e n $. 0 . B u t t h e n ( n - 1 ( - l ) n ) 2 = 1 , s o by ( a ) we g e t n - ' ( - l ) n E { l , - l } . n-'(-l)n = 1 implies -n = n , so char N = 2 . B u t then the r e s u l t i s t r i v i a l . I f n -1 ( - l ) n = - 1 , n ( - 1 ) = ( - 1 ) n . s o n'EN: n(-n') n ( - 1 ) n ' = (-1)nn' = - n n ' .

v

T h e f o l l o w i n g f a m o u s r e s u l t was f i r s t shown f o r f i n i t e n f . ' s by Z a s s e n h a u s i n 1 9 3 6 , f o r i n f i n i t e n f . ' s by B . H . Neumann i n 1 9 4 0 . T h e r e e x i s t s e v e r a l e s s e n t i a l l y d i f f e r e n t p r o o f s . The f o l l o w i n g (due t o K a r z e l ) seems t o be t h e most s i m p l e o n e .

8.11 T H E O R E M ( D i c k s o n ( l ) , Z a s s e n h a u s ( I ) , B . H . Neuriann ( l ) , L i g h (6), ( 1 3 ) , L i g h - M c Q u a r r i e - S l o t t e r b e c k ( l ) , K a r z e l ( l ) , Zemmer ( 2 ) ) . T h e a d d i t i v e group o f a n f . i s a b e l i a n and c h a r a c t e r i s t i c a l l y simple.

V n E N : n ( - 1 ) = - n . Hence by 1 . 1 0 9 ( a ) , P r o o f . By 8 . 1 0 ( b ) (N,+) i s a b e l i a n . Consider f o r nEN* t h e a u t o m o r p h f s m a n : 11 + N x + xn

.

If M take

9 I01 EM*. (m)

a

i s a c h a r a c t e r i s t i c subqroup of

L e t n ' b e a r b i t r a r y i n Pi*. whence M = N .

(PI,+)

Then

= n'EM*,

m-ln' Recall t h a t from 1 . 8 8 ( f ) and ( a ) one s e e s t h a t e v e r y n f . i s i s o m o r p h i c t o a n f . o f b i j e c t i v e m a p p i n q s ( p l u s t h e z e r o map) on a n a b e l i a n group r . S t a n d a r d g r o u p t h e o r y g i v e s us t h e s t r u c t u r e o f t h e a d d i t i v e groups of n f . ' s : 8.12 COROLLARY (Ligh-McQuarrie-Slotterbeck ( l ) , Heatherly (11)). L e t N be a n f . (a) If

char N = 0

then

(N,t)

i s torsion-free divisible,

so the d i r e c t sum of copies of (b)

I f c h a r N = p t h e n (N.t) t h e d i r e c t sum o f c o p i e s o f

(4,+).

i s elementary abelian, so

(a,,+).

253

8a Near-fields

The o r d e r s o f f i n i t e n f . ' s fields (cf.

a r e t h e same a s t h o s e o f f i n i t e

a l s o number 4 . ) ) , f o r 8 . 1 2 ( b )

8.13

COROLLARY I f N i s a f i n i t e n f .

8.14

COROLLARY ( H e a t h e r l y ( 2 ) ,

{Ol

Nd

with

= {Ol

or

some

n'n"

be

4 (N,t)

(0:n")

dn" 4

D = 8.4,

r)

and

{O}.

-8.1.5

i s a finite

r-

Take

(0:n")

dEN$. =

(0:n)

(Nd.t)

4 (N,+),

so

8.16

Hence so

= {Q].

By

Nd = N

and

field. 49,

f o r a characterization

Cohn) o f t h o s e g r o u p s w h i c h c a n b e t h e See a l s o L i r J h ( 1 3 ) .

m u l t i p l i c a t i v e aroup o f a n e a r - f i e l d .

3.)

IOI.

(nENldn = 0 ) 4 ( t J , t ) , ncli":

THE CENTER A N D THE KERHEL O F A N E A R - F I E L D

tIOTATIO:4 center o f

Let

C(N):

(N,.)

= ( n s l i j Wn'EN: and c a l l

Nd

nn'

= n'nl

be t h e

t h e "kernel o f N"

( K a r z e l e t a1 . ) .

8.17

THEOREM ( K a r r e l generated by of

C(N)

( 1 ) ) . The s u b n e a r - r i n n

C(PI)

Proof: 8.18

I o f the nf. N

c o n s i s t s o f a l l sums o f e l e m e n t s

and i s an i n t e o r a l domain.

o f N g e n e r a t e d by

C(N)

straiqhtforward

.

N be a f i n i t e nr.

i s a commutative f i e l d .

$: 0.

R E M A R K See k r a h l i n g ( 6 ) , p . ( d u e t o P. M .

Let

IN[=[, k

kEIN:

hence a b e l i a n .

{ O l -C.

But then

N

N

Consequently,

N i s a nf.,

~ E I P3

simple.

implies

n"kD:

3

then

Liqh (13).

(N,t)

Pi2

Then e i t h e r Proof. Let

and

implies

The s u b n e a r - f i e l d

i s t h e f i e l d of q u o t i e n t s of c a l c u l a t i o n s u s i n s 8.10

I.

and 8.11.

COROLLARY E v e r y n f . N c o n t a i n s a c o m m u t a t i v e s u b f i e l d F .

$8 NEAR-FIELDS AND PLANAR NEAR-RINGS

254

T h e r e i s a ( p o s s i b l y ) d i f f e r e n t s u b f i e l d i n 14: 8 . 1 9 THEOREM ( Z e m m e r ( 1 ) ) . I f N i s a n f . t h e n

Nd

i s a subfield

o f N. 8.20 REMARK I f P I i s a s u b f i e l d o f a n f . N t h e n N can be c o n s i d e r e d a s a v e c t o r s p a c e o v e r M. be denoted b y

I t ' s dimension w i l l

dimb,N.

8 . 2 1 REMARK C l e a r l y

C ( N ) t lid'

More e x a c t l y ( b u t w i t h o u t

p r o o f ) t h e r e i s t h e f o l l o w i n o r e l a t i o n between c e n t e r and kernel o f a near-field: 8.22

THEORErl ( A n d r i !

(2), Idahlino (2)).

i s no p r o p e r s k e w - f i e l d .

See m o r e on t h a t e . q .

L e t N be a n f . w h i c h

Then

i n Gfahlin? ( 6 ) .

We c l o s e t h i s n u m b e r w i t h t h e f o l l o w i n g 8 . 2 3 R E M A R K I f 11 i s -~

4.)

a nf. then

c h a r N = c h a r Nd

(by 8.9(a)).

D I C K S C I I J l4EAR-FIELDS

D i c k s o n o b t a i n e d t h e f i r s t p r o p e r n e a r - f i e l d s i n 1905 b y "distortinq" the multiplication i n a f i n i t e field. We a x i o m a t i z e t h i s p r o c e d u r e , t r a c k i n n t h e p r e s e n t a t i o n o f Wahling (6).

P r o o f s ( o r r e f e r e n c e s where t o f i n d them) can be

found there. F o r t h i s number, c o n f e r a l s o t h e c h a p t e r on D i c k s o n n e a r - r i c o s i n 5 9 d ) . U n l e s s o t h e r w i s e i n d i c a t e d , PI w i l l a l w a y s d e n o t e a n f

255

8a Near-fields

8.24

0: N"

D E F I N I T I O r J A map

n c o u p l i n g map i f

tf

A u t (!I,+,=)

+,,

+ +

i s called a

bn*dm = Q + n ( m ) . n

m,nc.N:

8.25

EXAMPLE

+:

n

+

8.26

NOTATIO?I I f

+

i s a c o u p l i n q map o n FI t h e n

no4m: =

8 .27

[

PROPOSITIfi

idN

i s a c o u p l i n g map o n PI

+0

@,(n)'m

i f

rn

0

i f

m = 0

If 0 i s a c o u p l i n n map o n II t h e n

(N,t,06)

i s again a near-field. (The " c o u p l i n q p r o p e r t y " i n 8.24

c f the associativity o f 8.28

D E F I N I T I O N ( P i , + , oO)

(N,+,*)

o

i s j u s t the restatement

+).

i s then called the @-derivation of

and d e n o t e d as

N*.

i s c a l l e d the Dickson-Vroup o f 4.

{$,lnEN*I N i s s a i d t o be a Dickson n e a r - f i e l d i f N i s t h e + - d e r i v a t i o n

o f some f i e l d F :

F+ = N.

To t h e a u t h o r ' s k n o w l e d o e , a l l k n o w n n e a r - f i e l d s examples

(up t o 7

- see below) a r e Dickson n e a r - f i e l d s .

We g i v e a n e x a m p l e o f a c l a s s o f f i n i t e a n d i n f i n i t e D i c k s o n near-fields which are not f i e l d s : 8.29

EXAMPLE ( Z e m m e r ( 1 ) ) . L e t F b e a c o m m u t a t i v e f i e l d a n d F(x) the f i e l d o f rational

i s a c o u p l i n g map o n

F(x)

functions (7.113(h)).

and

(F(x),t,o$)

i s a Dickson

n e a r - f i e l d. F o r "most"

f i n i t e f i e l d s we q e t i m p o r t a n t c o u p l i n n m a p s . B u t

f i r s t we n e e d t h e f o l l o n i n o

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

256

8.30 DEFINITION (q,n)EIN2

i s called a p a i r of

D i c k s o n numbers

i f ( a ) q i s some p o w e r

o f a prime p.

p'

( b ) Each p r i m e d i v i s o r o f n d i v i d e s q E 3

(c) If

(mod n )

8 . 3 1 THEOREM L e t ( q , n )

then

be a p a i r o f D i c k s o n numbers.

L e t F be the ( G a l o i s - ) f i e l d

(F*,*)

elements.

q-1. 4 does n o t d i v i d e n.

i s cyclic.

l e t H be t h e subqroup o f

a be t h e (Frobenius-)

GF(qn) = GF(pEn)

L e t q be a a e n e r a t o r and

(F*,*)

qenerated by

automorphism

f

-+

F*/H

c a n b e r e p r e s e n t e d as k

~ ( H c J ~ ' ) := a k

Let

E

{ H q , Ho q;T

Aut ( F , t , . ) .

i s t h e canonical epimorphism then

If

0:

qn.

of

fq

q2- 1 Then

qn

with

(F,t,-).

,. . . ,Hg-

n : F*

= An

Let

-+

qn-l

I.

F*/H

i s a couplino

map o n F . rj:

(In = F$ = ( G F ( ~ I , + , ~ @ )

= {xEFI 'dfcF":

af(x)

i s a nf. with

C ( N ) = Nd =

X I= G F ( q ) .

The number o f n o n - i s o m o r p h i c D i c k s o n n e a r - f i e l d s d e r i v e d @(n) , where i n t h i s way ( b y d i f f e r e n t c h o i c e s o f o ) i s 7 0 i s t h e E u l e r - f u n c t i o n a n d i i s t h e o r d e r o f p (mod n ) . F o r more infor-

T h e i r m u l t i p l i c a t i v e groups a r e isomorphic.

mation

see Llineburn ( 1 ) .

8 . 3 2 THEOREM By t a k i n g a l l p a i r s o f D i c k s o n n u m b e r s ,

all finite

D i c k s o n n e a r - f i e l d s a r i s e i n t h e way d e s c r i b e d i n 8.31. T h i s makes t h e q u e s t i o n , w h i c h ( f i n i t e ) n e a r - f i e l d s near-fields,

even more i n t e r e s t i n q . O f course,

i t m i o h t be h a r d

t o v i s u a l i z e Dickson n e a r - f i e l d s w i t h naked eyes. i n s t r u m e n t (see e.q.

a r e Dickson

So we u s e a n

Wahling ( 6 ) ) :

8.33 THEOREM ( " Z a s s e n h a u s - c r i t e r i o n " ) : A f i n i t e n f . N i s a D i c k s o n n f . i f f G: = (N*,-) i s metacyclic (i.e. [G,G] and

G/[G,c]

are cyclic).

I n t h i s case, G has two q e n e r a t o r s where 4 = I N d / .

a,b

with

b - 1a b = a q ,

257

8a Near-fields

I t was Z a s s e n h a u s ,

t o o , who d e t e r m i n e d a l l f i n i t e n f . ' s :

8 . 3 4 THEOREM A l l f i n i t e n f . ' s

-

up t o 7 e x c e p t i o n a l cases

-

a r e Dickson n f . ' s . Now we a r e g o i n q t o d e s c r i b e t h e s e 7 " o u t s i d e r s "

. . "7

N1,.

(numbering i s t h e one o f Zassenhaus):

A l l o f them a r e o f o r d e r p 2 w i t h p = 5,7,11 ( t w o c a s e s ) , 23, 29 o r 59. Since a l l N E I N 1,...,N71 c a n b e c o n s i d e r e d as v e c t o r s p a c e s o v e r Nd o f d i m e n s i o n 2 a n d s i n c e f o r e a c h neN* t h e map x -+ x n i s a n e l e m e n t o f A u t (h') i t suffices t o describe 'I d (N*,*) v i a 2x2-matrices over Nd): i

P

Nd = GF(p)

over

o r d e r o f Ni

(NY,.)

(b! i s a v e c t o r s q a c e

i s generated by

1

5

52

1 ("1

2

11

112

( - 15

-2)

3

7

72

1 (-1

3 -2)

4

23

232

1 (12

-6

5

11

1l2

6

29

2g2

7

59

5g2

The s m a l l e s t D i c k s o n

9

f with

xoy:

-2)

4)

-7 -2)

and

(40

and

2 (0

and

(-13 0

0 2)

0 -13)

15

which i s not a f i e l d i s niven bv

n

(GF(3L),t,o)

-2

5

1 (-12

nf.

( 01 - 1 o)

-

and

--

A:=

=

X * Y

i f x i s a square i n (GF(32),+,*!

I t s m u l t i p l i c a t i v e groitp i s t h e q u a t e r n i o n nro~ii, ( o f order 8).

$8 NEAR-FIELDS AND PLANAR NEAR-RINGS

258

-8 . 3 5 T H E O R E M ( L i g h-Ne a l ( 1 ) ) . Le t N be a f i n i t e n f . s u c h t h a t ncN?nk = n where k i s o f t h e form ( p ~ : I p \ { Z ,l j E l N ) . Then N i s a f i e l d .

5.)

NEAR-FIELDS

k = pj+l

AND DOUBLY TRANSITIVE GROUPS

N e a r - f i e l d s ( a n d some s i m i l a r s t r u c t u r e s ) can be used t o d e s c r i b e " s h a r p l y t r a n s i t i v e " p e r m u t a t i o n o r o u p s . The f o l l o w i n q d i s c u s s i o n f o l l o w s Kerby ( 7 ) . 8.3 6 N O T A T I ON I f M i s a n a r b i t r a r y s e t , l e t S,, be t h e s y m m et r i c g roup o n r l ( i . e . t h e a r o u p of a l l 1-1-maps Y I f kEIN (ml,...ymk)~Mk i s c a l l e d a p r o p e r k - t u p le i f a l l mi's are d i s t i n c t .

r 5 SH i s called 8 . 3 7 D E F I N I T I O N~

-L

M).

_(sharply) k - t r a n s i t i v e (on M ) k i f f o r a l l proper k-tuples (ml,...,rnk), ,~{)EM t h e r e i s ( e x a c t l y ) one YE^ w i t h y ( m i ) = m i f o r a l l i ~ { l. ,. . y k l ( c f . 4 . 2 6 ) . 1 - t r a n s i t i v e groups a r e simply c a l l e d t r a n s i t i v e , the s h a r p l y 1 - t r a n s i t i v e ones a r e j u s t t h e r e g u l a r permutation groups.

(mi,.. .

8 . 3 8 N O T A T I O N L et ( m l , . . . , m k ) ~ M k be p r o p e r a n d r 5 S M . : = {yEI'I i i c f l , . . . , k l : y ( m i ) = m i l Then r m1 ,.. ' m k d e n o t e s t h e s t a b i l i z e r ( s u b q r o u p ) of (nil,...,rnk).

.

Near-fields are primarily applicable t o sharply k-transitive g r o u p s . T h er e i s n o need t o c o n s i d e r l a r q e k ' s : 8 . 3 9 T H E O R E M (C. J o r d a n ( 1 8 7 2 ) , M. H a l l ( 1 9 5 4 ) and o t h e r s - s e e e . g . Kerby ( 7 ) ) : I f k 2 4 then a l l s h a r p l y k - t r a n s i t i v e Dermutaticn n r o u p s a r e f i n i t e a n d i s o m o r p h i c e i t h e r t o S, ( n 2 4 ) , A n ( n 2 6 ) O f t o t h s " ' ? , t b i e u g r ? , ; p s o f d e g r e e 11 o r 1 2 .

259

8a Near-fields

Since r e g u l a r p e r m u t a t i o n qroups a r e w e l l - s t u d i e d [see.

e.q.

( W i e l a n d t ) o r ( P a s s m a n ) ) , we t u r n o u r a t t e n t i o n t o s h a r p l y 2- and 3 - t r a n s i t i v e

permutation qroups.

We s t a r t w i t h t h e s h a r p l y 2 - t r a n s i t i v e o n e s . O u r i n t e r e s t s t e m s from

N be a nf.. Then t h e g r o u p t r a n s f o r m a t i o n s " ( c f . $ 9 ~ ) ) x * xa+b

8 . 4 0 EXAMPLE L e t

T2(H) (a,bEN,

of a l l " a f f i n e a $. 0 )

i s

s h a r p l y 2 - t r a n s i t i v e o n PI. However,

n o t a l l s h a r p l y 2 - t r a n s i t i v e q r o u p s seem t o a r i s e i n

We h a v e t o c o n s i d e r a liew a l o e b r a i c s y s t e m w h i c h i s

t h i s way.

a " l i t t l e b i t " more g e n e r a l t h a n a n e a r - f i e l d .

-- i s

8.41 DEFINITION A near-domain operations (a) (b) (c)

(N,+)

v

subject t o the followinn conditions:

n,nlEN:

n+n'

= 0 ==> n ' t n = 0 .

i s a qroup.

nEH: n o = 0.

v (f)v (e)

''*'I

i s a loop (with zero 0)

(N",.)

(d)

"+" and

N w i t h two b i n a r y

a set

n , n ' ,n"EN: n,n'~pI

3

(n+n')n" = nn"+n'n". d

n,n

,EN*

n"EN: n t ( n l + n ' l )

(n+n')+d,,nl

Near-domains

can be viewed as " a d d i t i v e l y n o n - a s s o c i a t i v e near-

fields"

8.75):

8.42

(cf.

R E M A R K A n e a r - d o m a i n w i t h a s s o c i a t i v e a d d i t i o n i s a nf..

I t i s n o t k n o w n i f t h e r e e x i s t n e a r - d o m a i n s w h i c h a r e no n e a r fields.

Anyhow,

t h o s e ones must be i n f i n i t e :

8 . 4 3 THEOREM A f i n i t e n e a r - d o m a i n

We d e f i n e f o r a near-dcmtiin N

i s a near-field.

T2(N)

a s i n 8.4'3 a n d g e t

n!'

$8 NEAR-FIELDS AND PLANAR NEAR-RINGS

260

8.44

THEOREM

( a ) For each near-domain N ,

T2(M)

i s sharply 2-transitive.

( b ) C o n v e r s e l y , f o r each s h a r p l y 2 - t r a n s i t i v e p e r m u t a t i o n g r o u p r on a s e t 1.7, M c a n be made i n t o a n e a r - d o m a i n s u c h t h a t r = T2(E4). 8.45 C O R O L L A R Y All f i n i t e s h a r p l y 2 - t r a n s i t i v e

groups a r e e x a c t l y t h e

T2(N)'s,

permutation where N i s a f i n i t e n f .

.

S o by 8 . 3 1 , 8 . 3 2 a n d 8 . 3 4 ,

a l l f i n i t e sharply 2-transitive permutation qroups a r e determined.

T h e r e e x i s t many c o n d i t i o n s u n d e r w h i c h a n e a r - d o m a i n i s f o r c e d t o b e a n e a r - f i e l d . They a r e e x c e l l e n t l y p r e s e n t e d i n Kerby ( 9 ) . We m e n t i o n o n l y o n e :

r i s a g r o u p t h e n I T : = { y c r l y 2 = 11 t h e s u b s e t o f t h e " i n v o l u t i o n s " o f r. Let ( 1 ~ 3 ' : = c Y ~ Y ~ I Y ~ , Y ~ E I ~ I .

8 . 4 6 NOTATION I f

denotes

8 . 4 7 T H E O R E V L e t r be a s h a r p l y 2 - t r a n s i t i v e p e r m u t a t i o n o r o u p on M a n d ( M , t , . ) " i t ' s " n e a r - d o m a i n ( 8 . 4 4 ( b ) ) . T h e n M i s a n e a r - f i e l d <->

(I,)'

5

r.

8 . 4 8 R E Y A R K S h a r p l y 3 - t r a n s i t i v e o r o u p s c a n be c h a r a c t e r i z e d

i n a s i m i l a r , b u t more c o m p l i c a t e d way b y a r o u p s o f t h i n a s l i k e " f r a c t i o n a l a f f i n e t r a n s f o r m a t i o n s " on c e r t a i n n e a r domains ( s o - c a l l e d " K a r z e l - T i t s - f i e l d s " ) . See Kerby ( 7 ) . See a l s o a l l S " - l a b e l e d i t e m s i n t h e b i b l i o s r a p h y .

6 . ) N O R M A L NEAR-FIELDS A N D I N C I D E N C E G P O U P S In o r d e r t o be a b l e t o f o r m u l a t e the c o n n e c t i o n s between n f . ' s a n d g e o m e t r y we d r i v e i n a n o t h e r c o u n t r y a n d r e c a l l some geometry. For a d e t a i l e d account s e e P,ndr@ ( 4 ) . C f . a l s o t h e a p p e n d i x t o Thornsen ( 1 ) .

26 1

8a Near-fields

8.49 D E F I N I T I O N L e t P b e a s e t a n d $ 5 2 ' . T h e p a i r (P&) i s c a l l e d a n i n c i d e n c e s t r u c t ure. (P,$) i s an i n c i d e n c e space provided t h a t

The elements o f

P a r e t h e n c a l l e d " p o i n t s " and t h o s e o f

L o f (a) i s c a l l e d t h e " l i n e determined by p , a " a n d d e n o t e d b y p9. If L , M & , s e t L / / M : <-> ( L = M ) v Call (P$) deaenerated i f every s e t of 3 v (LnM = $). "lines".

p o i n t s i s o n a common l i n e . 8 . 5 0 D E F I i- Y I T I O t J Two i n c i d e n c e s p a c e s c a l l e d isomorphic if MsP:

h(M)Ef,'

P

or (if

= P'

c->

3 h:P

ME$.

and

+

and

(P,$) with

P'

h

b i j e c t i v e and

h i s t h e n c a l l e d an i s o m o r p h i s m

L = L')

an aut.omorphism.

8 . 5 1 DEFIPIITIOPI A s u b s e t S o f a n i n c i d e n c e s p a c e c a l l e d s u b s p a ce i f i t i s "convex",

s

=/=

t :

8.52 REMARK.

arc!

(PI,$')

-

if

i.e.

is

(P,t)

\I s , t ~ S ,

StES.

The subspaces o f an i n c i d e n c e space

an i n d u c t i v e Moore-system.

form

(P&)

Hence i t makes s e n s e t o s p e a k

about t h e "subspace aenerated by a subset of P".

8.53 DEFINITION A n o n - d e g e n e r a t e d i n c i d e n c e s p a c e

(P,$)

i s

c a l l e d an

(a) a f f i n e plane if (b) p r o j e c t i v e plane i f

: LoM

+ 8.

LE$

\

pcP

\I LEA:

3

ME&

ILI 1 3

EM and

A L//V.

L,M&:

Each a f f i n e p l a n e c a n be extended t o a p r o j e c t i v e p l a n e b y a d d i n g some p o i n t s .

Conversely,

one g e t s an a f f i n e

a p r o j e c t i v e one b y t a k i n g o u t one l i n e ,

p l a n e from

58 NEAR-FIELDS AND PLANAR NEAR-RINGS

262

8 . 5 4 DEFINITION A s u b s p a c e o f an i n c i d e n c e s p a c e (P,;d) g e n e r a b l e by 3 p o i n t s ( n o t o n a c o m m n l i n e ) i s c a l l e d a plane i n ( P , , f ) . 8.55 DEFINITION An incidence space space i f each plane i n ( P L )

i s called a projective i s a ,projective plane. (P&)

be a p r o j e c t i v e s p a c e . B C P i s 8 . 5 6 DEFINITION L e t ( P , A ) c a l l e d a base o f (P,;L) i f B i s a minimal qeneratincr s e t for

(P,Z).

8.5 7THEOREM -

Each p r o j e c t i v e s p a c e h a s a ( n o n - e m p t y ) b a s e a n d a1 1 b a s e s a r e e q u i p o t e n t .

~

8 . 5 8 DEFINITION - I f B i s a base for the projective space P : = (P,$) t h e n d i m P : = ( B ( - 1 i s c a l l e d t h e --__ dimension o f P.

8.59 P ROPOSITION -

The a u t o m o r p h i s m s o f

p r o j e c t i v e space P ( t h e " c o l l i n e a t i o n s " ) f o r m a group Call ( P ) . d

8 . 6 0 DEFINITION A p r o j e L t i v e s p a c e (P,;C) i s called D e s a r g u e s i a n i f , whenever t w o " t r i a n g l e s " { a l , a 2 , a 3 ) and I b l y b 2 , b 3 1 ( a l , a 2 , a 3 , b , b , b E P ) a r e " p e r s p e c t i v e

w.r.t. a center

3

L1,L2,L3d

then

1

OEP"

( t h a t means t h a t

i e { 1 , 2 , 3 ) : O E L ~A a i E L i

~-

mnm, a l a 3 n m

some common l i n e L :

and

A

biELi)

m o m are in

8a Near-fields

263

\

bl /

L1

/

/

b;62/';

'

' X,'/ ' 3 4

I

I" /

43\

h

\

\ \

\'


8 . 6 1 R E M A RK Each p r o j e c t i v e s p a c e o f d i m e n s i o n Desarguesi a n .

2 3

i s

8 . 6 2 NOTATION I f V i s a v e c t o r s p a c e o v e r some f i e l d K t h e n V*/K*: = IK*v(vEV*I and

d:

= { L

1

L i s a subspace of V of ( v e c t o r - s p a c e - ) dimen-

sion 21.

8.63 THEOREM (Karzel ( 1 ) ) . i n the notation of 8.62, (V"/,*,L) i s a D e s a r g u e s i a n p r o j e c t i v e s p a c e o f d i m e n s i o n dim V - 1 . C o n v e r s e l y , one q e t s a 1 1 Desarguesian p r o j e c t i v e s p a c e s o f d i m e n s i o n 2 2 i n t h i s way. 8 . 6 4 DEFINITION The t r i p l e

(P,X,.)

i s called a (Drojective)

incidence qroup i f

( a ) (P,$)

i s a projective space. (b) (P,*) i s a oroup. ECOll(P). ( C ) V pep: c p : P * P (2

2 , .1

the same.

il

+

PC'

i s c a l l e d 3 e s a r , i u e >_- i d n i f

(P,$)

h a p p e r i s t c 5,

58 NEAR-FIELDS AND PLANAR NEAR-RINGS

264 8.65

R E M A R K I t seems t o b e c l e a r how i s o m o r p h i c p r o j e c t i v e

dim P

i n c i d e n c e groups and t h e dimension

of

a projective

incidence group a r e defined. What h a s a l l o f t h a t t o d o w i t h n e a r - f i e l d s ? 8.66

D€FINITION L e t

N

be a n f . and F a p r o p e r s u b f i e l d o f

N.

N i s s a i d t o be normal o v e r F p r o v i d e d t h a t (a) (b)


(F",,)

v

(N",.).

v

f,f'EF

nEN: n ( f t f ' ) = n f t n f ' .

i s then called a nornal near-field.

(N,F)

_ _ I

ti c a n b e

c o n s i d e r e d a s a v e c t o r s p a c e o v e r F. 8.67

R E M A RK If

i s normal and N i s a f i e l d t h e n by

(N,F)

Cartan-Brauer-Hua's

F

c

C(N).

theorem (see (Jacobson),

p . 186)

T h i s does n o t h o l d f o r g e n e r a l n f . ' s

(see

( 6 ) , p. 76).

Wahlin?

The b a s i c f a c t i s i n 8 . 6 8 THEOREM ( K a r z e l

(N,F)

(a) L e t

(as i n 8.63)

( l ) , pp. 69

-

73).

be a normal n f .

.

Then

(N*IF++,d..)

i s a Desarquesian p r o j e c t i v e incidence

group. (b) Conversely,

a l l (up t o isomorphic copies) Desarquesian

p r o j e c t i v e p l a n e s P a r i s e i n t h i s w a y f r o m some n o r m a l

nf.,

which

(if

dim P 2 2 )

i s unique up t o i s o -

m o r p h ism.

8.69 THEOREM ( K a r z e l ( l ) , p p .

N*/F,,

commutative and

76 a n d 78). dimF(N) 2 3

If then

(N,F)

i s normal,

N i s a commuta-

tive field.

If

dimF(N) = 2

2 GF(p )

and N i s f i n i t e t h e n e i t h e r N i s a f i e l d

o r a Oickson nf. o f o r d e r 9 (see t h e remarks

p r e c e d i n 9 8.35)

o r o f o r d e r 64.

265

8a Near-fields

For g e n e r a l i z a t i o n s ("normal l o c a l near-modules") see e.o. P i e n e r ( l ) , A n d r i ! ( 4 ) , K u z ' m i n ( l ) , Maxson ( 1 5 ) a n d T h e o b a l d , ( l ) y ( 2 ) . F o r o r d e r e d n f . ' s s e e e . g . Kerbv ( 3 ) , Grijger ( 2 ) . Now we t u r n t o a f f i n e D l a n e s .

7 . ) P L A N A R NEAR-FIELDS

F i r s t a g a i n a l i t t l e b i t o f geometry.

8 . 7 0 DEFINITION

An a u t o r n o r p h i s m a o f an a f f i n e p l a n e

(Pyx)

i s called dilatation i f 1 L d : a(L)II L. A dilatation Q i s a t r a n s l z t a t i o n i f cz = i d o r a i s f i x e d - p o i n t - f r e e . _ _ _ I -

_ _ I _

8 . 7 1 DEFINITION An a f f i n e p l a n e (P,&) i s called a translation plane i f the s e t ( i t i s a qroup!) of a l l t r a n s l a t i o n s i n (P,f',) w o r k s t r a n s i t i v e l y or! P . -

"

~

-

_ I

Consider, f o r a nf. N, .in N2 = : P the " l i n e s " I(x,xa+b)lxENl =: L,,b (a,bEN). TWO of s u c h , L a , b

and

a = a'. In order t o g e t s o m e t h i n g l i k e ar, a f f i n e p l a n e , we w a n t t w o " n o n p a r a l l e l " " l i n e s " t o h a v e e x a c t l y o n e common p o i n t . T h i s i s t h e c a s e i f f every equation xa x a ' + c w i t h a $; a ' h a s e x a c t l y one s o l u t i o n .

L a ' ,b'

can be c o n s i d e r e d as " p a r a l l e l " i f

planar (or projective) i f 8 . 7 2 DEFINITLOA A n f . N i s c a l l e d -each equation

xa = xbtc

(a

+

b)

has e x a c t l y one s o l u t i o n . Evidently, every f i e l d i s planar. 8.73 REMARK (blahling ( 6 ) ) . I t s u f f i c e s t o w a n t h a v e a u n i q u e s o l u t i o n f o r e a c h a 4 1. I n T a c ~we g e t w i t h o u t t o o much w o r k :

xa

x+l

to

$8 NEAR-FIELDS AND PLANAR NEAR-RINGS

266

8.74 THEOREM L e t N b e a n e a r - f i e l d ,

L a , b as i n t h e m o t i v a and L c := { ( c , x ) I x ~ f . l }

t i o n a l c o n s i d e r a t i o n s p r e c e d i n g 5.72 (c E N) Let

the "vertical

d:

= {L

lines".

l a , b ~ N lu ! L c l c ~ I l l .

a,b a f f i n e p l a n e <==> N i s p l a n a r .

(N2,$)

I n t h i s case,

8.75 REMARK --

Thev

(PI2,$)

i s an

i s a translation plane.

l l o t a l l t r a n s l a t i o n p l a n e s a r i s e i n t h i s way f r o m

a planar nf.

.

One h a s t o u s e " m u l t i p l i c a t i v e l y n o n -

associative planar nf.'s"

(cf.

"Veblen-Wedderburn-systems"

2

(I1 ,$)

p l a n e s a s some

8.41

-

8.42), s o c a l l e d

bl t o n e t a l l t r a n s l a t i o n

( d e f i n i t i o n of $ as above)

("Each

t r a n s l a t i o n p l a n e c a n b e c o o r d i n a t i s e d by a \ ' e b l t . n -

Hall ( l ) , p. 362.

) e.g. ! 4 e d d e r b u r n s y s t ~ ! ~ " . See

F o r t h e more g e n e r a l q u e s t i o n .

which oeometric structures

c a n b e c o o r d i n a t i z e d by w h i c h t y p e s o f a l o e b r a i c s t r t , c t u r e s , see a l l G - l a b e l e d

items i n the biblioqraphy.

We l o o k a l i t t l e b i t a r o u n d t o f i n d some p l a n a r r i f . ' s .

8.76 T H E O R E P I ( f l a x s o n (10) e t a l . ) A n f . N w i t h i s planar.

-

~

8.77 CDROLLARY -(e.o.

Zemmer

(1)). A f i n i t e nf.

(N) f i n i t e

dim

I

Nd

i s planar

So o u r s e a r c h i s t u r n e d a r o u n d : d o e s t h e r e e x i s t n o n - p l a n a r nf.'s

at all ?

8.78 EXAMPLES (a)

(Zemmer ( 1 ) ) . T h e D i c k s o n n f . (char F

0)

(F(x),+,o@)

o f 8.29

dimN (FI)

infinite

i s not planar.

(b) There e x i s t p l a n a r n f . ' s (Maxson ( 1 0 ) ) .

N with

Hence t h e c o n v e r s e o f

8.76 d o e s n o t

hold.

The f o l l o w i n g c o n c e p t i s u s u a l l y o n l y d e f i n e d f o r f i n i t e g r o u p s . 5ee e . g .

( P a ~ ~ n 2 . nqr )

Kerhy ( 9 ) .

267

8a Near-fields

r

8 . 7 9 DEFINITION ( a ) Each

5 SM

YEr,

y

i s c a l l e d a Frobenius oroup i f

f id

h a s a t most one f i x e d - p o i n t .

( b ) Kr: = {yEI']y i s f i x e d - p o i n t - f r e e 1 u ( i d 1 p r o p e r normal s u b q r o u p o f r .

i s a transitive

--8 . 8 0 R E M A R K S

( a ) Kr

i s c a l l e d the Frobenius-kernel

of

r.

( b ) Other c h a r a c t e r i z a t i o n s o f f i n i t e Frobenius oroups a r e e.q.: (a)

i s a Frobenius-proup i f f r i s t r a n s i t i v e , b u t not sharply 1 - t r a n s i t i v e ( = reciular) a n d (rnl,rn2)E~4', m1 m2 : r {id} (8.3e).

r

I S,,,

+

v

"1 r m 2

( B ) The F r o b e n i u s o r o u p s a r e e x a c t l y t h e s e m i d i r e c t p r o d u c t s o f a clroup A w i t h a f i x e d - p o i n t - f r e e autonorphisin group { i d } of A .

+

Anyhow, i f r i s a f i n i t e F r o b e n i u s q r c j u p , c h a r a c t e r i s t i c , r e q u l a r and n i l p o t e n t .

is

Kr

( c ) The f i n i t e s h a r p l y 2 - t r a n s i t i v e p e r m u t a t i o n a r o u p s o f deqree 2 3 a r e e x a c t l y the 2 - t r a n s i t i v e Frobenius groups. The c o n n e c t i o n t o p l a n a r n f . ' s theorems.

i s q i v e n by t h e f o l l o w i n g two

8.81 T H E OR E M ( e . 7 . A n d r P ( 3 ) ) . L e t N be a p l a n a r n f . w i t h IN1 > 2 . Then T 2 ( N ) = : r ( 8 . 4 0 ) i s a F r o b e n i u s n r o u p and

Kr

i s the s e t of a l l mappinas

char M = 2

then

char N

then

2

IKT'

V

nEN

Kr

3

(8.46) ~

€

x

+

xtb.

......

Moreover, i f

("1

1 y ~( n )~ = : n . . . . ( * * ) .

C o n v e r s e l y , a l l F r o b e n i u s g r o u p s w i t h ( * ) o r ( * * ) c a n be o b t a i n e d i n t h i s way frrim a p l a n a r n f .

$8 NEAR-FIELDS AND PLANAR NEAR-RINGS

268

Of

course one can d e f i n e

f o r every permutation proup as Kr Then one g e t s one more c h a r a c t e r i z a t i o n o f a

i n 8.79(b). planar nf. :

8 . 8 2 THEOREM ( K e r b y ( 7 ) ) .

N is

p l a n a r <=+

(Ir)

I4e c l o s e o u r r o u n d - u p o f

L e t N be a n f . 2

Then

(Nk,$)

= T2(N).

near-field theory with

Consider t h e N-qroup = I(al, ak)tN(Sy

...,

:'I

= Kr.

L e t N be a n f . and k

8.83 REMARK

and

Nk

E

IN.

and d e f i n e $ b y

, . . . , b k ) j ( a l , . . . , a k ) , ( b l , . . . ,b k ) ~ kN) .

i s a " n e a r l y a f f i n e space".

F o r t h e r e p r e s e n t a t i o n o f " a f f i n e i n c i d e n c e groups" see Piepcr (2).

Cf.

Groger ( 1 ) and Theobald ( l ) , ( ? ) W . efelscheid (10)

showed t h a t e v e r y n f . c a n be embedded i n a m i n i m a l p l a n a r o n e .

b)

PLANAR

NEAR-RINGS

1.) T H E STRUCTURE OF P L A Y A R N E A R - R I r i G S The " p l a n a r i t y p r o p e r t y " 8 . 7 2 for near-rinqs.

can o f course a l s o be f o r m u l a t e d

B u t i t i s n o t v e r y w i s e t o do s o :

with this property i s a near-field g e t a n y t h i n g new. (Anshel-Clay 8.84

s o we w o u l d n ' t

T h e r e f o r e we n e n e r a l i z e t h i s c o n c e p t

(1)):

D E F I N I T I O N L e t Pi b e a n r . a n d a z b:

( s e e 8.91).

a near-rina

<->b

a,bcN.

ncN : na = n b .

I n t h i s c a s e , a and b a r e c a l l e d ( r i g h t ) e--o u i v a l e n t mu1 t i p 1 ie r s -----

.

269

8b Planar near-rings

O f c o u r s e , t h i s i s an e q u i v a l e n c e r e l a t i o n on N .

8 . 8 5 DEFIriITION A n r . Pi i s s a i d t o b e a p l a n a r n e a r - r i n o i f (Nlrl23

and i f e v e r y e q u a t i o n (a $ b)

xa = x b + c

has a unique s o l u t i o n ( i n N ) . 8 . 8 6 NOTATIOlJ - If

by

let

Ncn,

A:

01;

= InENln Z

denote

M A

N*.

By 8 . 8 5 ,

N*

h a s a t l e a s t two e l e m e n t s .

8 . 8 7 PROPOSITION ( A n s h e l - C l a y

( 1 ) ) . Every p l a n a r nr. i s z e r o -

symmetric. P r o o f . Take n E N . L e t a b e EN*. T h e n 0 a n d n0 b o t h s o l u t i o n s o f xa = xO+0, h e n c e e q u a l .

are

8 . 8 0 PROPOSITIQN ( A n s h e 1 ' - C l a y ( 1 ) ) . L e t N be p l a n a r .

( a ) aEN (b)

i s a r i q h t z e r o d i v i s o r <->

nEN*

3

mEN

xcN:

a E 0 <=+

aEA.

xn = m.

P r o o f . ( a ) He o n l y h a v e t o show t h a t n a = 0 ( n 0) i m p l i e s a :0 . I n f a c t , a f O i r o p l i e s t h a t 0 a n d n a r e s o l u t i o n s o f xa = x O + O , a contradiction. (b) If

xn

nEM*,

=

xOtm

has a unique s o l u t i o n .

The l a s t r e s u l t g i v e s r i s e t o t h e f o l l o w i n n d e f i n i t i o n . 8 . 8 9 :IOTATION F o r xa = a .

Evidently, These ring.

5,'s

Let

aEN*

Ba:

= tn

a e B a , la€N help t o

let

1,

and

N* =

* CXEN (1,

clirify

be t h e u n i q u e s o l u t i o n o f x = XI

.

u

,,+Ba.

aEN the s t r u c t a r e o f a planar near-

88 NEAR-FIELDS AND PLANAR NEAR-RINGS

270

( 1 ) ) . L e t N be a p l a n a r n r .

8 . 9 0 THEOREM ( A n s h e l - C l a y (a)

Each

( b ) A and t h e (c) (d)

(e)

aEN*:

If

Ba's

BaN*

then

$I:

1, (aeN*) SeN*

(f) If

f o r m a p a r t i t i o n o f PI.

(aEN*)

= B,.

a,bEN*,

Each

Then

1,.

i s a group w i t h i d e n t i t y

(Ba,*)

.

B,

+

Bb i s

x

+

IbX

a (qroup-) isomorphism.

(PI,+,.).

i s a right identity for iI+

SN E S

and

[I

S

then

Ba.

aE: S

( c ) ~ :L e t a b e

Proof.

la(bn)

--

(1,b)n

itt

ncN

EN*,

= bn,

( a ) By ( c ) ~ , ( B a , * )

and

whence

bcDa.

Then

bneBa.

i s c l o s e d w.r.t. m u l t i p l i c a t i o n .

So

l a l a a n d 1, a r e b o t h s o l u t i o n s o f x a = r.'l+a. lala = l a ~ B a i s a l e f t i d e n t i t y i n (gal*).

If

beB,,

Now

6 b = 1, whence

l e t 6 be t h e u n i q u e e l e m e n t o f N w i t h

(8.88(b)).

( c ) ~ :C o n v e r s c l y ,

*

fl

b

a,bcN*

either

Bb'

1,

hence

so

xn = xOtn;

1,

=

and

lb

lb

and

= Bb.

(d):

By ( c ) ,

a',a''EBa,

ncN,

and n s o l v e

nl,

0 r e a l l y yoes from

cp(a'a")

(Iba')(lba")

= lb(a4a'') =

= $I(a')b(a''),

lb

a c t s a s i d e n t i t y by ( a ) .

If

@ ( a ' ) = m(a")

a'

la,

xb

can be w r i t t e n as

b&Ba

I a n = n = 1b n ,

n c B a n Bb,

( e ) : F o r each

=

every

Ba

O r

are solutions o f

B,

solve

.

I t i s e n o u q h t o show t h a t

Ban Bb = If

1,6

and

IaF = 6 & B a .

b = labEBaN (b):

Then 6

= 1,a'

= 1,Iba'

then

=

into

Bb.

If

((lbab)lb)a" =

since

Iba'

= la!ba"

B,

x l a = xO+nla.

lba".

= a".

Iba'EBb,

By ( e ) ,

where

27 1

8b Planar near-rings

If

t h e n l a b ' t B a i s mapped o n t o b ' . Hence @ i s

b'EBb,

an i s o r n o r D h i s m . (f) i s clear.

8 . 9 1 COROLLARY ( A n s h e l - C l a y djscrete (i.e,

(1)).

If

N

i s p l a n a r and E i s

(N

i s the identity)

5

then f u l f i l l s 8.72)

t h e n PI i s a n f . . Proof.

W

1,

a,bEN*:

Hence ( 8 . 9 0 ( b ) ) Hence

EN, N p l a n a r

3

(by 8.90(e)),

lb

z

(PI*,*)

4Ii

(N

=

SO

= lb'

i s a group.

(Ba,*)

=

, a )

1,

N n f . . So p l a n a r n r . ' s a v o i d 1 as Duke

Drakula the s u n l i g h t !

8.92

COROLLARY

A p l a n a r n r . i s i n t e g r a l <-=> PIQ = PIm <=>

8 . 9 3 THEOREM ( A n s h e l - C l a y

o f (N,+).

( 1 ) ) . L e t N be an i n t e q r a l p l a n a r n r . = 6, u { O j

Fa:

such t h a t each

B,

Suppose t h a t no

B a + Z b = N.

k = {O).

i s a normal subqt-oup

(acN")

N,

=

a

but for a l l

b,

Then

( a ) (N,+)

= (Ba,+):(Bb,+)

( b ) Each

(B,,t,*)

for all

a,bcN*,

+ b.

a

i s a nf.

(c) N i s abelian.

$: = ( n + B a l n c N , (8.53).

(d) If

----Proof.

(N,S)i s

acN*),

an a f f i n e p l a n e

( a ) i s c l e a r f r o m 8.YO(b).

( b ) follows f r o m 8 . 9 0 ( a ) . ( c ) i s a consequence o f ( a ) , (d):

If

Then

3

p,qEN,

p

take

q,

ncN: pcn+Ba =:

L

+

Since p q, Ba = B, each ( n t B a 1 2 2 . Hence

*I

= (14*/:1>2

pcM

M(I

If

M

with If

then

'

p-qEEa. L'

= n'+Bal

p-qEBa

nEaI.

L = L'. Since 0,lac8 a' i s an i n c i d e n c e space

implies that

L = n + B a c g and L.

qcL.

p,q~L'

and (N,$)

Now t a k e and

acN"

and

a l s o has t h e p r o p e r t y t h a t

(8.49). IN deqenerated.

( b ) and 8.11.

PEN.

If

Y:

(N,&) = piBa

i s not then

E h~ a s t i l e s a T e ~ r 3 ~ e r t +y e n

58 NEAR-FIELDS AND PLANAR NEAR-RINGS

272

M ' = P+Bb M' = L

If

M = M'.

M'

If

Hence

+L 3

f o r some

M'n L = 0.

then

xcEa

3

beM".

p + g b = ntBa,

then

yeBb:

If

a = b, a

0,

p

-x+n =

a contradiction.

a = b

t1 = 1 4 ' .

and

(N,$)

8.94 REMARKS I t c a n b e shown t h a t t h e a f f i n e p l a n e 8.93(c)

r e s u l t can be o b t a i n e d i f t h e

Pa's

B a : = Ba ~ { o f u c o n n e c t i o n t o " @ ( I, I V ) - r i r o u p s " .

d e f i n e d as

i n

A similar

can be c o o r d i n a t i z e d b y a s k e w - f i e l d .

For a l l of

whence

PI = Ba+Bb.

b,

So

n - p = x+y.

= y + p c ( n + E a ) n (P+Bb) = Consequently again

so

are alternately

There i s a l s o a clos.

t h a t see Anshel-Clay

(2).

As C l a y p o i n t s o u t ,

t h e r e i s a l s o some r e l a t i o n t o " i n v e r s e p l a n e s "

(cf.

F e r r e r o ( 1 2 ) ) . I f N i s an i n t e g r a l o l a n a r n r . w i t h i d e n t i t y then N i s a s k e w - f i e l d o r isornornhic t o the n e a r - f i e l d {f : N

+

N l V m , n r N:

f(rnn)

rnf(n)}.

=

(2) o r Clay (10)).

8 . 9 5 E X A M P L ES (see Anshel-Clay

( a ) Every p l a n a r n f . w i t h more t h a n 2 elements i s a p l a n a r nr.. ( b ) L e t V be a normed v e c t o r space o v e r I R . =

11 w I I

v.

(V,+,*)

Then

Define

V ~ W :=

i s an i n t e n r a l p l a n a r non-

r i ncl. ( c ) L e t V b e a v e c t o r s p a c e o v e r IR the property that 4)(tv) Define

ta$(w). v*w: =

3

aeR*

II$(M) I 'Iav.

t j

and

tEIR,

Then

6:

V .,.

IR h a v e

t 2 o

(V,+,*)

VEV: i s a planar

n e a r - r i no. See A n s h e l - C l a y

o f t h e 6,'s ( d ) No

Vo(r)

(2)

f o r the oeonetric interpretations

as l i n e s ,

rays, hyperbolas

etc.

.

: i s d i s c r e t e , so (i-) i s a nf. with y(o) more t h a n 3 e l e m e n t s , w h i c h i s c e r t a i n l y n o t t h e c a s e . or M(r)

i s planar:

p l a n a r i t y would imply t h a t

So i n c o n t r a s t t o n e a r - f i e l d - t h e o r y , n o t p l a n a r i n aerieral

( c f . 3.77).

a finite

nr.

is

273

8b Planar near-rings

8 . 9 6 T H E O REM ( F e r r e r o (5), B e t s c h - C l a y

r

(a) Let

r.

automorphism aroup o f

N:

=

p

G

be a g r o u p and

I i d l

be a f i x e d - p o i n t - f r e e

If r i s f i n i t e t h e n e a c h

o f 1.4(b)

( r , + , a B )

inteqral iff

(1)).

i s a planar near-rinn.

N i s

i s t h e complete s e t o f a l l non-

IBilisIl

z e r o o r b i t s ( n o t a t i o n as i n 1 . 4 ) . l e t N be a p l a n a r n e a r - r i n n .

( b ) Conversely, aEN:

ga:

N

-+

n

N

+

na

point-free

(a):

Then

G:

flt

(ga\acN

=

%

C

=

1

$. { i d )

automorphism oroup

beti*,

F o r each

Proof.

.

Consider f o r i s a fixed-

o f (N,+).

Bb '

C o n s i d e r acidin t h e s i t u a t i o n o f 1 . 4 ( 5 ) .

IN/:( = [ G wiall23, s i n c e y E 6 <-> a y = q 6 . So i t r e m a i n s t o s h o w t h e " p l a n a r p r o p e r t y " : Assume t h a t qy(E)

or

= g6(5)+q

(with

But

S o y

-qtid

y

= S*6trl,

or

T h i s means t h a t

6.

= n w i t h 9, A 9 6 $: i d ) : ( - q t i d ) ( C ) = - q i l ( r l ) .

(-q6+oy)(S)

-'q6

?: = i s bijective,

so t h i s e q u a t i o n has e x a c t l y

one s o l u t i o n : Suppose t h a t = -9(B)tB

9(a-a)

i s fixed-point-free Since (b):

r

= (-q+id)(B)

(-otid)(a)

and

and

q A id,

-n+id

i s finite,

-g(a)ta =

then

Since g

= a(a)-cl(B) = a-B. a = 6.

i s bijective.

If aspi*, C E N 3 X E N : p a ( x ) = xa = c So q a c A u t ( N , t ) and G = Cqa(acN*}

by is

8.88(b).

a group. C o n s i d e r t h e map

$:

(Bb,*) a-

Evidently,

If lbal

$(al)

-+

G

,

where

bcN*.

ga

UJ i s a h o m o m o r p h i s m .

= $(a2),

= lba2,

so

then al

= a*,

XEN:

xal

= xa2,

whence

and $ i s shown t o be a

monomorphism. .RI

Now t a k e s o ~ e g c , c ~ l l.

$(Ib c ) = albc

= 9,

Since

l b ccBb

by 8.9r)(c),

a n d t~ i s a n i s o m o r p h i s m .

274

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS C;

i s fixed-point-free:

let

fulfill

qa(n) = n

f o r some n E N , n f 0, then 0 a n d n f u l f i l l x a = x * l a + O ( 8 . 9 0 ( e ) ) . S o a 5 1a ' which means that XCW: q a ( x ) = xa = x l a = x , from which we

v

deduce t h a t

ga = i d .

8 . 9 7 R E M A R K ( B e t s c h - C l a y ( 1 ) ) . T h i s shows t h a t ( s i m i l a r t o t h e I

s i t u a t i o n in planar n e a r - f i e l d s ) every f i n i t e planar nezrr i n q c a n be c h a r a c t e r i z e d by some p a i r ( ? , G ) o f q r o u p s , where G .C. { i d 1 5 A u t r i s f i x e d - p o i n t - f r e e . S o e v e r y f i n i t e p l a n a r n e a r - r i n a determines a Frobenius oroup (8.79) a n d conversely ( c f . also Ferrero ( 5 ) ) , a n d the c o n s t r u c t i o n o f a p i a n a r r i e c r - r i n o CI: a n i v e n a d d i t i v e

g r o u p 'I i s n o t h i n q e l s e t h a n t h e c o n s t r u c t i o n o f a nont r i v i a l f i x e d - p o i n t - f r e e automorphism n r o u p o n r . Cf. 8 . 1 2 4 , H e a t h e r l v - n l i v i e r ( 3 ) a n d Adler ( 1 ) .

-8 . 3 8 C O R O L L A R Y ( B e t s c h - C l a y ( 1 ) ) L e t Id he a f i n i t e p l a n a r n e a r _ I

r i n g a n d l e t G be a s i n 8 . 9 6 . T h e n

( a ) IGj d i v i d e s (b) (N,t)

Proof.

I F 1 1 - 1.

i s nilpotent,but not necessarily abelian.

( a ) i s c l e a r from 8 . 9 6 ( b ) a n d 8 . 9 0 ( b ) ,

( b ) follows from ( T h o m p s o n )

and ( c f . 0 . 3 3 ( b ) -> ( a ) ) .

See a l s o 8 . 1 2 4 . The l a s t r e s u l t i s i n some o t h e r way r e m a r k d b l e :

planar nearr i n g s a r e " n o t f a r away from b e i n n n e a r - f i e l d s " ( c f . 8 . 8 8 l b ) ) . B u t t h e y a r e f a r enouqh t o have n o n - a b e i i a n members i n c o n t r a s t t o 8 . 1 1 . We need

8 . 9 9 DEFINITI9N ( F e r r e r o (5), S z e t o ( 3 ) ) . A nr. N i s c a l l e d s t r o n q l y u n i f o r m i f ij n E N : ( 0 : n ) = ( 0 1 o r ( 0 : n ) = M , but

3

mEN:

(0:m)

=

{O).

For t h e f o l l o w i n g r e s u l t , c f . a n d Olivier ( 3 ) .

Ferrero ( 5 ) ,

Heatherly-Olivier ( 3 )

8b Planar near-rings

8.100

THEOREM ( F e r r e r o ( 5 ) ,

Clay (ll),

275

S z e t o (3)).

( a ) L e t N be a p l a n a r n r . . Then N i s s t r o n g l y u n i f o r m ,

the multiplication i s not t r i v i a l !1.4(b)) a n d a l l non-zero o r b i t s o f G ( s e e 8 . 9 6 ( b ) ) a r e p r i n c i p a l ( t h a t means t h a t f o r a l l x , y o r b i t t h e r e i s e x a c t l y one

i n t h e same n o n - z e r o r j ~ G with o(x) = y).

( b ) C o n v e r s e l y , i f N i s a f i n i t e nr. which i s s t r o n q l y uniform, has n o n - t r i v i a l m u l t i p l i c a t i o n a n d the p r o p e r t y t h a t e v e r y n o n - z e r o o r b i t u n d e r C; ( d e f i n e d a s i n 8 . 9 6 ( b ) ) i s p r i n c i p a l , t h e n ii i s p l a n a r .

a€/\, qa = 6 a n d ( 0 : a ) = N . I f aEN", and (0:a) = { O l , hence N i s strorinly 1N/:i23, t h e m u l t i p l i c a t i o n cannot uniform. Since be t r i v i a l . G i s fixed-point-free (8.96), so a l l o r b i t s are principal (cf. 4.28).

Proof.

_I.-I

( a ) If

gaEAut

(b)

9,:

FI

S i n c e PI i s f i n i t e a n d s t r o n r l l y u n i f o r m , a l l x + xa a r e e i t h e r = 6 o r a u t o m o r p h i s r n s .

(Observe t h a t

Ker q a = ( 0 : a ) ) .

L e t G be t h e orolJp

Since a l l o r b i t s are p r i n c i p a l , G i s f i x e d - p o i n t - f r e e . Since

o f a l l t h o s e automorphisms.

trivial, R E tlARK -8 . 1-0 -1 ~

G

{id}.

$.{O!

i s not

Now a p p l y 8 . 9 6 ( a ) .

( S z e t o ( 3 ) ) . 8 . l O O ( b ) does n o t hold i n the i n f i n i t e

( Z , + ) @ (??,+) and d e f i n e (n,m)c(n',m'): T h e n TI: = ( T x Z , + , * ) i s a n i n f i n i t e s t r o n o l y uniform P r . . * i s not t r i v i a l a n d a l l non-zern o r b i t s + r e principal. On t h e o t h e r h a n d , N i s n o t p l a n a r , f o r (2,O) (0,O), but x(2,O) = x(O,O)+(l.l) has no s o l u t i o n . C f . a l s o r,. Betsch's report i n the " Z e n t r a l b l a t t f u r Mathematik".

c a s e : Take

2

= nl(n,m).

I n F e r r e r o - C o t t i - P e l l e g r i n i ( 1 ) i t i s shown t h a t 2 i f N i s p l a n a r t h e n U =N. F o r I
8.102 REMARK

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

276

F i n a l l y , we d e s c r i b e a c o n s t r u c t i o n m e t h o d f o r f i n i t e i n t e q r a l planar near-rings f o r f u t u r e use ( i n t h e next number). 8 . 1 0 3 T H E O R E M ( C l a y ( 1 1 ) ) . Let F be t h e f i e l d o f o r d e r pn ( P E P ) and t a n o n - t r i v i a l d i v i s o r o f p n - 1 . (F",.) pn-l. There i s a c y c l i c suboroup B o f o r d e r t . Choose r e p r e s e n t a t i v e s

i s cyclic o f order

F*

of

ul

=

l,u2,.

fQtg: =

Define Then

.. , u m

(F,+,nt)

for the cosets if

{

Buly...

,Bum

o f B i n F".

g = O

i f q .P 0 , ~ E B u ~ ,(1 = b u i : - b !I 9 i s a n iiiteriral planar n e a r - r i n n .

.

Proof: s t r a i g h t f o r w a r d a n d hence o m i t t e d .

2 . ) P L A r i A R NEAR-RINGS A H D B A L A f j C E D I N C O t l P L E T E B L OC K DESIGNS -

-8 . 1 0 4 DEFINITION L e t PI b e a n r . ,

~ E N *

and

bEN.

Then t h e s e t

aN+b

i s c a l l e d t h e b l o c k d e t e r m i n e d by a , b . B l o c k s o f t h e f o r m aN ( a 0 ) a r e c a l l e d b a s i c b--? o c k s .

+

-

G = i q x l x ~ I I * } i s a s i n 8 . 9 G ( b ) and i f = G ~ { b } then a N t b = G'atb.

8.105 REMARKS I f Go:

Ga = aN* = a N \ i O l f o r m a p a r t i t i t i o n o f ?I*, f o r they a r e e x a c t l y t h e non-zero o r b i t s of (N,+) under G.

The

F o r a p p l i c a t i o n s i n t h e ( n e a r ) f u t u r e we remark 8 . 1 0 6 PROPOSITION -L e t N be p l a n a r a n d 0 laPltb[ = ] G I = ]N/-123. Proof.

V

bcN.

Then

a n + b = a n ' + b <==> sn = a n ' <=> x = a a n d f u l f i i l x n = x n ' <==> n : n ' <-> g, = 9 , ' .

n,n'EN:

x = 0

aENQy

Observe 8 . 8 5 .

8b Planar near-rings

277

I n o r d e r t o b e a b l e t o f o r m u l a t e o u r p r i n c i p a l r e s u l t s we n e e d 8 . 1 0 7 D E F I N I T I O N aEN i s c a l l e d an " e l e m e n t o f t h e f i r s t c a t e v o r y " ( F e r r e r o ) i f aN = ( - a ) N t a = - ( a N ) t a = : - a N t a . C1(N): = I a E N l a i s of the f i r s t category]. Clearly

OcC1(N).

8 . 1 0 8 T H E O R E M ( F e r r e r o ( 8 ) , ( 1 9 ) ) . L e t N be p l a n a r and aEN. Then aEC1(N) <-> aN 5 ( N , t ) . I n t h i s c a s e a n d i f aN i s f i n i t e , aN i s e l e m e n t a r y abel ian. P t-0 0 f

.

.

L e t a be cC1(N). L e t bcaN*, b'caN. We w a n t t o show t h a t b-b'EaN. Now 3 EN*: b = a n . Hence u s i n g 8 . 9 0 ( e ) a n d 8 . 8 8 ( b ) , bN = aN = aNn = = ( - a N t a ) n = - a N n + a n = - a N + b = -bFI+b. Now 3 n ' E N : b ' = b n ' . Thus we o e t "=>

" *

b ' - b = b n - b E b l l - b = - b N t b - h = - b N = -aN. Me c l a i m t h a t aN = - a N . Me may a s s u m e t h a t a $. 0 . By 8 . 1 0 6 , 3 c,dEaH*: c ?= d . From a b o v e we q e t Opt-d E -aN*. Hence 3 n"E:N*: c - d = - a n " . T n e r e f o r e a n " = d - c c ( a N * ) n (-aN") s i n c e we h a v e shown a b o v e t h a t t h e d i f f e r e n c e o f a n y two e l e m e n t s O F all* i s in -aH*. S o ( a N * ) n ( - a f i * ) 4 P , , whencc. aN* = -aN* by 8 . 1 0 5 , f r o m w h i c h a N = -aN. Now o u r c o n s i d e r a t i o n s i m p l y t h a t aPI 5 ( N , + ) . "<="

i s obvious, since

a = a.laEaN

(8.90).

The r e m a r k f o l l o w s f r o m t h e o b s e r v a t i o n t h a t t h e automorphisms o f G , r e s t r i c t e 4 t o a N , form an a u t o T o r p h i s m g r o u p on aN w h i c h a c t s t r a n s i t i v e l y on a N \ ( O l = aN* a n d f r o m t t l e o r e m 1 1 . 1 o f (Plielandt).

§a

278

NEAR-FIELDSAND PLANAR NEAR-RINGS

8 . 1 0 9 COROLLARIES ( F e r r e r o ( 1 9 ) ) . L e t N b e p l a n a r a n d

aEN.

Then ( a ) b bcaN*: bb! = aN i f acC1(H). ( b ) A l l bcaN a r e o f t h e s a m e c a t e o o r y . ( c ) C l ( N ) i s a u n i o n o f - say u - o r b i t s o f under G. ( d ) I f 3 x E C l ( N ) : xT4 = N t h e n C l ( N ) = t i . 8.110 REMARK See Ferrero (12), "difference sets".

(N,t)

(19) f o r t h e connection t o

bltiich b l o c k s c o i n c i d e ?

8.111

THEOREI.1 -- ( F e r r e r o ( 1 9 ) , c f . F e r r e r o ( 8 ) a n d C l a y ( 1 1 ) ) Let N be p l a n a r , a , a ' ~ N " a n d b , b ' E N . Then a N t b = a'r!tb'<=) ( a ) o r ( b ) , where (a) b (b) b

=

+

b'

a n d aN = a " , nt tlt b', - a r t . = a ' ,~ b'EaV*t+b

and

a,a'ECl(N).

P r o o f . ->: F i r s t l e t a N t b = a ' N t b ' . If b = b' then aN = a'FI a n d we a r e i n c a s e ( a ) . So s u p p o s e t h a t b b'. F r o m a11 = a ' r i t ( b ' - b ! we g e t s o m e n E N w i t h 0 = a ' n t ( b ' - b ) . b b' i m p l i e s t h a t nEN*. So a ' n = b - b ' . Sinilarly, 3 n'EN'+: a n ' = b ' - b . Hence 0 b - b ' E ( a ' I i * ) n (-ail*), whence a'N* = iw = -aN by 8.105. So a N t b = - a N t b ' . Consequently 1 n"EN*: b ' = a n " + b , s o a N t b = - a N t b ' = - a N t a n " + b , w h e n c e aN = - a N t a n " = - a N t q n , , ( a ) . A p p l y i n q a n- ,1. g i v e s aN = - a N + a , s o aEC1(N). By s y m m e t r y , a'EC1(N) and ( b ) i s shown.

+

+

aN+b = a ' N t b ' . So a s s u m e ( b ) . L e t x E a l N t b ' . We h a v e t o show t h a t tlt xEaNtb. I f x = b'EaK t b , x ~ a N t b . I f x f b ' , 3 n,n',n":Y*: x -- a ' n s b l = - a n ' t ~ ' = - a n ' t a n ' ' + ~ . S i n c e aN I ( N ,+) , b y 8 . 1 0 8 , - a n ' + a n " E a N , whence xca'l-b. Ti.e c n n v e - q c i n c ' l ~ 5 3 7 1 - 5 shol.pn s i ~ i l a r l y . <-:

(a) t r i v i a l l y implies

279

8b Planar near-rings

These b l o c k s prove u s e f u l f o r c o n s t r u c t i n o block d e s i o n s . F i r s t we d e f i n e t h e s e i t e m s . P

8.112 D E F I N I T I O N An i n c i d e n c e s t r u c t u r e ( P , B ) (BE2 - 8 . 4 9 ) i s s a i d t o be a t a c t i c a l c o n f i o u r a t i o n w i t h p a r a m e-.t e r s (v,b,r,k)eIN if _

_

l

_

_

_

l

~

_

( a ) IP/ = v . (b)

IBI

( c ) Each (d)

Each

= b.

pcP

i s i n e x a c t l y r elements of

8.

BEB

BEB:

c o n t a i n s e x a c t l y k elements of P, [BI = k.

i.e.

A t a c t i c a l c o n f i o u r a t i o n i s a b a l a n ..-l___l__ ced block desinn i f

( e ) Each p a i r of

(p,q)EP

2

, p+q,

i s i n e x a c t l y A elements

B

a n d complete i f (f) b =

o t h e r w i s e in c o m p 1 e t e The e l e m e n t s o f

B

.

are called blocks.

"Balanced incomplete

BIBD"; (v,b,r,k,J) are b l o c k d e s i a n " i s a b b r e v i a t e d by " --XV t h e p a r a m e t e r s of t h e BIUD and E : = (11) is called i t s efficiency. -___I

~

8. 113 EXAt.lPLES

( a ) L e t P be a s e t w i t h v e l e m e n t s , k E I N , k s v . L e t B: = { B T P / ~ B I = k } . Then i s a t a c t i c a l confinuration with V v- 1 (v, ( k ) , (k-l), k) and ( i f k 2 2 ) 2 parameters

(P,a)

complete balanced block dcsian with ( b ) Consider the f i e l d

zp

(pelP\!21)

X = ( kv -- 22 ) . and the affine

(zi,$) a s i n 8 . 7 4 . Then (i?:,g) i s a B I B D plane with parameters (p2,p2+p,p+l,p,l). (For p = 2 we q e t a c o m p l e t e b a l a n c e d b l o c k d e s i g n . )

_

_

280

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

For t h e t h e o r y o f b l o c k d e s i r l n s we r e f e r t h e r e a d e r t o ( H a l l ) o r (Dembowski), where a l s o 8 . 1 1 3 a n d 8 . 1 1 4 can be f o u n d . The p a r a m e t e r s o f a BIBD a r e n o t i n d e p e n d e n t a t a l l : 8.114 P R O P O S I T I O N L e t

(P,B)

be a B I B D . Then

( a ) k b = vr = l ( ( p , B ) l p ~ B ,

BEBII

( t h i s holds f o r every

tactical confiquration!). ( b ) r ( k - 1 ) = X(v-1) = I { P E P \ P

$.

(1 A

3 BEB:

~ E AB q E B I ) ,

where q i s a r b i t r a r y i n P . ( c ) If b > l a n d and r r k .

ksv-1

then

btv

( "Fisher's inequality") A -

BIBD's a r e an e s s e n t i a l t o o l i n e x p e r i m e n t a l d e s i o n s . The f o l l o w i n g example s h a l l i l l u s t r a t e t h i s a n d p r o v i d e enourlh motivation f o r t h e r e a d e r t o endure a l s o t h e next paoes.

8.115

A P P L I C A T I O N Suppose you have b k i n d s of f e r t i l i z e r s and w a n t t o t e s t some c o m b i n a t i o n s o f r f e r t i l i z e r s always o n t h e same number k o f e x p e r i m e n t a l f i e l d s . Take some B I B D with parameters (v,h,r,k,X), a n d d i v i d e t h e whole e x p e r i m e n t a l a r e a i n t o v p a r t s .

(P,B)

S i n c e IBi b = number o f f e r t i l i z e r s , c a n be w r i t t e n a s 3 = C B 1 , B 2 , . . . , B b l . Give t h e f e r t i l i z e r number i o n e v e r y f i e l d o f t h e b l o c k B i . Then: ( a ) every f i e l d c o n t a i n s e x a c t l y r d i f f e r e n t f e r t i l i z e r s , ( b ) every f e r t i l i z e r i s applied on e x a c t l y k d i f f e r e n t f i e l d s , and ( c ) e v e r y p a i r of d i f f e r e n t f i e l d s has e x a c t l y h k i n d s o f f e r t i l i z e r s i n common. i t i s a non-trivial 8 . 1 1 6 R E Y A R K S Of c o u r s e , g i v e n b , r , k , problem how t o q e t a BIBD w i t h s u i t a b l e p a r a m e t e r s . I n g e n e r a l , i t i s a n open q u e s t i o n w h e t h e r f o r e v e r y quintuple ( v , b , r , k , h ) o f n a t u r a l numbers which f u l f i l l the c o n d i t i o n s o f 9.!14 t h e r e E X - s t s a BIEO with t h e s e p a r a m e t e r s . We w i l l n o w a p p l y p l a n a r n e a r - r i n o s t o q e t nc?w c l a s s e s o f E 1 I : O ' s .

281

8b Planar near-rings

The e f f i c i e n c y o f a B I B D c a n be i n t e r p r e t e d e c o n o m i c a l l y i n t h e example above. B I B D ' s o f e f f i c i e n c y 1 0 , 8 5 a r e u s u a l l y c o n s i d e r e d t o be " g o o d " . Vany o f them a r e l i s t e d i n (Cochran-Cox) Balanced complete block designs a r e u s u a l l y " r a t h e r i n e f f i c i e n t " . T h i s i s the reason f o r lookinq a t t h e incomplete ones.

.

8 . 1 1 7 THEOREll ( F e r r e r o ( 1 2 ) ) . L e t N b e p l a n a r w i t h I N [ = : V E N , Denote b y 3 t h e s e t o f a l l b l o c k s ( 8 . 1 0 4 ) . L e t a l (a2) be t h e number o f n o n - z e r o c r b i t s o f (N,t) under G c o n s i s t i n o of elements of C 1 ( N ) (not o f C1(N), r e s p e c t i v e l y ) ( c f . 8 . 1 1 0 ) . Then (11,B) i s d t a c t i c a l configuration w i t h parameters

P-r o o f . The f i r s t p a r a m e t e r i s c l e a r . c o m p u t e t h e number o f d i f f e r e n t b l o c k s a n d a p p l y 8.111: The number o f b l o c k s a N t b w i t h a c C l ( N ) i s

We

y -, lGOl

t h e one of t h o s e w i t h

atCl(N)

(case ( b ) ) i s

u2-v.

Now a p p l y 8 . 1 0 6 t o net k = \ G o / i n 8 . 1 1 2 ( d ) . N e x t o b s e r v e t h a t t h e number r n o f b l o c k s c o n t a i n i n r r an e l e m e n t n E N i s t h e same f o r e a c h ncN, s i n c e i t e q u a l s t h e number o f b l o c k s c o n t a i n i n o 0. Now we know t h a t ( N , B ) i s t a c t i c a l a n d we c a n a p p l y 8 . 1 1 4 ( a )

c o u r s e , t h i s c o u l d be a c c o m p l i s h e d d i r e c t l y , t o o ) . Observe t h a t v = ( a l + a 2 ) [ C 1 t 1 . N o t h i n q i s more n a t u r a l now t h a n t o a s k , u n d e r w h i c h c o n d i t i o n s (ti,%) i s a 3!93. T + e T e f c t ) i e o r o i a n s w e r s t i i i : q u e s t i o n , t t l u s b r i n q i n a j o y and happin,ess i n t o o u r l i f e .

282

5 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

8.118 THE9REI.I ( F e r r e r o ( 9 ) - ( 1 2 ) ) . Let ( N , ? 3 ) be a s a b o v e . (N,B) i s a B I B D <-> C 1 ( N ) i.1 ( t h e n h = 1 ) o r (then X = l G o l ) . C1(N) = {Ol

P r o o f . - > : I t d o e s n o t seem t o be p o s s i b l e t o d e d u c e r Ck - 1 ) E Z ( 8 . 1 1 4 ( b ) ) . t h i s f r o m t h e f a c t t h a t -vz-So we have t o work. C a l l ( f o r a , b E F I ) a , b e q u i v a l e n t i f aH = b l I ( a a n d b a r e t h e n i n t h e same o r b i t i l n d e r G ) a n d d e n o t e t h i s by a x h . Ide n e e d a lemma. N be p l a n a r a n d n ' , n " be E N , n ' ?= n " . L e t A : = A~ be t h e number o f b l o c k s B w i t h n',n"cR ( " b l o c k s thr-ouoh n ' a n d n " L e t 11 b e t h e number o f d i f f e r e n t r e p r e s e n t a t i o n s of n: = n'-n" as 3 d i f f e r e n c e of two eqcriva1ent elements n o t contained in C l ( N ) . Then: If nEC1(N) then h = p + l . I f ntC1(FI) t h e n A = 1 1 + 2 .

-Lemma: L e t

'I).

P r o o f o f t h e Lemma: F i r s t o b s e r v e t h a t i f t h e block aNtb contains D a n d n ( = n ' - n " ) then

aI.I+(b+n")

contains n ' and n".

X i s t h e number o f b l o c k s t h r o u q h

r)

hence and

n (+ 0). {O,nir'aN+b be s u c h a b l o c k .

How many d i f f e r e n t b o c k s w i t h

e x i s t ? Let

aN+b

whence nii = C a s e ( 1 ) : I f b = 0 , nEaH*, ) a n d t h ere i s only = aN by 8 . 1 0 9 ( one p o s s i b i l i t y t o have I 9 , n l E a ' R f o r some a'Er4'. Case ( 2 ) : b = n . Then OEali+n, n E -aiJ, whence aN = -nN. So t h e r e i s a a a i n j u s t one block t h r o u o h 0 and n .

*

n. 3 nlEN : 0 = anl+b. C a s e ( 3 ) : 0 -f b Hence a ' i = - b r l , a n d d H + h = - b N + b . So i f n i s a d i f f e r e n c e a s s t a t e d i n t h e l e m m a , t h e blockc, ? n t o r i s i ? e r ? ' l ~ ?

283

8b Planar near-rings

have t h e form -cN+c. Conversely, f o r the block -bN+b +It we a e t , s i n c e n E a N t b , 3 n 2 E i l : n = = -bn2+b, which i s a r e p r e s e n t a t i o n o f n a s a d i f f e r e n c e o f two e q u i v a l e n t e l e m e n t s of bN. If bEC1(N), bFls(i4,t) i m p l i e s nEb1.I. S i n c e a l s o O c b N , we a r e i n c a s e ( I ) , a c o n t r a d i c t i o n t o 0 $. b $. n . S o l e t b be & C 1 ( N ) . Then - b l . I t b i s n e i t h e r in case (1) nor ( 2 ) n o r equal t o sortie o t i e r - b ' X : t i ' containinn 0 2nd n, b u t w i t h b 4 b ' by 8 . 1 1 1 . S o i n c a s e ( 3 ) a r e j u s t a s many b l o c k s n o t i n ( 1 ) and ( 2 ) a s t h e r e a r e r e p r e s e n t a t i o n s of n of the described kind, namely u . So t h e r e s u l t f o l l o w s i f one o b s e r v e s t h a t the two blocks i n (1) a n d ( 2 ) coincide i f f nEC1(N).

P r o o f o f t h e t h e o r em. 8y t h e planar n r o p e r t y , +t nEN* n',n"tFi , n o n" 3 XE~I: n = xnl-xn". So n has ~ G l - ( l ~ \ - l s)u c h r e p r e s e n t a t i o n s (when varyirtq n ' , n " ) . Now t a k e some a r b i t r a r y q E G . -1 Then n = x n ' - x n " = q n l ( ~ ) - q n , , ( x )= ( a n r o a ) ( r l ( x ) ) -(qn.oa)(q-l(x)), p r o v i d i n c r a l l o t h e r ways t o w r i t e n a s a d i f f e r e n c e of e q u i v a l e n t elements. So t h e r e

+

are j

lrst

' G " ( l G ! 1GI - q-

1

d i f f e r e n t ways t o

IGI w r i t e n as s u c h a d i f f e r e n c e .

( a ) If n&C1(Y) a n d n = a-b (azb) then a , b a r e b o t h t$Cl(Fi). For i f e.o. aEC1(N) then bEC1(N), whence a - b c C 1 ( N ) by 8 . 1 0 8 a n d 8 . l I n . By o u r l e ~ r , ; , h = : , t 2 = ( I G ' - l j t 2 = , G ' t I .

284

38 NEAR-FIELDS AND PLANAR NEAR-RINGS

(b)

If

as

n = a-b

ncC1(N)

\GI-1

then the

with

a%b

ways t o w r i t e n

n = a-(a-n)

are exactly

w i t h aEnW*\H\In). F o r nN i s ( 8 . 1 0 8 ) a n a b e l i a n group of order l G o l ; s o V acnN*\(\(nl; a - n = = - n t a Q a. Observe t h a t a and (a-n) a r e i n C1(N). So none o f t h e ! G I - 1 differences o f equivalent e l e m e n t s g i v i n g n a r e a s d e s c r i b e d i n t h e lemma, p = 0,

whence

and

X = 1.

I t may h a p p e n t h a t ( i n 8 . 1 1 7 a n d 8 . 1 1 8 )

nor

Cl(rl)

{Ol

=

neither

C1(N)

= N

(see Betsch-Clay ( I ) ) .

One c a n e v e n s a y m o r e ( s e e F e r r e r o ( 1 2 ) ) : 8.119

(N,p)

REMARK L e t

Cl(N)

If

be t h e BIBD o f 8.117/8.118.

then

= N

(N,t)

i s elementary a b e l i a n (8.108)

a n d t h e r e i s some f i n i t e f i e l d F s u c h t h a t

= (F2,X) space,

o f 8.74;

(N,B)

(N,$)

=

can be considered as affine

and t h e b l o c k s a r e j u s t t h e l i n e s o f t h i s space.

L o o k i n g a t t h e o t h e r c a s e ( w h i c h b r i n c l s u p p o s s i b l y new d e s i c r n s )

yields f i r s t 8 . 1 2 0 COROLLARY ( F e r r e r o ( 1 2 ) ) . Let

pa/[NI. = lGol.

Then

pa, w h e r e pciP a n d o f 8.117 i s a BIBD w i t h k = X =

(N,3)

P r o o f . Assume t h a t

3

i s PEP, ( 8 . 1 0 8 ) , a c o n t r a d i c t i o n . Hence and 8.118 g i v e s t h e r e s u l t . nEN*:

elementary abelian, and

pa/lN!

CI(N) See F e r r e r o ( 1 2 ) , planes.

Cf.

L e t N be a f i n i t e p l a n a r n r .

\ G o \ have n o t t h e f o r m

=

{Ol

ncC1(N).

so

l n N l = pa

Then

nN

with

Teorema 8 f o r t h e c o n n e c t i o n t o f i n i t e f48bius

a l s o Anshel-Clay

A n o t h e r way t o r e a c h t h e c a s e

(1).

C1(N)

=

!Ol

i s the followino.

8b Planar near-rings

8 . 1 2 1 COROLLARY ( c f .

285

Ferrero ( 8 ) ) . L e t N be a f i n i t e i n t e g r a l

planar nr. without subnear-fields.

T h e n t h e same c o n c l u -

s i o n as i n 8.120 h o l d s . P r o o f . Suppose t h a t

ncN*

i s i n

an a b e l i a n subaroup o f (nN)"z

C1(N)

E

beOn c a n

n:I

a contradiction.

{Ol

=

Bn:

b y 8.90 ( a ) . i s a a r o u p and nll i s a

n

((ntl)",.)

s u b n e a r - f i e l d o f Id,

i s

(nN)*

while every

b = 1 b = nn-lb

Consequently,

nN

Then

b y 8.108.

(N,t)

B n i s c l c a r f r o F 8.89,

be w r i t t e n as

tience

C1(N).

and t h e r e s u l t f o l l o w s f r o m

8.118. I n (ti),

Ferrercj constructs

( I N / ,6) IN1 = v ) .

BIBD's

f r o m n e a r - r i n g s 14 w i t h v v 1) ( v , --&., 3, 3) (where

having parameters

1,

9,

B o t h c a s e s i n 8.118 c a n b e o b t a i n e d by t h e f o l l o w i n q n e a r - r i n o s : 8 . 1 2 2 COROLLARY ( C l a y ( 1 1 ) ) .

Consider t h e planar nr.

(F,+,*t)

o f 8.103. (a)

m t = p -1

If

8.117 (P (b)

n 9

-

msn

f o r some

pn(pn-l) --m p (p"-1)

pn-1

P

1

9 -

pm-1 n

(p

Proof. F i r s t observe t h a t t + l = 1G

0

aeF*.

= pm-1

Then

pn(pn-l) t

rn

= x,

(msn). (F,+):

follcws

is a

(t+l)(pn-l) t

9

1

( o f 8.103),

Set has

B

E:

=

B uf'l}.

t + l = pm

from 8 . 1 ? 3 ,

elements

consists o f a l l

hence b e i n q a subrrroup o f

T h i s i s easily t r a n s i e r r e d t o (b)

(F,B)

t = / G I = i6l

a*F = a B

and i s a subaroup o f

xp

,

then

1.

( a ) L e t t be

wit4

as i n

.pm4

BIBD w i t h p a r a m e t e r s ttl, t + l ) .

Take

-

(F,B)

m , 1).

I f t i s n o t o f t h e form

so

then

i s a BIBD w i t h parameters

aE.

XCF

(F,+).

Now a p p l y 8.!!8.

286

$ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS

8.123 REMARK Observe t h a t one can efficiency

E =

pn-pn--"

qet

RIBD's o u t o f 8.122 w i t h

(in (a) )

and

E =

pn- 1

p n . t_(pn-1) ( t+l)

( i n ( b ) ) , which i s c l o s e t o 1 f o r l a r g e n . 8.124 R E M A R K B I B O ' s c a n a l s o be c o n s t r u c t e d f r o m n o n - a b e l i a n f i n i t e p l a n a r n e a r - r i n q s ( s e e Clay ( 1 1 ) ) . D e f i n e on B 7 ~ H 7 x i Z 7 an a d d i t i o n "8" by

(a,b,c) 8 (d',b',c'):

=

(a+a', b t b ' , c+c'+a'*b).

N be d e f i n e d v i a q ( a , h , c ) : = ( 2 a , 2 b , 4 c ) . L e t q:N T h e n ( B . H . rleuriann ( 2 ) ) (N,$) i s a non-atelian n ! - ~ u p 2 and G : = ! i d , q , o 1 i s a f i x e d - p o i n t - f r e e automorprlisri group of ( N , @ ) . Clay goes 8 . 9 6 ( a ) q i v e s some p l a n a r n e a r - r i n n ( N , G I , s B ) . on t o p r o v e t h a t i s a B I R D , o f c o u r s e v!ith k = X a n d C 1 ( N ) = {Ol ( t h i s f o l l o w s f r o m 8 . 1 2 0 ) . Clay a l s o q e n e r a l i z e s t h i s example. -+

(N,a)

S e e B e t s c h - C l a y ( 1 ) f o r a n e x c e l l e n t summary o f t h e t h e o r y o f p l a n a r n e a r - r i nqs t o c l e t h e r wi t h new r e s u l t s ( e . 0 . c o n n e c t i o n 5 t o p a r t i a l l y balanced incomplete block desiclns) a n d h i n t s f o r f u r t h e r r e s e a r c h . See a l s o C l a y ( 1 7 ) , ( 1 3 ) .

287

3 9 MORE CLASSES OF NEAR-RINGS

a ) c o n t a i n s commutativity theorems s i m i l a r t o the "n(x)-theorem" o f J a c o b s o n and t h e " n ( x , y ) - t h e o r e m " o f H e r s t e i n i n r i n q t h e c r y . Our d i s c u s s i o n i s d o n e i n t h e w o r l d o f I F P - n e a r - r i n a s ( t h a t a r e n r . ' s N w h e r e a b = 0 i m p l i e s a n b = 0 f o r a l l nc:i). A d q n r . with the "n(x,y)-property" i s a convutative r i n o . p - n e a r - r i n q s a n d Boolean n r . 3 a r e a l s o c o n s i d e r e d ( a s s p e c i a l cases). N e x t , we s t u d y n r . ' s w i t h o u t n i l p o t e n t e l e m e n t s . They a r e ( i f in s u b d i r e c t p r o d u c t s o f i n t e n r a l n r . ' s which a r e s t u d i e d i n p a r t 2 ) o f b ) . The f i n i t e i n t e a r a l n e a r - r i n a s a r e planar i f f they are n o t " t r i v i a l " . Special inteqral n r . ' s ire c a l l e d " n e a r - i n t e g r a l domains". T h e i r c h a r a c t e r i s t i c i s zero o r a prime. c ) c o n t a i n s a d i s c u s s i o n of a f f i n e n r . ' s ( i . e . a G e n e r a l i z a t i o n o f n r . I s o f t y p e M a f f ( V ) ) . We e x a m i n e t h e i d e a l s t r u c t u r e , the radicals and n r . ' s constructed o u t of affine n r . ' s . F u n d a m e n t a l f o r t h e s e n r . ' s i s t h e f a c t t h a t ri0 i s a r i p 9 and N c a n ideal of N. d ) b r i n q s ( f o r c e r t a i n c l a s s e s o f oroups) answers t o t h e q u e s t i o n s , which n r . ' s ( n r . ' s w i t h i d e n t i t y , . . . ) a r e d e f i n a b l e on a q i v e n a d d i t i v e o r o u p . F o r i n s t a n c e , e v e r y fir. w i t h i d e n t i t y on a c y c l i c o r o u p i s a commutative r i n n . S e v e r a l g r o u p s a r e e x p l o r e d w h i c h c a n n o t be t h e a d d i t i v e o r o u p o f a nr. w i t h i d e n t i t y .

no)

We g o o n b y d i s c u s s i n g o r d e r e d n r . ' s i n e ) ( a n d d i s c o v e r t h a t v e r y few n r . ' s c a n be f u l l y o r d e r e d ) . R e g u l a r n r . ' s a r e s t u d i e d i n f ) , tame n r . ' s i n g ) , w h i l e h ) c o n t a i n s i n f o r m a t i o n o n MS(r), where S i s n o t a f i x e d - p o i n t - f r e e automorphism group. We c i o s e w i t h t h e c o n n e c t i o n s b e t w e e n n r . ' s a n d a u t o m a t a i n i ) a n d a survey on o t h e r tonics i n j ) .

288

$ 9 MORE CLASSES OF NEAR-RINGS

a)

IFP

-

NEAR-RINGS

I n r i n g t h e o r y , t h e f o l l o w i n s two t h e o r e m s a r e c e r t a i n l y among t h e m o s t f a m o u s c o m m u t a t i v i t y t h e o r e m s ( s e e e . q . ( P r o c e s i ) ) : T H E O R E M 1: L e t R b e a r i n o w i t h

!X E R 3

n ( x ) c D W l ~ :x ~ ( ~ x) =

Then l? i s c o m m u t a t i v e . ---T H E O R E M 2 :

L e t R be a r i n g w i t h x , y ~ R3 n ( x , y ) E I N \ C l J : ( x y - y x ) n ( x , v !

= x y - yx.

Then R i s c o m m u t a t i v e . ( T h e f i r s t o n e w a s o b t a i n e d by N. J a c o b s o n ; t h e s e c o n d o n e i s due t o I.N. H e r s t e i n . ) We w i l l g e n e r a l i z e t h e s e r e s u l t s t o c L r t s i n c l a s s e s o f n e a r r i n g s ( i n c l u d i n s t h e d a n r . ' ~ )u s i n o s u b d i r e c t d e c o m p o s i t i c n s . I n o r d e r t o g e t a s a t i s f a c t o r y t r e a t m e n t we s t a r t w i t h a m o r e general c l a s s of near-ri nss:

1.1 lFP-!iEAR-RINGS

9 . 1 DEF1:IITIOti -

A n r . :t i s s a i d t o f u l f i l l t h e i n s e r t i o n - o f -

factors-property

(I F P ) provided t h a t

a , b , n & N : ( a b = 0 ->

anb = 0).

N h a s t h e s t r o n r i I F P i f e v e r y homornorphic i m a o e o f I

11 h a s

the IFP. The n e x t i s a n i n t r i n i s t i c c h a r a c t e r i z a t i o n o f t h e s t r o n q I F P :

289

9 a IFP-near-rings

9.2

PROPOSITION ( P l a s s e r ( 1 ) ) . N h a s t h e s t r o n a I F P : <=> IsN V a , b , n ~ N : ( a b E I =-> a n b c I ) .



v

The p r o o f i s s t r a i a h t f o r w a r d a n d h e n c e o m i t t e d . We w i l l s o o n q e t e x a m p l e s o f I F P - n e a r - r i n n s . characterize these near-rinas.

9.3

B u t b e f o r e we

PROPOSITION ( B e l l ( l ) , P l a s s e r ( 1 ) ) . The f o l l o w i n q assertions are equivalent: ( a ) N has t h e IFP-property. (b)

'd

(c)

ncN:

(0:n) 9 N.

S5i.I:

(0:s) A

N.

Again, t h e proof i s obvious. Observe t h a t e v e r y I F P - n e a r - r i n q N w i t h l e f t i d e n t i t y e i s i n eO = 0 i m p l i e s t h a t e n 0 = 0 , whence n O = 0 f o r a l l nEN.

q ,f o r 9.4

DEFINITION C o n s i d e r t h e f o l l o w i n a p r o p e r t i e s :

(PI):

(Po)

(P2):

W NE

9.5

3

XEN

(Po):

n ( x ) > l : xn(x)

and N i s

=

x.

EQ.

3 n(x,y)>l: ?lo. X,YEN

(P3):

x , y , z c N : xyz = x z y

(P4):

x,yEN

(xy-yx) n ( x y y )

ISN: X Y E I ->

=

xy-yx

and

("weak c o m m u t a t i v i t v " ) . YXEI.

REMARKS ( a ) The ' ' x n ( x ) = x " - p r o p e r t y d o e s n o t i m p l y t h a t Ncno, f o r e v e r y Ncnc f u l f i l l s i t . N r . ' s w i t h ( P l ) a r e c a l l e d " L - n e a r - r i n a s " i n L i a h ( 1 1 ) . 5 e e S z e t o (6), ( 8 ) f o r a characterization v i a sheaf representations. N with xcil: x 2 = x a n d ( P 3 ) s t u d i e d b y R d t l i f f ( 1 ) an: Subr;hnanyay ( 1 )

( b ) Abelisn n r . ' s

were

$ 9 MORE CLASSES OF NEAR-RINGS

290

("Boolean s e m i r i n q s " ) . Abelian n r . ' s w i t h (P,) a r e c a l l e d " s e m i r i n o s " t h e r e . The n r . ' s I{ w i t h 1 x c N : x T x a n d ( P 3 ) a r e t h e "8-near-rinos" of Ligh ( 1 4 ) .

(c) ( P 4 ) was c o n s i d e r e d by B e l l (11, ( 2 ) a n d P l a s s e r ( 1 ) . Every nr. w i t h ( P 4 ) i s in B u t , on t h e o t h e r hand, every c o n s t a n t nr. has ( P 3 ) .

no.

9.6

PROPOSITION (Re11 ( 2 ) , ( a ) ( P I ) =>

(5) =>

( b ) Each o n e o f

Lioh (16)).

(Pa). to

(P1)

(P4)

implies the strong IFP-

property. P r o o f . ( a ) : ( P 1 ) -> ( P 2 ) i s i m m e d i a t e . Assume ( P 2 ) a n d X Y E I . Then y x - x y 5 yx ( m o d I ) a n d 3 n c I N \ { l I : y x - x y = ( ~ x - x y )z ~ ( Y X ) ~= y x y x . . . y x : 0 (mod I ) . Hence Y X E I . ( b ) : S i n c e ( P l ) - (P,) a r t i n h e r i t e d t o homomorphic i n a g e s i t s u f f i c e s t o show t h e I F P - p r o p e r t y i n t h i s

c a s e . By ( a ) , we o n l y h a v e t o l o o k a t

(P3): I f

ab = 0

and

( P 4 ) : I f a b s I and whence anbcI

nEN

nEN

by

then

then (P4).

(Pg)

a n b = abn

bas1,

hence

and

(P4).

!In = 0 . b(an)cI,

See e . g . Liph ( 1 6 ) and ( T h i r r i n ) f o r t h e c o n n e c t i o n t o " L u L r i n o s " ( i . e . r i n n s , i n w h i c h e v e r y o n e - s i d e d i d e a l i s twos i d e d ) . C l e a r l y each duo r i n ? i s a stronn I F P - n r . ( b u t n o t c o n v e r s e l y ) , For a d e t a i l e d s t u d y o f " d u o - n e a r - r i n a s " s e e Choudhari ( I ) , ch. V I I I , Choudhari-coyal ( I ) a n d Parnakotaiah-Rao ( I ) . For e a s y r e f e r e n c e , i t rewards t o d e f i n e f o r t h i s chaptet. 9.7

D E F I N I T I O N L e t a n r . H be o f

type I t ype I1

if

N s i m p l e and s i r o n a l y uniform ( 8 . 9 9 )

FIE%,

i f ;icng i s n o t s i r i p l e , b u t t ' l e i n t e r s e c t i o n o f a l l n o n - z e r o ; d e a l s c o n t a i n s no n o n - z e r o i d e PI r) o t E. n t

.

9a IFP-near-rings

29 1

N$n0, fJgnc

t y p e 1 11 if a n d i f P ( a s a b o v e ) h a s a nonzero idempotent then P = N 0' t y p e I V i f NE?'),. type v i f V X , Y E N : xy = 0 . The s t r u c t u r e o f s t r o n q I F P - n e a r - r i n o s i s o i v e n by 9.8

THEOREM ( i i o h ( 1 6 ) ) . Every s t r o n q I F P - n e a r - r i n q N i s a

subdi r e c t product of subdi r e c t l y i r r e d u c i b l e IFP-near-rinas o f type I , I I , I I I , I V or V .

-P- r o o f . L e t N be t h e s u b d i r e c t p r o d u c t o f some s u b d i r e c t l y i r r e d u c i b l e near-t-inns (1.62(a)).

The

lJi

(icsomc index s e t I )

N i l s h a v e t h e I F P - p r o p e r t y by 9 . 1 .

i s simple, use 9 . 3 t o net ( a ) I f Nismo a n d N i i n t o type I o r type V .

Ni

( b ) Now l e t By 1 . 6 0 ( c ) ,

b e n o t s i m p l e and P be as i n 9 . 7 . tlic% P -f; { O l . Assume t h a t P c o n t a i n s t h e

idempotent e p 0. If 3 x € N i : x e -1 x t h e n 0 xe-xe(0:e) 9 N i , so Pc(O:e), ee(o:e) a n d e = e 2 = 0, a contra-

4

d i c t i o n . Hence e i s a r i o h t i d e n t i t y , c o n t a i n e d i n P , whence P = N i , a c o n t r a d i c t i o n . (c) If

Ni

l o ) $. ( N i ) o

i s neither (0:0)

e v e r y i d e m p o t e n t e $. 0 I f x ~ ( O : 0 ) , x = xeEP,

nor

then As i n ( b ) , i s a rinht identity in PIi. hence P = (0:O) = ( N i ) o .

F?&


so

cnc

Pc(0:r)).

I n s p e c i a l c a s e s o n e a e t s more o u t o f 9 . 8 : 9.9

C O R O L L A R Y ( R a t l i f f ( I ) , L i q h ( 1 6 ) ) . A nr. w i t h (Po) and t h e strong I F P i s a s u b d i r e c t product of s u b d i r e c t l y irreducible near-rinqs Ni E0) w i t h r i q h t i d e n t i t y o f t y p e I ( i n which c a s e N i i s s i m p l e and i n t e q r a l j . t y p e I 1 1 ( i n which c a s e t h e a n p i h i l a t o r i d e a l s a r e e x a c t l y { O I and ( : i i ) o ) o r t y p e I V .

$9 MORE CLASSES OF NEAR-RINGS

292

P r o o f . C l e a r l y , t y p e V cannot o c c u r . Suppose t h a t N i h a s t y p e 1 1 . By t h e s u b d i r e c t i r r e d u c i b i l i t y , P {Ol. If XEP*, Xn(x)-l i s a n o n - z e r o i d e m p o t e n t i n P, a c o n t r a d i c t i o n . By t h e same a r n u m e n t , we g e t P = N 0 i n the case that Ni i s o f t y p e 111. I f t h e n ( 0 : n ) $. { O l , we d e d u c e f r o m 9 . 3 t h a t N o c ( O : n ) . I f m = m 0t m , ~ ( O : n ) , m c i s z e r o . Hence ( 0 : n ) i s e i t h e r lo1 o r = (Pii)o a n d t h e same f o l l o w s for all (0:s) (SgNi). Now l e t N i be of t y p e I . (Po) f o r c e s N t o be i n t e g r a l . Now p i c k u p some e f 0 . e n ( e 1 - 1 = :

i s a rioht identity, for = ze"(')-ze = ze-ze = 0 ,

v

ZENi:

whence

(zr-z)e = zre-ze =

z r = z.

9 . 1 0 C 3 R B L L A K Y- I f N h a s ( P 1 ) t h e n N i s isrrrnorphic t o a s u b d i r e c t product of simple i n t e o r a l near-rinos with a r i g h t i d e n t i t y . Corollary 9.9 c r i e s f o r

9.11 D E F I- N I T I O N ( R a t l i f f ( I ) ) . A n r . PI i s c a l l e d a l m o s t sn& i f PI h a s a t m o s t 2 d i f f e r e n t a n n i h i l a t o r i d e a l s . 9.12 C O R O-L L A R Y ( R a t l i f f ( l ) , Liqh ( I I ) , ( 1 6 ) ) . Every s t r o n n IFP-nr. with ( P o ) i s r e p r e s e n t a b l e as s u b o i r e c t product of almost small n e a r - r i n q s . S e e a l s o L i g h ( 1 1 ) and S z e t o ( 1 ) f o r more d e t a i l e d v e r s i o n s o f 9.8 f o r n e a r - r i n g s w i t h ( P o ) o r ( P 3 ) . Now we t u r n t o

(P2).

9 . 1 3 PROPOSITION L e t

F i r s t we n e e d N = No

( c f . Pamakotaiah-Pao ( 2 ) )

be s u b d i r e c t l y i r r e d u c i b l e .

N h a s t h e I F P , b u t n o n i l p o t e n t e l e m e n t s b e s i d e 0 -' N i s integral. ( b ) I f I.4 h a s ( P 2 ) , i s i n t e o r a l a n d h a s :Id ;b i r ) l then N f u l f i l l s b o t h c a n c e l l a t i o n l a w s , i s a b e l i a n and e i t h e r t o m c l u t a t i v e o r EW,.

(2)

=>

29 3

9a I FP-near-rings

*

P r o o f . We may a s s u m e t h a t N (a) Consider any x~I.1".

101. (Cx k

JkEIN

does n o t c o n t a i n 0 and i s c o n t a i n e d ( Z o r n ! )

i n a

The s e m i o r o u p

m u l t i p l i c a t i v e semiqroup c o n t a i n i n g 0.

maximal f o r n o t

tlx

Ix:

Consider

I,*)

U

=

(0:m).

IFP

The

mEtYX

Ix4 N .

implies that

w.r.t.

Since

n

multiplication!),

(!Ix i s c l o s e d

x$Ix

{O}.

I, =

ZEN*

Since

3

4 I i s subdirectly irreducible,

If

nEN

by

My

i s n o t i n PI

YEN":

I

Y t h e subsemiaroup oenerated

Y'

a n d n c o n t a i n s 0.

=

CO).

So some p r o d u c t c o n t a i n i n n

a t l e a s t o n e t i n p s n ( a n d p o s s i b l y e l c i l e n t s o f rl ) Y must be z e r o . Such a p r o d u c t has one o f t h e f o l l o w c n o forms: = 0,

rnlnm2

nm' = 0 , m " n = 0,

n = 0

n = 0

An a p p l i c a t i o n o f

(P4)

3

i n which case

meMy:

My = N"

Thus

1) I f

with

yields

nEI

n = 0.

aqain

(b)

{Ol,

nm =

3

and

arbitrary d(en-n)

EN.

$. 0,

xy-yx

= xy-yx.

zero idempotent.

Y

or = {01,

so

N i s inteoral.

x,yCN:

(xy-yx)'

,m"~Il ). Y

(ml,m2,m'

Let

dENd

be

= d e n - d n = d n - d n = 0,

e

i s a non-

a n d let n b e

0

( n e - n ) e = 0,

Then

kEIN\Il)

take

( ~ y - y x ) ~ -= :l

so

whence

ne =

n

e n = n.

and

50

N h a s a n i d e n t i t y 1 a n d e a c h n o n - z e r o i d e m p o t e n t = 1.

2 ) how l e t

a,b,ccN

c e n t r a l , we y e t

3

fEr!*:

af-fa

with

b = c.

li.

0,

!2~IN\{ll be s u c h t h a t (a(fa)-(fa)a)'-'

Then

ab = ac,

a

0.

If a i s

If a i s not central,

hence

(af-fa)a

0,

Let

(a(fa)-(fa)a)L = a(fa)-(fa)a. = 1 b y 1) a n d a h a s a l e f t

inverse which aoain r e s u l t s i n

b = c.

3 ) W E now show t h a t ( N , + ) i s a b e l i a n . L e t 1+1 = : 2 . If 2 = 0, e a c h e l e m e n t o f N i s o f o r d e r 2 a n d r.1 i s abelian. If

p

0, but 2 i s c e n t r a l then erpandino (n+m)(l+l) i n both ways gives IT^ = q + n =or a l l m.ncN, 5 3

2

again N i s abelian.

8 9 MORE CLASSES OF N E A R - R I N G S

294

If

2

$. 0

a n d 2 i s n o t c e n t r a l , we h a v e t o e x a m i n e

the conditions of

N i s

above,

By t h e c o n s i d e r a t i o n s

1.109(c).

cVl

n(-1)

and

=

n

= n.(l)

yields

n = 0. Then u i s a r i r r h t f o r ( 1 - 2 u ) Z = 2-21.12 = 2 - 2 . 1 = 2 - 2 = 0 2 u = 1 . L e t rEN b e a r b i t r a r y ; c a l l

2 has a l e f t i n v e r s e u (say), inverse, too, implies

h: = u - r . = lr = r.

Then

h+h

urtur

r = hth = h'th'

Finally,

(utti)r = (2u)r =

gives

2h = Z h ' ,

whence

h = h'. An a p p l i c a t i o n o f ' l . l r ) 9 ( c ) shoi.;s t h a t

i s

(PI,+)

abel i an. 9.14

THEORE!?

(Bell

I

( I ) , (2), I . i ? h ( 1 2 ) , L i a h - L u h ( ( 1 ) ) .

L e t ii b e a d o n r .

Proof.

.

N has ( P 2 j

as i n 9.8,

Decompose,

'I-->":

Ni

irreducible rtr.'s

Ni

(P2)

has a l s o

I4 i s c o n m u t a t i v e .

<=>

N into subdirectly

$- { O l ,

and

C o n s i d e r some

Ni.

!Oi.

(Ni)d

( a ) We f i r s t s h o w t h a t e a c h n i l p o t e n t e l e m e n t i s

central

i n C(P4)

(i,e.

-

8.16).

!.!e w i l l

accomplish

t h i s by i n d u c t i o n o n t h e d e n r c e k c f n i l p o t e n c e . k = 1

b u t we a l s o n e e d k = 2 :

i s trivial,

n2 = 0.

Suppose t h a t

= xnxn-nxxn = 0-0 = 0, nxn = 0 (xn-nx)nx xn-nx

= 0

mc!i

mk-'cC(N);

(xn-nx)'

b

nE.14: n k - l

with

n k = Q.

hence

= c),

0

implies

Similarly, whence

. . . . . . . . ( mx-xm)m

= 0 -->

Then

ncC(N), and k-1 2 (rn ) = 0, SO

n k x - x n k = mxm'-'= ( I F P ! ) = (mx-xm)n(xm-xn)m.. k- 1 k- 1

XEN:

- xmm k -=l ( m x - x m ) m k - '

0

((mx-xm)m)

=

(m(xm)-(xmjm)

( P 2 ) a o a i n y i e l d s m(xrn)-(xm)m = 0. aSovc, i t tut-r,; G I ; t +-hat ; r : x - x r 1 2 = O , :vhe?r,?

Applyinq AS

nr! =

by t h e I F P .

=

(P2).

by

Now 8ssuc:e t h d t take

so

= 0,

(xn-nx)xn

XEN:

since

nxxn = 0

and

V

Then

mx-xm = 0 .

.

9a I FP-near-rings

29 5

( b ) From ( a ) a n d t h e I F P - p r o p e r t y o n e a e t s ( a s f o r

rinqs) that the s e t Np%(Ni) of N i f o r m s an i d e a l . ( c ) I f N p t ( N i ) = Ni, (P2) commutativity of (N,. ) . N p t ( N i ) = {O}

(d) If

and

of a l l n i l p o t e n t elements

instantly results the

N F ~ , , N i i s inteqral

by 9 . 1 3 ( a ) ,

h e n c e a b e l i a n b y 9 . 1 3 ( b ) , c o n s e q u e n t l y a r i ~ q( 6 . 9 ( c ) a n d 6 . 6 ( c ) ) a n d t h e r e f o r e a c o m m u t a t i v e otie ( T h e o r e m 2 ) . I f Pi&?,, N i s commutative by 9 . 1 3 ( b ) .

-+ f i p t ( V i )

(e) If

$- I J i ,

consider

TTi:

= ''i/Npt(Xi)'

Ri

h a s no n o n - z e r o n i l p o t e n t e l e m e n t s , b u t i s a n a i n dg. w i t h ( P 2 ) . By ( d ) , fli i s a commutative r i n q . So f o r a l l n',n"EN, n ' n " - n " n ' c N p t ( N i ) , from which

we cret

n'n"

= n"n'

by

P2)'

( f ) S i n c e s l i rIi a r e commutative n e a r - r i n q s , same a p p l i e s t o t i . (91

" <=I'

the

i s t r i vi a1

9 . 1 5 R E H A R K ( L i g h - L u h ( 1 ) ) . The a s s u m p t i o n i n 9 . 1 4 t h a t " N i s d g . " c a n be r e l a x e d by " N i s a 0 - n r . " w h i c h mean5 t h a t e a c h homomorphic i m a q e fT o f NE??~ h a s '17,

C O R O L L A R Y ( B e l l ( 2 ) , Liqh ( 1 2 ) , T' w i t h Pi = N. Then N h a s (P2)

( 1 G ) ) . L e t N he a d o n r .

i f f N i s a conmutative

rinq. P r o o f : a p p l y 9 . 1 4 a n d l.l07(C). 9 . 1 7 C O R O.I_I_ L L A P Y ( B e l l (11, L i o h ( 7 ) , ( 1 1 ) ) . L e t PI h a v e I F P , ( P o ) and non-zero d i s t r i b u t i v e cleinents i n e v e r y non-;ero homomorphic imarie. Ther! N i s ~ r )a n~d a s u b d i r e c t n r o d u c t of

hf.'s

(hence abelian),

$ 9 MORE CLASSESOF NEAR-RINGS

296

P r o o f . No = ( 0 : O )

so

N b y t h e I F P . Now N / N o {OI, whence N/Ho = {Ol

5l

(N/No)d =

(P1)

T h u s we h a v e

IFP

and hence

i s constant,

N = No~??o.

and

and t h e s t r o n q

(P2)

available.

By 9.10,

N i s the subdirect product o f simple intearal

w i t h r i c r h t i d e n t i t i e s and no n i l p o t e n t B y 9 . 1 3 ( b ) , e v e r y Ni i s a b e l i a n and e i t h e r

n e a r - r i n q s Ni elements.

commutative (then a simple commutative r i n a w i t h hence a commutative f i e l d ) o r

identity, (Po)

Ni

implies that

Observe from 2.52(b)

i s a nf.).

DCCI

that the

(then

E??,

i n FI w i l l t u r n !l i n t o a

f i n i t e d i r e c t sun1 o f n f . ' s .

9 . 1 8 COROLLARY ( B e l l ( l ) , L i q h ( 7 ) ,

I

(Po)

with

(by 9.17

( l l ) , (16)). Every danr.

i s a s u b d i r e c t p r o d u c t of

and 9 . 1 4 )

9.19 COROLLb,RY ( R a t l i f f ( 1 ) ) . A n r . w i t h

If

Ni

i s of

Ni

type I then

(Po)

+ {Ol

I -

a subdirect product o f nr.'s

IY.

commutative f i e l d s

and hence a c@mmlrtative rinrr.

Ni

and

(P3)

i s

o f t y p e I , I 1 1 or

i s a commutative f i e l d

o r has more t h a n one r i q h t i d e n t i t y . Proof.

N c a n n o t b e o f t y p e 11.

Accordinc: t o 9.9,

If

I+

i s o f t y p e I and h a s j u s t one r i o h t i d e n t i t y e t h e n

v

XENY : x ' ( ~ ) - '

Hence

= yn(y)-lyx i d e n t i t y and domain,

(as i n t h e p r o o f o f 9.13(b)).

= e

x , y ~ N q : xy = xn ( x ) - l x y = yx.

tii

so

N~

i s

= exy = eyx =

commutative,

i s (by 9.13(b))

e i s an

a simple intenral

hence a f i e l d .

9 . 2 0 COROLLARY ( R a t l i f f ( 1 ) ) . L e t N h a v e

(Po),

(P3)

and n3n-

z e r o d i s t r i b u t i v e elements i n e v e r y non-zero homomorphic image ( t h i s happens e.g. Then

N

i f

Ncql

o r i f N i s do.).

i s a s u b d i r e c t p r o d u c t o f commutative f i e l d s and

hence a ccmmutative r i n g ,

9 a I FP-near-rings

P r o o f . By 9.17, o f 9.9

all

N&wo,

so i n t h e s u b d i r e c t d e c o m p o s i t i o n

are o f type I.

Ni

Ncnl.

By 9 . 1 3 ( b ) ,

29 7

Due t o 9.19,

Ni

i s a commutative

field. ~9.21 R E- M A R K S - Now we m e n t i o n ( w i t h o u t p r o o f , o t h e r c o m m u t a t i v i t y theorems.

(a)

l l ~ ? ? ~be a d q n r . w i t h n ( x ) E D u : X - X " ( ~ ) E C ( N ) ( 8 . 1 6 ) . T h e n PI i s a Let

(Bell ( 6 ) , ( 7 ) ) .

XEN

but w i t h reference)

3

c o m m u t a t i v e r inq.

( b ) (Licrh-Luh

w i t h i d e n t i t y i n which

(1)). A f i n i t e D-nr.

a l l n i l p o t e n t elements a r e c e n t r a l i s a commutative

ring. (c)

( L i g h (11)). A f i n i t e dqnr.

w i t h o u t n i l p o t e n t elements

i s a commutative r i n o .

(d) (Ligh (12)).

A f i n i t e dqnr.

i s commutative i f f a l l

zero divisors are central.

(e) (Plasser ( I ) ) . If N Then

P~E??~

and a l e f t i d e n t i t y e.

(Po)

hds

xcN : e-xO

i s i d e m p o t e n t c=->

i d e m p o t e n t s a r e d i s t r i b u t i v e <=> nf.'s.

( P 1 ) ->

N has

c e n t r a l <->

Anyhow,

(N,t)

ring.

Each a - n r .

(9) ( L i g h ( 1 5 ) ) .

w

Call

i s torsion

N F ) ~

Every a-nr.

-dENd.

ncIN

with

A nr.

b

(N,t).

an a - n r .

if

(Po)

dENd

implies

i s a commhtative r i n o .

(Pg)

N with

: ( x Y ) ~= x n y n .

this property (or with = xn+yn)

and e a c h e l e m e n t

w i t h o u t n i l p o t e n t elements i s a

Each n r .

x,yeN

a l l idempotents dre

N i s subdirect product of

has a s q u a r e - f r e e o r d e r i n

(f) (Liqh (8)).

fulfills

Every a-nr. X,YEN:

n&IN

=

(N,.).

w i t h o u t n i l p o t e n t elements and w i t h

x , y ~ N : ( x Y ) ~= x 2 y 2

( h ) (Liqh-Utuni

N with

(xty)"

has o n l y n i l p o t e n t commutators o f

NcWl

all

i s abelian.

( 1 ) ) . 9 i s a C!-nr.

n E N : nN = nPin

( N n = nNn,

one irnDiies t h e o t h e r :

(C2-nr.)

if

respectively).

Neither

59 MORE CLASSES OF NEAR-RINGS

298

I f F i s a f i e l d then n o t C2-nr. .

MafF(F)

(1.4) i s a C1-,

A f i n i t e i n t e g r a l nr.

has C1,

but not

but

C2.

E v e r y C2-nr. ( b u t n o t e v e r y C 1 - n r . ) h a s t h e I F P . N i s C1- a n d C 2 - n r . i f f N i s C1-nr, a n d e v e r y idempotent i s central. S e e t h i s p a p e r f o r d e c o m p o s i t i o n t h e o r e m s f o r C 1 - and C2-nr.'s with f i n i t e n e s s conditions. ( i ) A r i n g R i s called a P 1- r i no i f for all rrR t h e r e i s a c e n t r a l idempotent ro with r r 0 = r a n d e 2 = r&R : ( e r = r e ==> r 0 e = r * ) ( t h i s P I h a s n o t h i n ? t o d o :.rith o u r ( P 1 ) ) . See P l a s s e r ( 1 ) f o r a s i m i l a r c o n c e p t f o r n e a r - r i n ? s . ( j ) F o r more r e s u l t s s e e B e l l (!I)% L i g h ( l ? ) , F'arin R a m a k o t a i a h - R a o ( 2 ) , ( 5 ) a n d Kim-Park ( 1 ) .

(2),

2 . ) p-NEAR-RINGS L _

I ON L e t p b e a p r i m e . I! n r . --9 . 2 2 D E F I N I T--

r i n q provided t h a t

_ i

Evidently, every p-nr.

~

~ E N :xp

=

has p r o p e r t y

x

PI i s c a l l e d a ---p-nearA px =

0.

(Po).

9.23 PROPOSITIOIJ ( P l a s s e r ( 1 ) ) . A p - n r .

w i t h l e f t identity i s

zero-symnetric. P r o o f . L e t e be t h e l e f t i d e n t i t y . T h e n i t i s e a s i l y shown by i n d u c t i o n t h a t X E N k j kcM : ( e + x O ) k = e + k ( x 0 ) . Hence e t x O = ( e + x O ) P = e + p ( x O ) :: e , whence x0 = 0 . 9.24 REMARK ( P l a s s e r ( 1 ) ) . 9.23 does not h o l d f o r yenera1 o r . ' s with (P,).

9.25

CORQLLARY ( R a t l i f f ( 1 ) ) . A p - n r .

with (P3) a n d non-zero d i s t r i b u t i v e e l e ~ e f i t si n ever;; h o m c n 3 r ~ k i c i n a p e i s i s c morphic t o a s t i b d i r e c t p r o d u c t of c o p i e s of t h e f i e l d 8,, h e n c e a ring.

9a IFP-near-rings

299

-Proof.

By 9 . 2 0 , N i s a s u b d i r e c t p r o d u c t o f s i m p l e Ni w i t h i d e n t i t y . Rino t h e o r y commutative p-rinqs

t e l l s us t h a t

%

Ni =2,.

9.26 THEOREM (Plasser (1)). A f i n i t e p-nf. the field

PI i s i s o m o r p h i c t o

ZP.

-.P r o o f . N i s a D i c k s o n n f . , f o r N c a n n o t b e o n e o f t h e 7 e x c e p t i o n a l c a s e s ( 8 . 3 4 and t h e s u b s e q u e n t d i s c u s s i o n ) , s i n c e i n each one o f t h e s e c a s e s A 5 = A , b u t Ei5 R.

n

q E p 3 nEIN : IIJI = q b v 8.13. Since ( N , + ) i s a f i n i t e p-qroup, / N I i s some power o f p , c o r ~ s e q u e n t l y q = p . Now ( N " , . ) h a s q e n e r a t o r s a , b w i t h b - 1a b = a q = = ap ( 8 . 3 3 ) . Thus a b = b a P = h a , N i s commutative !OW

3

(P3). and h e n c e h a s Now t h e r e s u l t f o l l o w s f r o m 9 . 2 5 .

9-27 REMARKS ----___ ( a ) Cf. a l s o 8 . 3 5 . ( b ) The f i n i t e n e s s c o n d i t i o n i n 9 . 2 6 i s i n d i s p e n s a b l e , f o r

there exist infinite p-fields.

A n a p p l i c a t i o n o f 2 . 5 2 ( b ) n i v e s w i t h 9 . 1 7 and 9 . 2 6 t h e f o l l o w i n n 9 . 2 8 C O R O L L A R Y ( P l a s s e r ( 1 ) ) . L e t :i be a f i n i t e p - n r . w i t h I F P and w i t h s o n - z e r o d i s t r i b u t i v e elein.ents i n e v e r y nonz e r o homomorphic i m a o e . Then 14 i s i s o m o r p h i c t o a ( f i n i t e ) hence a f i n i t e p - r i n o . d i r e c t sum o f c o p i e s o f zP ' 9.29 R E M A R K R a t l i f f ( 1 ) s t u d i e d p - n r . ' s

N ( e s p e c i a l l y Tor

p=3

a n d P = 5 ) , w h i c h c a n be d e r i , J e d f r o m a p - r i n c l R i n a way that (N,+) = ( R , + ) and the'product i n N i s defined v i a a f i x e d p o l y n o m i a l f u n c t i o n o v e r R . The n r . ' s c o n s i d e r e d

i n this dissertation fulfill

(P,)

and

(P3).

69 MORE CLASSES OF NEAR-RINGS

300

3.)

B O O L E A N NEAR-RINGS

I t d o e s n o t seem t o be q u i t e c l e a r how t o d e f i n e a B o o l e a n n e a r r i n g . S o we t a k e w h a t s e e m s t o be t h e m o s t q e n e r a l p o s s i b l e d e f i n i t i on.

9.30 DEFINITION A nr. N i s B-o o l e a n : <=> --

XCH : x2 = x.

Hence a B o o l e a n n r . i s a ( P o ) - n e a r - r i n q w i t h all x.

n(x) = 2

for

9 . 3 1 R E M~ARKS

( a ) Every c o n s t a n t nr. i s a Boolean nr. w i t h (Pj), h u t

(Po)

and

not a 2-nr. i n g e n e r a l .

( b ) A Boolean nr. w i t h

and non-zero d f s t r i b u t i v e e l e m e n t s i n e v e r y n o n - z e r o homomorphic image i s a s u b d i r e c t p r o d u c t o f c o p i e s o f iZ2. This r e s u l t o f Ligh ( 1 4 ) f o l l o w s from 9 . 2 5 .

( c ) (Ligh ( 5 ) , (14),

(P3)

(a 6-nr.)

( 8 ) , ( l o ) , H e a t h e r l y ( 7 ) ) . The same

a s s e r t i o n h o l d s f o r d q . B o o l e a n n r . l s . Clf c o u r s e , t h i s f o l l o w s from 9 . 1 8 , b u t t h e r e i s a l s o a d i r e c t e l e m e n t a r y proof i n Liqh ( 1 0 ) . ( d ) See p . 4 1 8 / 4 1 0 f o r a l i s t o f a l l B o o l e a n n r . I s d e f i n a b l e on t h e two n o n - a b e l i a n a r o u p s o f o r d e r 8 .

( e ) F e r r e r o - C o t t i ( 2 ) , ( 3 ) c o n s i d e r e d nr.'s w i t h t h e i d e n t i t i e s abc = acbc = a b a c . These a r e t h o s e ones w h i c h c o n t a i n a n i d e a l I w i t h I' = I 0 1 a n d N / I is a Boolean r i n q . ( f ) A Boolean n r . w i t h l e f t i d e n t i t y i s a Boolean r i n g with

i d e n t i t y (Ligh ( 5 ) ) . ( 9 ) More r e s u l t s a r e c o n t a i n e d

Ramakotaiah-Rao ( 2 ) .

i n Heatherly-Stone

( 1 ) and

301

9b Near-rings without

9 . 3 2 C O R O L L A R Y ( H e a t h e r l y ( 7 ) ) . A Boolean n r . i s a f i n i t e d i r e c t sum o f i d e a l s which a r e

.no

n r . ' s with the t r i v i a l ~ u l t i n l i c a t i o n xy =

with DCCI inteoral simple

[

x

Y 9 0

O

y = o

P r o o f . Apply 9 . 1 0 , 2 . 5 2 ( b ) a n d t h e f a c t t h a t e v e r y n o n zero element i s (as a n idempotent s e e t h e proof o f 9.13(b)) a right identity.

-

9 . 3 3 EXAMPLES

( a ) ( C l ay - La wve r ( I ) ) . Le t

be a Boolean r i n o w i t h i d e n t i t y 1. Le t a ' : = a t 1 and a v b : = ( a ' A b ' ) ' . I f XEB, define for a,bEB a b : = a A ( b v x ) . Then (B,+,ax) i s a Roolean n r . w i t h (Pj) which i s a r i n q i f f x = 0. N r . ' s d e r i v e d from Boolean r i n g s a r e c a l l e d " s p e c i a l B o o l e a n n e a r - r i n n s " i n t h i s p a p e r . Their- i d e a l s t r u c ture i s considered. (B,+,A)

Q~

I _

_ _ _ I _ _ _ I -

( b ) Subrahmanyam ( 1 ) c a l l e d a n a b e l i a n Boolean n r . w i t h

(P3)

"Bool e a n s e m i r i n n " . Every P 1 - r i n g ( 9 . 2 1 ( i ) ) ( B , t , = ) g i v e s r i s e t o a Boolean s e m i r i n a ( B , t , * ) , where a s b : = ab 0 Every c o n s t a n t a b e l i a n n r . i s a Boolean s e m i r i n a . A Boolean s e m i r i n g can be r e p r e s e n t e d as a d i s j o i n t u n i o n of " n e a r l y d i s t r i b u t i v e " l a t t i c e s . S ee t h i s p a p e r f o r more d e t a i l s .

.

b ) NEAR-RIPiGS WITHOUT

1. \ N E A R - R I N G S l.IITtiC)UT N I L P O T E N T E L E M E N T S

N r . ' s w i t h o u t n o n - z e r o n i l p o t e n t e l e m e n t s came u p a t s e v e r a l d i f f e r e n t p l a c e s i n o u r d i s c u s s i o n o f n e a r - r i n o s . We c o l l e c t some of t h e r e s u l t s c o n c e r n i n g t h e s e n e a r - r i n o s .

09 MORE CLASSES OF NEAR-RINGS

302

9.34 -

REMARKS Let N be a n r . w i t h o u t n o n - z e r o n i l p o t e n t e l e m e n t s . Then ( a ) N h a s no n i l ( p o t e n t ) s u b s e t s ( 2 . 9 6 ) .

NEW,

has D C C N t h e n e v e r y non-zero N-subgrou? c o n t a i n s a non-zero idempotent ( 3 . 5 1 ) . Moreover, i n this case = Jo(N) = V(N) = = I01 (5.40).

(b) If

P(N)

( c ) I n any c a s e ,

T(N) = p(X) =

{01.

9. 3 5 E X A M P L ES ( a ) Every c o n s t a n t n r . has n o non-zero i i i l c o t e n t e l e m e n t s . ( b ) Every i n t e g r a l n r .

(hence e v e r y n f . ) has t h i s p r o p e r t y ,

too. The c o n n e c t i o n t o t h e p r e v i o u s c h a p t e r i s g i v e n by 9 . 3-6 - T H-E O- R E M ( B e l l ( l ) , M a r i n ( l ) , Q a r n a k o t a i a h - ' J a o ( 2 ) ) ~ Let N be zero-symmetric.

Equivalent are:

( a ) N has no non-zero n i l p o t e n t e l e m e n t . ( b ) N i s a subdirect product of interlral n r . ' s .

~Proof. ( a )

( b ) i s nothinn e l s e trran i n t h e proof o f 9 . 1 3 ( a ) : N has a family o f i d e a l s I x ( ~ E N " ) w i t h z e r o i n t e r s e c t i o n and each N / I x i s integral. =>

( a ) : I f x " = 0 , i n e a c h component n i ( N ) o f t h e s u b d i r e c t r e p r e s e n t a t i o n o f N we a e t i ~ ~ ( x ) = Q , whence x = 0 . ( b ) =>

Hence w e w i l l d e v o t e t h e n e x t nur?ber t o i n t e o r a l n e a r - r i n a s . B u t b e f o r e , some more r e s u l t s m i n h t be a p p r o p r i a t e . 9 . 3 7 PROPOSITION(Bel1 ( l ) , H e a t h e r l y ( 7 ) , M a r i n (l), R a m a k o t a i a h Rao ( 2 ) ) . A n r . N E % 0 w i t h o u t n o n - z e r o n i l p o t e n t e l e m e n t s i s an IFP-nr.

9b Near-rings without

xy = 0

If

Proof.

(x,y~N)

so

( y x 1 2 = 0,

= n ( y x ) = nO = 0,

9.38

then

nr.

nEN:

xny

whence

(ny)x =

s o N has t h e IFP.

COROLLARY ( H e a t h e r l y ( 7 ) ) .

Every s u b d i r e c t l y i r r e d u c i b l e

w i t h o u t non-zero n i l p o t e n t s i s i n t e o r a l .

riEQ

non-zero

y x y x = yOx = 0,

ow

0.

yx

303

Every

idempotent i s a r i o h t i d e n t i t y .

P r o o f . The f i r s t a s s e r t i o n h o l d s by 9 . 1 3 ( a ) .

$. 0

e

If

whence

XEN:

i s idempotent,

(xe-x)e = 0,

xe = x.

To 9 e t m o r e , we h a v e t o i n p o s e s o m e f i n i t e n e s s c o n d i t i o n s o n P I . 9 . 3 9 P R O P OS ITION

( H e a t h e r l y ( 7 ) ) . Let

irreducible nr.

(a)

(b)

N N

be a s u b d i r e c t l y

w i t h DCCN and w i t h o u t non-zero

{O}

n i l p o t e n t elements.

tlE'TIo

Then

i s i n t e q r a l and 2 - p r i m i t i v e on

N.

has a r i g h t i d e n t i t y .

( c ) l i d =f. 0

==,

N

( d ) I f N i s dg,

-Proof.

i s a nf.

.

t h e n ti i s a f i e l d .

( a ) Consider,

T h e r e i s some

XEN*,

for

... .

nE.DJ w i t h

i n t e g r a l by 9.36,

so

3

t h e c h a i n YCx?Nx2?Plx 3. n+l P! i s fixn Nx

Nxn = ( N x ) x n

T h e r e f o r e Pi i s 2 - p r i m i t i v e o n (b) h o l d s by 4.46

implies

..

PI = N x .

N.

o r by 9.38.

( c ) By t h e same a r g u m e n t a s i n t h e p r o o f o f 3 . 1 3 ( b ) ,

N

Now a p p l y 4 . 4 7 ( a )

c o n t a i n s an i d e n t i t y .

and 9 . 1 7 .

( d ) i s obvious, 9 . 4 0 REMARK ( H e a t h e r l y ( 7 ) ) . abelian nr.'s elements,

N

with

There e x i s t even f i n i t e simple (PI)

and w i t h o u t non-zero n i l p o t e n t

which are not n f . ' s .

We c a n r e d u c e t h e t h e o r y o f n e a r - r i n a s w i t h D C C E l i r n d n o n o n z e r o n i l D o t e n t e l e m e n t s t o t h a t ~f

9.39:

59 MORE CLASSES OF NEAR-RINGS

304

9 . 4 1 THEOREM ( H e a t h e r l y ( 7 ) ) . zero n i l p o t e n t elements. 2-semisimple

Let

NEnO

Then

N

h a v e DCCN a n d n o n o n -

has a r i r l h t i d e n t i t y ,

a n d t h e f i n i t e d i r e c t sum o f n r . ' s

f u l f i l l a l l c o n d i t i o n s o f 9.39.

is

which

If e v e r y n o n - z e r o homo-

Pi h a s n o n - z e r o d i s t r i b u t i v e e l e m e n t s t h e n N i s a f i n i t e d i r e c t sum o f n f . l s ; i f N i s d q . t h e n Pi i s a

morphic imaqe of fi'nite

d i r e c t sum o f f i e l d s .

P r o o f . Decompose nr.ls

Pii

N into subdirectly irreducible inteqral ( 9 . 3 6 ) . I n f a c t , N i s a f i n i t e d i r e c t sum

o f t h e s e ones

Now a p p l y 9 . 3 9 ( b ) ,

(2.52(b)).

9.39(a)

and 5.49.

9 . 4 2 C0RrJLLAP.V ( t l e a t h e r l y ( 7 ) ) .

If

i s a f i n i t e nr.

NFno

w i t h o u t non-zero n i l p o t e n t elements. T h i s i s c l e a r by 9 . 3 8 that

Cf.

n(x)

N

Then

has

(PI).

a n d 9 . 4 1 ( I l e a t h e r l y a o e s on t o show

can be chosen t o be c o n s t a n t f o r a l l xzN.

(9.4)

a l s o L i g h (11)).

Moreover,

we h a v e some i n f o r m a t i o n c o n c e r n i n q t h e n e a r - r i n o ;

i n d i s c u s s i o n , which guys belono t o t h e c e n t e r

( I ) ) . Let

9 . 4 3 PROPOSITION ( B e l l potent elements.

o f N:

C(N)

have no Eon-zero n i l -

FIE'Q,

Then

(a) Every d i s t r i b u t i v e idempotent i s c e n t r a l . all

(b) I f Proof.

idempotents ?re i n

C(K).

F i r s t we s h o w t h a t f o r e a c h i d e m p o t e n t e ,

XEN: e x = e x e . rl

e(ex-exe)

Now

and

(-exe).(ex-exe)

(ex-exe)e

ex(ex-exe)

= ( - e x ) O = 0.

= ex(ex-exe)t(-exe)(ex-exe)

= '3,

= 0

so (9.37)

(IFP).

Therefore

= O t 0 = 0,

Hence (ex-exe!*

whence

e x - e x e = 0.

(a) If

ecNd,

= exe-exe

= 0,

xe = exe = ex.

v

XEN: e(xe-exe) hence

(xe-exe!e

= exete(-exe) = 9,

whence

=

305

9 b Near-rings without

( b ) I f N h a s a n i d e n t i t y 1 , c o n s i d e r a g a i n some i d e m p o t e n t e . ( 1 - e ) e = 0 , s o \1 xcN: ( l - e ) x e = 0 . Also, ( x e - e x e ) e = xe-exe and (1-e)xe = x e - e x e , therefore (xe-exe)' = (xe-exe)e(xe-exe) = = (xe-exe)(l-e)xe = 0 , so xe = exe = ex f o r a l l XEN.

9 . 4 4 REMARKS ( a ) See Marin ( 1 ) f o r c h a r a c t e r i z a t i o n s o f t h o s e n e a r r i n g s w i t h o u t non-zero n i l p o t e n t e l e m e n t s which a r e ( f i n i t e l y or n o t ) c o n p l e t e l y r e d u c i b l e i n t o c e r t a i n o t h e r n e a r - r i n a s . See a l s o Szeto-Nong ( 1 ) . ( b ) Recall 9 . 2 1 ( f ) .

( c ) A g a i n , l e t N ~ 9 j ' ~ h a v e no n o n - z e r o n i l p o t e n t e l e m e n t s . Then ( B e l l - L i g h ( I ) ) : a ) I f N i s d g . w i t h f i n i t e l y many s u b n e a r - r i n o s ,

N i s

a f i n i t e commutativc rinq.

B ) I f 14 h a s a t m o s t 2 i d e n p o t e n t s a n d no p r o p e r ( f i n i t e l y many) s u b n e a r - r i n q s , N i s a f i n i t e f i e l d (a near-fi el d , respectively). ( d ) Don't forget t o observe 9 . 5 4 .

2 . ) N E A R - R I N G S !.IITHOUT Z E R O D I V I S Q R S ( I P J T E G R A L N E A R - R I ! i G S )

I

9 . 4 5 EXAMPLES

( a ) Every c o n s t a n t nr. i s i n t e c r r a l .

d e f i n e s a n i n t e g r a l nr.

(r,+,*)

(cf.

1.4(b)).

306

$ 9 MORE CLASSES OF NEAR-RINGS

So o n e c a n s a y n o t h i n c l a b o u t t h e a d d i t i v e a r o u p o f a n i n t e q r a l n e a r - r i n g . To o v e r c o m e t h i s we w i l l g i v e t h e f o l l o w i n q

9 . 4 6 DEFINITION A n i n t e g r a l n r . N i s n o n - t r i v i a l i f i t s m u l t i p l i c a t i o n i s not one of 9 . 4 5 ( a ) o r ( b ) . These n o n - t r i v i a l i n t e g r a l n r . ' s a r e sometimes c a l l e d " n e a r i n t e g r a l d o m a i n s " ( s e e L i g h (131, H e a t h e r l y - O l i v i e r ( l ) , ( 2 ) , Adams ( l ) , ( 2 ) ) . B u t t h e y a r e n o t a l w a y s e m b e d d a b l e i n t o a n e a r - f i e l d , s o we r e s e r v e t h i s d i s t i n q u i s h i n q name t o a more s p e c i a l c l a s s o f n o n - t r i v i a l i n t e g r a l n e a r - r i n a s ( s e e 9.52 and 9 . 6 5 ) . See a l s o O l i v i e r ( 2 ) . 9 . 4 7 PROPOSITION ( C l a y ( a ) , H e a t h e r l y - O l i v i e r I f N i s i n t e g r a l t h e n NEQ o r N~fl,.

( l ) , Plasser (1)).

P r o o f . S u p p o s e t h a t 3 X E N : x0 ;f; 0 . T h e n f o r a l l n E N h a v e (nxO-n)xO = n x 0 - n x 0 = 0 , w h e n c e n x O = n . Hence nO = nxOO = nxO = n , a n d N i s c o n s t a n t .

we

T h u s every n o n - t r i v i a l i n t e g r a l near-ring i s zero-symmetric. Integral near-rinqs also appear in previous chapters. I n order t o p r e s e n t a good a e r i a l v i e w on t h i s t o p i c we c o m p i l e t h e s e facts: 9 . 4 8 R E M A R K S R e c a l l t h a t an i n t e y r a l n e a r - r i n a N h a s t h e followinq properties: ( a ) N has t h e r i g h t c a n c e l l a t i o n law ( l . l l l ( a ) ) , ( b ) I f N i s f i n i t e and n o n - a b e l i a n t h e n e a c h e l e m e n t o f

N has a u n i q u e s q u a r e - r o o t ( 1 . 1 1 2 ) . ( c ) N i s a prime near-rincj ( 2 . 6 6 ) . ( d ) I f N i s n o n - t r i v i a l i n the s e n s e o f 9.46 and h a s the DCCI t h e n N i s s u b d i r e c t l y i r r e d u c i b l e ( 2 . 1 0 7 ) . A p p l y i n g 9 . 3 9 we q e t : I f PI h a s m o r e o v e r t h e D C C N t h e n t h e r e e x i s t s a r i a h t i d e n t i t y , !id A implies t h a t N i s a n f . (Ligh-Malone ( l ) ) ,N i s d q . i m p l i e s t h a t N i s a f i e l d ( c f . a l s o 6 . 1 4 ( b ) ) , N i s 2 - o r i m i t i v e 9~ N

307

9b Near-rings without

a n d s o N i s s i m p l e ( H e a t h e r l y ( 7 ) ) . See a l s o G r a v e s -

Malone ( 1 ) . ( e ) On t h e w h o l e , 9 b ) l ) i s a p p l i c a b l e , f o r N h a s n o nonzero n i l p o t e n t elements. So i f N i s n o n - t r i v i a l , i t has the I F P .

9 . 4 9 R E M A R K t o 9 . 4 8 ( d ) . Llithout c h a i n c o n d i t i o n s one c a n n o t conclude t h a t a n i n t e q r a l n r . N with N d f {O} i s a near(7.78). N i s f i e l d : t a k e a f i e l d F a n d form N : = Fo[x] i n t e g r a l ( 7 . 6 8 ( c ) , l . l l l ( a ) ) , each a x ( a E r I ) i s in lid, b u t N i s n o n f . ( 7 . 6 8 ( b ) ) . I n f a c t , f o r e v e r y kcIN,

k + l ~ ~ + l t . . . [ a ~ ,, .a. .~E +F I ~ i s a n i d e a l a n d 11=12=13= i s a s t r i c t l y descending chain (Heatherly ( 7 ) ) . C f . a l s o Graves-Malone ( 1 ) .

Ik:

= { a k xk +a

...

9 . 5 0 T H E O R E M ( F e r r e r o ( 8 ) ) . L e t N be a f i n i t e i n t e q r a l n e a r - r i n ? N i s n o n - t r i v i a l <=> N i s p l a n a r . P r o o f . -->: C o n s i d e r 6 o f 8 . 9 6 f b ) . S i n c e ncN*: Nn = N ( 9 . 4 8 ( d ) ) . Each g a (aEN") i s a monomorphism s i n c e N i s i n t e g r a l . N i s f i n i t e , so I; I A u t ( N , t ) . N i s n o n - t r i v i a l , so 6 {id]. G i s also fixed-point-free (Heatherly-Olivier ( 1 ) ) : Let g a (aEN*) have a f i x e d - p o i n t no 0. Let x be a r b i t r a r y i n N . S i n c e N n o = N , 3 yxcN: x = y x n o

+

Hence p a ( x ) = x a = y x n o a = y x c l a ( n o ) = y x n o = x , so g a = i d . S i n c e N i s t r i v i a l l y s t r o n q l y uniforni we may a p p l y 8.100 a n d a r e t h r o u g h . <-:

i s immediate, s i n c e

We a p p l y 8 . 9 8 t o g e t

(cf.

!I{/:/

1 3

(8.85).

Adams ( 1 ) , ( 2 ) , O l i v i e r ( 2 ) ) :

9 . 5 1 C O R O L L A R Y ( L i a h ( 1 3 ) ) . The a d d i t i v e o r o u p of a f i n i t e non-trivial integral nr. i s nilpotent, b u t n o t necessarily abel i a n .

$9 MORE CLASSES OF NEAR-RINGS

308

Mathematics i s a c r a z y job: t h e a d d i t i v e groups of t h e s e n r . ' s without n i l p o t e n t elements i s nilpotent. Anyhow, we can use 8.11 t o g e t :

9.52 C O R O L L A R Y Not e v e r y n o n - t r i v i a l i n t e g r a l n r . can be embedded i n t o a n e a r - f i e l d . 9.53 REMARKS ( B e t s c h ) . I f one r e c a l l s ( r , + , o R ) o f 1.4(b), 1.15 a n d 8.97, t h e f o l l o w i n q r e s u l t s a r e s i m p l e c o r o l l a r i e s from group t h e o r y , s i n c e a a r o u p G i s t h e a d d i t i v e oroup o f a n o n - t r i v i a l i n t e g r a l n e a r - r i n r r i f f C, h a s a n o n - t i v i a l g r o u p o f f i x e d - p o i n t - f r e e automorphisms. ( a ) (Adams (l), (2)). For each k E I N there e x i s t both f i n i t e a n d i n f i n i t e n o n - t r i v i a l i n t e q r a l n r . ' s I.4 s u c h t h a t (N,+) i s n i l p o t e n t of derlree k ( s e e ( H u p p e r t ) , p . 499). C f . B l a c k e t t (7). (11)

(Ligh (13), H e a t h e r l y - O l i v . i e r ( 2 ) ) . N e i t h e r t h e f o r c e FI c o m m u t a t i v i t y nor t h e n i l p o t e n c y of ( N , + ) t o be i n t e q r a l

.

( c ) ( H e a t h e r l y - O l i v i e r ( 2 ) , Adams ( l ) , (2)). I f !I i s i n f i n i t e and i n t e g r a l , (N,+) i s n o t necessarily n i l p o t e n t ( n o t even f o r d g n r . ' ~ w i t h b o t h c a n c e l l a t i o n 1 aws)

.

The f o l l o w i n g r e s u l t i s somewhat n o s t a l q i c i n n a t u r e , f o r i t c o n c e r n s n r . ' s o f 9b)l).

9.54 C O R O L L A R Y ( H e a t h e r l y - O l i v i e r (2)). L e t N E ~ , b e a f i n i t e n r . w i t h o u t n o n - z e r o n i l p o t e n t e l e m e n t s , such t h a t no is homomorphic image i s t r i v i a l (9.46). T h e n ( N , + ) ni 1 p o t e n t . P r o o f . 9.36 a n d 9.51.

9.50, 9.51 a n d 9.54 e x c l u d e many a r o u p s f r 3 n b e i n o a d d i t i v e qroups of n o n - t r i v i a l i n t e q r a l n r . ' s :

309

9b Near-rings without

9.55 REMARKS

(a)

( L i g h - M a l o n e ( 1 ) ) . C o m p l e t e g r o u p s , t h e d i h e d r a l crroup o f o r d e r 8 and t h e q u a t e r n i o n g r o u p c a n n o t be t h e a d d i t i v e group o f a n o n - t r i v i a l

(b)

(Betsch,

Heatherly-Olivier (1)).

a l l f i n i t e groups integer

> 1

.

r

i n t e g r a l nr.. The same a p p l i e s t o

o f o r d e r 2 n , wher.e n i s a n o d d

r

For

i s n i l p o t e n t i n t h i s case,

havinq

an element o f o r d e r 2 which i s f i x e d b y a l l automorphisms

o f r . Hence t h e r e i s no n o n - t r i v i a l f i x e d - p o i n t - f r e e automorphism group on r . (c) If

l(I',t)] = p + l

non-trivial o n l y one i s

9.56

( p ~ l P \ I 2 1 ) t h e n e i t h e r t h e r e i s no

i n t e g r a l n r . d e f i n a b l e on GF(2'),

EXAMPLES F o r n o n - t r i v i a l

where

(r,t)

or the

2n = p t l .

i n t e g r a l n r . ' s on

Z5, Z,

see

C l a y (6), f o r o n e s o n Z31,Z121 c o n s u l t F e r r e r o ( a ) , f o r some o n Z9, Z3@Z3 a n d B15 s e e H e a t h e r l y ( 5 ) , f o r a l l

o f them H e a t h e ' r l y - 0 1 i v i e r ( 2 ) . U h i t t i n g t o n (1) g i v e s a computer-aided d e s c r i p t i o n o f a l l n o n - t r i v i a l i n t e o r a l nr.'s

on groups o f " l o w " o r d e r . A d d i t i o n a l i n f o r m a t i o n can

b e f o u n d i n Adams ( l ) , ( 2 ) . Lawver ( 3 ) and O l i v i e r ( 2 ) . Clay ( 7 )

raised the question,

See a l s o p . 346 and p . 3 4 8 ,

i f every n o n - t r i v i a l

h a s as c h a r a c t e r i s t i c z e r o o r a p r i m e ( c f .

8.9).

inteoral nr.

Some o f t h e

e x a m p l e s a b o v e a n s w e r t h i s n e g a t i v e l y . So we q i v e a c o n d i t i o n where t h i s i s t h e case:

9 . 5 7 THEOREM ( H e a t h e r l y - O l i v i e r ( 1 ) ) . L e t N b e a n i n t e q r a l n e a r r i n g w i t h Nd Then t h e c h a r a c t e r i s t i c o f N i s e i t h e r 0.

4

0 o r a prime. Proof.

Take

dEN2.

Let

have a f i n i t e o r d e r , say

XEN

.

p * j , w h e r e P E P ,j E I N T h e n 0 = dO = d ( p j x ) = = (pd)(jx), whence p d = 0 . So d ( p x ) = (pd)x = = Ox = 0 and h e n c e D X = 0.

89

310

M O R E CLASSES OF N E A R - R I N G S

9 . 5 8 R E M A R K ( H e a t h e r l y - O l i v i e r (1)).

T h e same c o n c l u s i o n h o l d s i f t h e n o n - t r i v i a l i n t e g r a l nr. N h a s no n o n - t r i v i a l r i q h t ideals. Confer a l s o 9.17.

9.59 -

REMARKS

( a ) ( H e a t h e r l y - O l i v i e r (2)). I f a n o n - t r i v i a l i n t e q r a l n r . h a s t h e p r o p e r t y t h a t 3 nEN* mEN: m ( - n ) = -(mn). o r i f N i s f i n i t e with I N \ = p l . . . p n q l2 . . . q m 2 p 3 2 k ( p I , . . . , p n , q l , ...,q,, p d i s t i n c t odd p r i m e s , w h e r e p i s o f t h e form Z r + l a n d 2 k -1 i s a prime o r k = 0 ) then N i s a b c l i a n . The same f o l l o w s f o r i n t e q r a l n r . ' s N w i t h a n o n - z e r c i d e m p o t e n t e s u c h t h a t 3 h s N : h t h = e ( B . H . Neumann (1)). Cf. Olivier ( 2 ) . ( b ) ( F e r r e r o ( 8 ) ) . Given a p r i m e power p b , t h e r e e x i s t s

v

a n o n - t r i v i a l i n t e q r a l n r . on

i f t h e r e i s some pP kEIN s u c h t h a t t h e s m a l l e s t o f a l l numbers k x - 1 which a r e d i v i s i b l e by p i s a l s o d i v i s i b l e by p b . F e r r e r o c a l l e d N B - d i s t r i b u t i v e i f \1 a,bEN \ r c B : a ( z b ) = z ( a b ) . A f i n i t e n o n - t r i v i a l intetrral 8d i s t r i b u t i v e nr. h a s an e l e m e n t a r y a b e l i a n a d d i t i v e

-

group. ( c ) See H e a t h e r l y - O l i v i e r ( Z ) , S z e t o ( 2 ) ar,d R a m a k o t a i a h Reddy ( 1 ) f o r a d e s c r i p t i o n o f t h e m u l t i p l i c a t i v e s e m i oroup ( N " , . ) of a n o n - t r i v i a l i n t e g r a l n e a r - r i n g N . Fluch more i s i n H e a t h e r l v - O l i v i e r ( 2 ) , ( 3 ) a n d O l i v i e r ( 2 ) . H a v i n g 9 . 5 2 i n mind we l o o k f o r i n t e o r a l n r . ' s w h i c h c a n b e embedded i n t o a n e a r - f i e l d . 9.60 D E F I N I T I O N A nr. N is c a l l e d n e a r - i n t e a r a l domain ( n i d . ) i f N f u l f i l l s %he l e f t c a n c e l l a t i o n law a n d t h e l e f t Ore condition (1.64). G e n e r a l i z a t i o n t o "H-monoQenic n r . ' s " c a n be f o u n d e . a . i n O l i v i e r ( 2 ) . See 9 . 2 7 5 .

9b Near-rings without

9.61 WARNING N i d . ' s

31 1

a r e c a l l e d "near-domains"

b y Graves-Valone

(1) - ( 3 ) . S i n c e t h i s c o l l i d e s h e a v i l y w i t h 8 . 4 1 , n o t u s e t h i s name. Near-integral

we w i l l

domains a r e r e a l l y i n t e o r a l :

(I)). Let

9.62 TREOREM (Graves-tlalone

{Ol

II

.

be a n i d .

Then

(a) N i s int,eqral.

( b ) NcVo (c)

{Ol

N

If

Prctof.

I N 1 > 2,

and i f

N i s not trivial.

has t h e DCCN t h e n N i s a n e a r - f i e l d .

Suppose t h a t

ncnc = nc

rig = n

C

we q e t

ncO

+0

f o r some

nc = 3

nEN.

Since

by t h e l e f t

c a n c e l l a t i o n l a w , a c o n t r a d i c t i o n . H e n c e N€fl0. I f nm = 0 , b u t n $. 0, we a o a i n u s e l e f t m = 0

cancellation t o oet

So i f N i s t r i v i a l ,

I N 1 2 3,

If then

nml

diction

take

out o f

n,mEN:

&.

n,ml,m2cN*

n

m + O

0

m = O

(9.45).

+

w i t h ml m2. But i s aaain resultino i n a contra-

= n = nm2

m1 = m 2 .

IN\

= 2

Otherwise N i s n o n - t r i v i a l .

u s t h a t TI h a s a r i q h t i d e n t i t y e . one and

c

nm =

To s h o w ( c ) , o b s e r v e t h a t f o r N =

nm = 0 = n o .

xEN*

B y 1.112,N

xe = x = x e '

then

i s a nf.

we n e t

9.48(d)

tells

If e' i s another implies e = e'.

.

9 . 6 3 REMARKS ( a ) O b s e r v e t h a t we d i d n o t u s e t h e O r e c o n d i t i c j n i n 9 . 5 2 . (b)

(Graves-Malone

(1)). 9.62(c)

does n o t h o l d w i t h o u t

a chain condition: Let ( r , + ) be t h e f r e e a r o u p on two g e n e r a t o r s x,y. F o r n € N 0 l e t T n b e t h e map

r

+

r

sendinq a word

L e t N be t h e dqnr. (Tnjnc"Mol. Then

w(x,y)

i n x and y i n t o w(nx,ny).

nenerated a d d i t i v e l y by t h e s e t ~IE?, !I h a s t h e l e f t c a n c e l l a t i G n

law but n e i t h e r the l e f t nor t h e r i o h t Ore property. N i s not a near-field.

09 MORE CLASSES OF NEAR-RINGS

312

9 . 6 4 T H E O R E M ( G r a v e s - N a l o n e ( 1 ) ) . Let N be a n i d . a n d S : = (N",.). Then t h e nr. N s o f l e f t q u o t i e n t s o f N w.r.t. S is a nf.

.

Proof. S f u l f i l l s t h e c o n d i t i o n s of 1.65, so Ns exists. I n N,, each non-zero element i s i n v e r t i b l e , hence N, i s a nf.

Now we g e t t w o c o r o l l a r i e s due t o G r a v e s - M a l o n e ( 1 ) ( s e e 8 . 9 , 8 . 1 0 and 8.11). 9 . 6 5 C O R O L L A R Y E v e r y n i d . c a n be embedded i n t o a n e a r - f i e l d .

_ I -

9 . 6 6 C O R O L L A R Y L e t N be a n i d . . Then (a) N i s abelian. (b)

tf n,mtz.N: n ( - m )

= -nm.

( c ) N has a s c h a r a c t e r i s t i c e i t h e r z e r o o r a prime.

This i s a s a t i s f a c t o r y r e s u l t , w h i c h may c o n c l u d e o u r c o n s i d e r a t i o n s o f i n t e q r a l n e a r - r i n q s . lrle o n l y makr? some remarks: 9 . 6 7 REMARKS ( a ) ( G r a v e s - M a l o n e ( 1 ) ) . I f I i s an i d e a l o f t h e n i d . p1 then the n e a r - f i e l d s o f q u o t i e n t s o f I a n d N c o i n c i d e . ( b ) B e r m a n - S i l v e r m a n ( 1 ) d e f i n e d a n r . N t o be a D-rincr i f N i s i n t e g r a l and neN 3 ne,nr&.N*: n e n s C ( N ) and nn,cC(N). (C(N) i s a q a i n t h e c e n t e r o f PI). G r a v e s - M a l o n e showed i n ( 1 ) t h a t e a c h D - r i n q i s a n i d . a n d i n ( 3 ) t h a t t h e c o n v e r s e d o e s n o t h o l d by l o o k i n n a t t h e n r . N o f f o r m a l power s e r i e s o v e r IR w i t h a2" = 0 f o r a l l nclNo. N i s a n i d . , b u t no D - r i n c .

( c ) See Graves-Malone ( 2 ) f o r a d i s c u s s i o n o f n i d . ' s which h a v e a n Eucl i d e a n a 1 o o r i t h i n ( " E u c l i d e a n n e a r - d o n a i n s " ) . Each s u c h n i d . N h a s t h e A C C N , a n i d e n t i t y , o n l y monog e n i c Il-subnrouos a n d a u n i a u c f a c t o r i s z t i o n i n t o lrrlits and primes f o r e v e r y non-zero e l e m e n t 3 f Y , Confer 7 . 7 2

313

9c Affine near-rings

C)

AFFINE

NEAR-RINGS

Now we s t u d y a c l a s s o f n e a r - r i n q s w h i c h a r e i n a c e r t a i n s e n s e t h e " m o s t e l e m e n t a r y n o n - z e r o - s y m m e t r i c n e a r - r i n y s " . The dominant p r o p e r t y i s t h a t t h e c o n s t a n t s form an i d e a l . Let F be a f i e l d , V a v e c t o r s p a c e o v e r F a n d M a f f ( V ) t h e n e a r - r i n g o f a f f i n e t r a n s f o r m z t i o n s on V a s i n 1 . 4 ( c ) . 9 . 6 8 PROPOSITION ( B l a c k e t t (2)). following properties: (a) (b) (c) (d)

Maff(V) = : N

has t h e

N i s abelian.

No

= Nd.

Nc 9 N . % No N/IJ,

-

is a ring.

The p r o o f i s s t r a i g h t f o r w a r d a n d o m i t t e d . O b s e r v e t h a t a l l s u b n e a r - r i n g s o f Maff(V) a l s o f u l f i l l these p r o p e r t i e s . Now we c o n s i d e r " a f f i n e t r a n s f o r m a t i o n s " o v e r r l r o c p s : 9 . 6 9 NOTATION Let

Ma(r):

r

be a n a b e l i a n g r o u p .

= Hom(r,r)tMc(r).

Again, the proof i s obvious. Ma(r) a l s o enjoys t h e p r o p e r t i e s ( a ) m o t i v a t i o n f o r an a x i o m a t i c t r e a t m e n t :

-

( d ) of 9.68. This q i v e s

9 . 7 1 DEFINITION ( G o n s h o r ( 1 ) ) . A n r . N i s c a l l e d an a b s t r a c t a f f i n e near-rinn (=: a . a . n . r . ) provided t h a t

(a) N i s abelian. ( b ) No = Nd.

314

8 9 MORE CLASSES OF NEAR-RINGS

Evidently,

Ma(r), M a f f ( V ) ,

polynomials

( l i n e a r p o l y n o m i a l f u n c t i o n s ) a r e examples o f

a.a.n.r.'s.

See a l s o 9 . 8 1 .

e v e r y r i n g and e v e r y n r . of l i n e a r

There i s no need t o p o s t u l a t e ( c ) and ( d ) o f 9.68 s i n c e

N be an a . a . n . r .

9 . 7 2 PROPOSITION L e t (a)

(N,+)

=

.

Then

(~~,+)h~,+).

( b ) N c A N. ( c ) No

%

i s a ring.

N/Nc

The p r o o f o f ( a ) i s done b y rememberinq 1.13, t h e r e s t i s established by s t r a i n h t f o r w a r d computations ( f o r

(c)

one can use 2.8). The m a i n t y p e s o f s u b s t r u c t u r e s o f an a . a . n . r .

are easily

characterized:

9 . 7 3 PROPOSITION ( G o n s h o r ( 1 ) ) . L e t N b e a n a . a . n . r . .

(a) A l l N-subgroups S o f N a r e o f t h e form with

Then

S = So:Nc

S sN N o . 0

(b) A l l r i g h t i d e a l s R o f N are

R = RofR

C

and P.,N,sR,. Ro 4, N o , ( R , , + ) a (Pi,,+) ( c ) A l l i d e a l s I o f N a r e g i v e n by I = Io:I

I,

A N o , No!,

E

where

C

with

Ic, Io N c c I,.

( d ) A l l two-sided i n v a r i a n t subgroups T o f N a r e T = TofNc, w h e r e T o A No. Again,

t h e p r o o f c o n s i s t s o n l y of

2.18).

The f o r m u l a t i o n o f ( a )

every

N;,s,

with

< N o No

9 . 7 4 COROLLARY E v e r y t w o - s i d e d

-

standard arquments (observe

(d)

i s meant i n t h i s way t h a t

i s a n N - s u b a r o u p o f N,

and so on.

i n v a r i a n t subqroup o f an a.a.n.r. i s an i d e a l , and e v e r y No-subaroup i s a l e f t i d e a l ( a n d conversely, o f course).

315

9c Affine near-rings

I n the care of 9.75

Maff(V)

one c a n i m p r o v e 9.73

REMARK N o t e v e r y l e f t i d e a l o f a n a . a . n . r . decomposed s i m i l a r t o 9.73:

L: = ((O,O):(O,l)).

and

= (xty-1,xty-1)

for 9.76

(see 9.76).

Lc(x,y)

can be d i r e c t l y 2 N = Haff(R ) defined by L(x,y): =

take f o r instance &EN,

Lo

i s i n L, b u t

= (-1,-1)

and

a,

and

Lo(x,y)

=

a r e n o t i n L,

(x+y,xty).

PROPOSITION ( W o l f s o n ( 1 ) ) . F o r e a c h o r d i n a l

TI: = ( h E H o m F ( V , V ) l d i m (a) A l l i d e a l s o f

Im h

Maff(V)

are given by

In particular,

(c)

dimFV <

unique i d e a l o f

m

HomF(V,V)

9 . 7 7 THEOREM

Then

(A>-1);

TA+Mc(V)

.

.G

so e v e r y

implies that

Mc(V)

i s the

Maff(V).

TA

i s some

and t h e f a c t t h a t e v e r y

(see e.g.

L e t N be an a.a.n.r.,

(Baer)).

l ( N o ) t h e Jacobson-radical

7(,iolc) t h e

o f t h e r i n c r No a n d N,

let

{a).

i s a rino.

E s s e n t i a l l y , t h i s f o l l o w s from 9.73(c) ideal i n

=

= HomF(V,V)/TX,

x c Maff(V)

homomorphic image

X 2 0

%

+!I

Maff(V)/T

v2-1,

T-l:

Let

M c ( V ) = T o t t ~ l c ( V)sTltIlcfV)=.

hence t h e i d e a l s a r e

(b) For


r a d i c a l o f t h e No-module

(= t h e i n t e r s e c t i o n o f a l l maximal No-suboroups o f Nc).

Then

yo(N)=.. . =

P r o o f . BY 9 . 7 4 ,

3 (,,,Nc). ... = 7 2 ( ~ )I.f

r2(II)= ?(No)

?,(:I)

=

t a k e some N o - q r o u p

r

+

o f t y p e 2.

i s a n Pi-qroup o f t y p e 2 b y 3.14

so

nE a 2 ( N o )

by 5 . 2 5 ,

= ?(No).

9 . 7 8 REMARK

Since (b).

Therefore

The r e s t f o l l o w s from

v a t i o n s i n 5.67

n E r d 2 ( 1 i ) n N,, No 2 N/Nc, r Hence nE(o:T),

/d2(N)nNo

= Y(No)

5.32 and t h e obser-

(t).

I t i s n o t known i f one c a n o b t a i n s i m i l a r r e s u l t s

f o r t h e p r i n e and n i l r a d i c a l s o f an a.a.n.r.. C o n c e r n i n g p r i m e i d e a l s o n e c a n say t h a t i f P o 9 t h e n Po i s p r i m e i n N o i f f

PotNc

i s nriine i n N.

II,

316

$ 9 MORE CLASSESOF NEAR-RINGS

To g e t a t h e o r e m o n c o n s t r u c t i o n s o f a . a . n . r . ' s

i t w i l l be

( c f . Dasic ( l ) , Natarajan ( 4 ) )

c o n v e n i e n t t o have

N and

9.79 DEFINITION F o r any n r . D(n;n' , n " ) :

n,n',n"~N

let

= n(n'tn")-nn'-nn''

denote the " d i s t r i b u t o r o f n

w.r.t.

n'

and

n"

'I.

N be a n r . Then t h e f o l l o w i n q s t a t e m e n t s a r e

9.80 T H E O R E M L e t equivalent:

.

( a ) N i s an a . a . n . r . ( b ) PI i s a b e l i a n a n d (c)

(PI,+)

= (Nd,t)i(Nc,t).

No ( r i n g - ) nodule (where

i s an

(Nc,+)

i s

nOnc

d e f i n e d as i n N ) .

( d ) There e x i s t s a n a b e l i a n group r w i t h N s M a ( r ) . n,n',n"~N: D(n;n',n") = -nc ( e ) N i s a b e l i a n and

v

( a ) => ( b ) ->

Proof.

r

No-module

( c ) i s obvious.

N to a faithful No ( t h i s can always be done). C o n s i d e r

Ma(r),

+

I t i s e a s y t o see t h a t

v

implies n = rn

C

faithful

r+ r

h ( n ) = h ( n 0t n c ) = : f n :

where

i s a nr.-homomorphism.

C

-nO

( d ) (Gonshor ( 1 ) ) : E x t e n d

( c ) ->

h: N

-

j nEN: f n E ! l a ( r )

h i s i n j e c t i v e since

ycr: fn(y) = fm(y); and t h i s i n t u r n t h a t

,

y+noy+nc and t h a t h f n = f,

taking y = o yields no = ?no, s i n c e r i s

.

( d ) ==> ( e ) Suppose t h a t

N

5 Ma(r).

Then

N i s abelian

a n d D ( n ; n ' ,n") = n ( n ' + n " ) - n n ' - n n " = n o ( n l t n " ) + n c = n n ' + n o n " - non ' non " - n = -nc. -non'-nc-n0n''-n C C

0

-

( e ) =-> ( a ) :

I t s u f f i c e s t o show t h a t

T a k e n,EN,. D(n,;n',n"j

= -noO = 0,

Then f o r a l l n'.n"EN s o n,EN d '

( c ) i n 9 . 8 0 has some s o r t o f a c o n v e r s e :

No

C Nd.

we n e t

9c

317

Affine near-rings

9 . 8 1 PROPOSITION L e t R be a r i n a a n d R M a n R - m o d u l e . T h e n t h e r e i s e x a c t l y o n e way t o e x t e n d t h e m u l t i p l i c a t i o n RxM + f.1 t o a m u l t i p l i c a t i o n " 0 " i n (N,t): = = (R,+) @ (It,+) such t h a t ( N , + , o ) i s a nr. w i t h lid = N o = Reto) a n d N c = t O l $ M , n a m e l y 0

:

(r,m)o(r',m') Moreover, ( N , + , o ) a r i s e i v t h i s way.

(rr',rm'+rn) ( c f . Clay ( 1 ) and 9 . 7 8 ) . i s t h e n an a . a . n . r .

and a l l a . a . n . r . ' s

i s an Proof. I t i s a r o u t i n e check t o s e e t h a t (N,+,o) a.a.n.r. with the indicated properties. Suppose t h a t ( N , t , * ) i s a nr. w i t h (N,+) = = (R,+)@(M,+), Nd = N o = BQiO) a n d N c = I0)eM. Then (r,m),(r',m')EN: (r,m)*(t-',rn') = = ((r,O)+(O,m))*((r',O)+(O,rn')) = (r,O)*(r',O)+ +(r,O)*(O,m')+(O,m)*(r',m') = (rr',O)+(O,rrn')+(O,m) = = (r,m)o(r',m'). Since Nd = No, 11 i s a n a . a . n . r . The r e s t f o l l o w s f r o m 9 . 8 O ( c ) .

9.82 C O R O L L A R Y ( a ) The c l a s s o f a l l a . a . n . r . ' s i s a v a r i e t y . ( b ) S u b n e a r - r i n ? s , homomorphic i m a q e s and d i r e c t p r o d u c t s o f a.a.n.r.'s a r e again a . a . n . r . ' s . ( c ) Every a . a . n . r . i s a s u b d i r e c t p r o d u c t o f s u b d i r e c t l y irreducible a.a.n.r.'s. P r o o f . 9 . 0 0 ( e ) shows t h a t t h e c l a s s o f a l l a . a . n . r . ' s i s e q u a t i o n a l l y d e f i n a b l e , hence a v a r i e t y . ( b ) and ( c ) a r e well-known c o n s e q u e n c e s .

The l a s t a s s e r t i o n o f the preceding r e s u l t c r i e s f o r a description of subdirectly irreducible a.a.n.r.'s: N i s subdirectl:) 9.83 T H E O R E M Let H b e an a . a . n . r . w i t h FI =L N o . i r r e d u c i b l e <==> N, has a s m a l l e s t under a l l No-subaroups CO).

+

318

0 9 MORE CLASSES OF NEAR-RINGS

P r o o f . By 1 . 6 0 , N i s s u b d i r e c t l y i r r e d u c i b l e i f f FI h a s a s m a l l e s t n o n - z e r o i d e a l 1 . By 9 . 7 3 ( c ) I h a s t h e f o r m I = I o + I c w i t h I, 4 N o , NoIc C N C a n d I o 11c c I,. Since Nc 9 N , I c Nc, w h e n c e I. = {Ol. S o I i s and of c o u r s e the s m a l l e s t j u s t an No-subgroup o f N, one. h a v e a s m a l l e s t n o n - z e r o No-subC o n v e r s e l y , l e t N, g r o u p M , a n d l e t J be a n i d e a l p ( 0 1 o f P I . A q a i n , J = J o + J c , w h e n c e M c J c C_ J . By 9 . 7 3 , rl i s an i d e a l o f N and hence t h e s m a l l e s t u n d e r a l l non-zero ones,

9.84 R E M A R K S ( a ) N e i t h e r 'IN, A N " n o r ''No i s a r i n g " a l o n e i m p l y t h a t a n r . N i s an a b s t r a c t a f f i n e o n e ( t a k e e . g . any n o n - a b e l i a n c o n s t a n t n e a r - r i n g a s counterexample). ( b ) A l s o , n o t e v e r y a . a . n . r . N c a n b e embedded i n t o some M a f f ( Y ) ( e a c h element o f any M a f f ( V ) h a s a s c h a r a c t e r i s t i c 0 o r a prime; t h i s does not n e c e s s a r i l y hold i n the M a ( r ) ' s . ) . We w i l l d i s c u s s t h e s e p r o b l e m s now.

9 . 8 5 THEOREM Let N be a nr. w i t h

and

NoNc = N c

a base.

I!,

Then:

N i s an a . a . n . r .

<->

N,

sl N .

P r o o f . => h o l d s b e c a u s e o f 9 . 7 2 ( b ) . T o p r o v e "<=" , l e t N, b e an i d e a l .

( a ) F i r s t we show t h a t N i s a b e l i a n . By N, 4 N 1 . 1 3 , t h e e l e m e n t s o f N, a n d N c commute. By

N,N,

So

v

= N,,

nc,n;cNc:

n,€NC

3

n;eNc

n c + n "C = n o n 'c+ n ;

3

nocN : n ,

= (no+n;)n'

C

=

and

n,nl.

=

( n ; + n o ) n ; = n c" + "0"; = ng + n C , pr3ving that is ahelian. S i n c e N c i s a b a s e , we g e t N 4 M ( N c ) . Vence PI i s abel i a n .

=

(Fee,+)

b

9c Affine

319

near-rings

( b ) Now we s h o w t h a t N o S N d (Nd C No is always t r u e ) . T a k e noEN 0' nAENo b n c c N c : x : = n o ( n '0+ n c ) - n 0 n '0E N , , s i n c e N, Sl N . no". Therefore H e n c e x = x0 = n ( n b O + n c O ) - n n ' O 0 0 0 n 0 ( n b t n C ) = n 0 n 0' + n o nc ' F u r t h e r m o r e , l e t nc,n; be a r b i t r a r y a N c . T h e n 3 nbcN 0 3 n;cNc: n; = Thus no(nc+n;) =

"ni.

= (n n t n n')n; = o c 0 0 c o c = no(nc+n;)n; = n n +n n ' n " = n n t n o n h . o c o c o o c C o n s e q u e n t l y , f o r a l l n;,n;feN 0 a n d n c e N C w e o e t noiiAnc+noii;nc = ( n , ( n b t n g ) ) n C = n o ( n A n c + n "on c )

= n (n +n'n") 0

= (non;+non;)nc.

S i n c e N, forms a b a s e , n o ( n ; + n g ) = n 0 n 0' + n 0 n0" P1 u g g i n q t h e s e r e s u l t s t o g e t h e r y i e l d s

'd

n',n"EN

V

noEbto: n o ( n ' t n " )

'

= n ((n;+nA)+(n;+n;)) 0

=

= n 0 ( ( n 0' t n ' 0' ) + ( n 'c+ n "c) ) = n 0 ( n 0' + n ; ) + n o ( n ~ + n ~ ) = n o n A t +n n " + n n ' + n n " = n ( n ' + n ' ) + r ; ( n ; + n i ) = n o n ' + n n " . 0 0

Hence

o c no

o c o o c 0 is d i s t r i b u t i v e and

0

No =

bid.

more i s p o s t u l a t e d , o n e Nc I f i n s t e a d o f N,Nc i n f o r m a t i o n o u t o f 9.85 c o n c e r n i n g 9 . 8 4 ( b ) :

n e t s m u c h more

be a b a s e a n d i d e a l We m a y a s s u m e ncE14Z: N , n c = N,. of a n r . N, s u c h t h a t t h a t NM(N,) (1.96). Then t h e r e e x i s t s a f i e l d D makinq i n t o a v e c t o r s p a c e o v e r 0, N i s 2 - p r i m i t i v e a n d d e n s e N, i n M a f f ( H C ) a n d No i s a p r i m i t i v e r i n o , d e n s e i n HomD(NcyNc).

9 . 8 6 THEOREM ( c f . H e a t h e r l y ( 2 ) ) . Let N,

b

is a P r o o f . 9.85 t e l l s u s t h a t N i s a n a . a . n . r , so No i s a p r i m itive r i n g . The a s s u m p t i o n s i m p l y t h a t No r i n g , h e n c e a d e n s e s u b r i n g o f Horn,, N c , ? J C 1 , w h e r e D i s t h e c e n t r a l i z e r H o m ~ ~ o ( ~ i c , ~ ~An c ) . a p p l ic a t i o n o f 3 . 2 7 ( a ) f i n i s h e s t h e proof. Since

320

$ 9 MORE CLASSES OF NEAR-RINGS

W e c l o s e t h i s s e c t i o n w i t h t h e o r e m 9.88 f o r w h i c h w e n e e d 9.87 DEFINITION A s e t N together with two binary operations +,. i s c a l l e d a g e n e r a l i z e d r i n a ( B e a u m o n t (1)) i f ( a ) (N,+) is an abelian group. (b) (N,*) is a semioroup. ( c ) 3 r,seIN\{lI n l ,..., nr,ni r s = 1 1 n.n!. i = l j=1 1 J

,...,n;&N:

(

r

S

i=l

J=1

1 ni) (.I

nj) =

9.88 T H E O R E M

( B e a u m o n t ( l ) , F e r r e r o (3)). L e t N b e a nr. w i t h Then: bounded order o f the elements of ( N c , + ) .

N i s a n a.a.n.r.

<=>

N i s a generalized ring.

P r o o f . =>: L e t N be a n a.a.n.r. . T h e n , b y 9 . 8 0 ( e ) , n,n',n"&N: n(n'tn") = nn'tnn'l-no. Hence k k W kEIN b nl,...,nkcN: n( 1 n i l = 1 n n i - ( k - 1 ) no. i=l i=l L e t s ' b e t h e 1.c.m. o f t h e o r d e r s o f t h e e l e m e n t s o f (N,,+), a n d s e t s ' t l -. : s . T h e n s 2 2 and

V n,nl

,...,n S E N :

5

n(

j

5

nil =

1

nni. i=l i=l F r o m t h i s o n e a e t s ( c ) in 9.87 f o r a r b i t r a r y <=:

v

rEIN\{l).

--

F r o m 9.87(c) w e c a n c o n c l u d e t h a t n,n' , n " E N :

n(n'+n") = (ntO+. . .+O) (n'+n"+O+.

..+O)

r-summands s-summands = n n 1 + n n " t ( s - 2 ) (nO)t(r-I) (I) (n'+n"+O+ ...to)) = = nn'+nn"+(s-2) (no). B u t n O = ( n + O + ...+ 0 ) ( O t O + ...t o ) = nO+(s-1) (no). T h u s (s-l)(nO) = 0 a n d (s-2)(n0) = - n Q . C o n s e q u e n t l y w e g e t f o r all n,n',n"EN: D(n;n' ,no') = nn'+nn"-no-nn'-nn" = - n o . By 9.89, N i s a n a b s t r a c t a f f i n e n e a r - r i n o .

=

321

9d Near-rings on given groups

9.89 REMARKS ( a ) (Maxson ( 1 ) ) .

Maff(V)

h a s a u n i q u e maximal i d e a l .

( b ) ( H e a t h e r l y ( 3 ) ) . HomF(V,V) r i n g o f Maff(V).

i s a maximal s u b n e a r -

( c ) Malone ( 5 ) d e s c r i b e s how a u t o m o r p h i s m s o f N o a n d ( w h e r e N i s a n a . a . n . r . ) c a n b e " m a t e d " t o q i v e an a u t o m o r p h i s m o f PI ( c f . 1 . 1 1 4 ) .

Nc

( d ) See B l a c k e t t ( 3 ) , ( 4 ) f o r matrix r e p r e s e n t a t i o n s of a f f i n e t r a n s f o r m a t i o n s over a f i n i t e - d i m e n s i o n a l v e c t o r space V. ( e ) Observe the c o n n e c t i o n s t o n e a r - f i e l d s a n d doubly t r a n s i t i v e groups ( s e e e.q. 8.40). ( f ) By 9 . 8 2 ( a ) , f o r e v e r y s e t A t h e r e e x i s t s t h e f r e e a . a . n . r A o v e r X . F o r l a t e r u s e ( s e e 9 9 i ) we d e s c r i b e , h o w A l o o k s l i k e : A c o n s i s t s o f a l l f i n i t e sums o f e l e m e n t s + a i , w h e r e e a c h o l i i s a n e l e m e n t o f t h e f r e e monoid o v e r A u COI ( s e e 9 . 2 4 5 ) . ( 9 ) I f R i s a r i n g a n d M E A , t h e v a r i e t y o f a l l R-modules a then M [ X I = ( r x + m l r E R , m s M 1 i s an a . a . n . r . . In f a c t , e a c h a . a . n . r . N i s i s o m o r p h i c t o s u c h a n MIc[ X I ( t h e u n d e r l y i n g r i n g R can be chosen a s R = N o ) .

See 7 . 1 4 9 ( e ) .

( h ) S e e C l a y ( 1 8 ) f o r an e x c e l l e n t s u r v e y on a . a . n . r . ' s .

d ) NEAR-RINGS O N GIVEN G R O U P S

1 . ) MULTIPLICATIONS O N A G R O U P

-

9 . 9 0 DEFINITION S i m i l a r t o 8 . 2 4 8 . 2 8 , we c a n d e f i n e couplina maps on a n r . N a s maps 0 : N + End ( N , t , - ) with 4 n O Q m = @gn(m).n f o r a l l n,mEN.

89

322

As i n 8 . 2 6 / 8 . 2 7

MORE CLASSES OF N E A R - R I N G S

o n e c r e t s a new n r .

(N,t,06)

c a l l e d t h e 6 - d e r i v a t i o n o f N, w h e r e

n

o6

N*,

=:

aaain

m : = b,(n).m.

I f f o r a nr. P I t h e r e e x i s t s a r i n g R a n d a c o u p l i n a mac we c a l l Pi a D i c k s o n n e a r - r i n o . R' = N ,

@

on R such t h a t

More on D i c k s o n n e a r - r i n q s can be f o u n d i n 9.153.

(r,+)

9 . 9 1 PROPOSITION L e t multiplication" Every n r .

(r,+,*) y

9.92

-

4:

by

r

$y:

6

*

A +

r

6.y

r

b e a group and d e f i n e t h e " c o n s t a n t

as i n 1,4(b).

(T,t,*) +

We o n l y n e e d

Then

(r,+,*)c?&.

c a n be c o u p l e d t o t h i s

3n

("+)

End

(r,+,*),

given throuqh

.

THEOREM ( C l a y ( 2 ) , ( 4 ) ,

(6), ( 8 ) ) . T h e r e i s a l - l - c o r r e s p o n -

dence between a l l n e a r - r i n g m u l t i p l i c a t i o n s on b in a r y o p e r a t i o n s

9.93

"

'I

makinq

r

(that are

) i n t o a near-ring)

(N,t,

and

NOTATION (a) I f 4 i s defined by (b) I f

we w r i t e

.

6'

i s d e f i n e d v i a Q we w r i t e

(as i n 8.26).

'6

W i t h s t r a i n h t f o r w a r d p r o o f s one a e t s

-

9 . 9 4 PROPOSITION ( C l a y ( 4 ) ) . L e t

(a)

e 4

be a s b e f o r e .

g i v e s a d i s t r i b u t i v e n r . <==>

tf

y , 6 ~ r:

i s c o m m u t a t i v e <->

(c)

produces a zero-symmetric nr.

o6

y,6cr

= 4 t&

Y

: oY(6) = b 6 ( y ) .

(b) * +

We h a v e t o l o o k , w h i c h

-9 . 9 5

r,@

<=>

b

0

= 6.

m u l t i p l i c a t i o n s y i e l d isomorphic n r . ' s :

D E F I N I T I O N Two m u l t i p l i c a t i o n s i f t h e r e i s some a . s A u t T w i t h

and

e 2

are s imilar

y , b ~ r :a(y.la)=a(y)*2a(a).

6'

323

9d Near-rings on given groups

9 . 9 6 P R O P O S I T I O l i ( C l a y ( 4 ) ) . The f o l l o w i n o c o n d i t i o n s a r e equivalent: (a)

and

a 1

(b) (r,+,*l)

are similar.

e 2

QJ

(T,+,*z).

9 . 9 7 PR3POSITION ( e . q .

Clay (8)).

(I-,+,*@)<=>

(a) y i s a zero d i v i s o r i n

i s interiral

( b ) (I-,+,.,?)

<==,

all

(p

4,

Y

i s not injective

a r e i n j e c t i u e (-{ST*:.

o r y E K e r B f o r some B . (c) If

'd

r

(r.+,-$)

i s finite,

YETF:

i s intenral iff

$ y ~ ( r~, t )~. t

A t t h i s p l a c e i t m i g h t be a p p r o p r i a t e t o remark t h e c o n s t r u c t i o n

(r,+)

method o f 1 . 4 ( b ) o f a n r . on automorphism group.

(cf.

v'ia a f i x e d - p o i n t - f r e e

a l s o Theorem G o

We a l s o m e n t i o n w i t h o u t t h e e v i d e n t p r o o f

9.98

P.ROPOSITIOtl

( C l a y ( 4 ) ) . The m u l t i p l

i s t r i v i a l ( i n t h e sense o f 1 . 4 ( b ) )

9.99

THEOREM ( C l a y ( 4 ) ) . T h i s h o l d s by 9 . 9 1 .

(r,t,*)

i s constant i f f

I n t h i s case,

4;

$iEAut(r).

= id.

9.100 REYARKS ( a ) See t h e a p p e n d i x f o r a l l n r . ' s I

7

on g r o u p s o f o r d e r

a n d o f many o n e s o f o r d e r 8 and 1 2 .

Examples o f n r . ' s

on n o n - a b e l i a n g r o u p s o f o r d e r 12

a n d 18 a r e i n M a l o n e ( 1 ) . Y e a r b y ( 1 ) c o n t a i n s many m o r e e x a m p l e s .

(b) T h e r e a r e g r o u p s ( Z i , f o r i n s t a n c e ) , on w h i c h no r i n g s except the zeroring i s definable. Abelian groups w i t h t h i s property a r e c a l l e d " n i l qroups".

$9 MORE CLASSES OF NEAR-RINGS

324

L a w v e r ( 1 ) , ( 2 ) has shown t h a t t h e r e m i g h t e x i s t n o n trivial

n e a r - r i n g s on n i l groups.

Ligh-Malone

(1) have

shown t h a t n e a r - r i n g s w i t h o u t z e r o d i v i s o r s on c o m p l e t e 0 i f b=O groups a r e constant o r f u l f i l l ab = a i f bSO ( a g r o u p i s s a i d t o be c o m p l e t e i f i'ts c e n t e r = Co3

[

and i f a l l automorphisms a r e inner;

a l l S,,

(n { 2,6)

are complete). ( c ) Lawver ( 3 ) s t u d i e d n r . ' s sums o f g r o u p s . (d) Clay (1),(6)

Cf.

o n f r e e groups and on d i r e c t

1.22(a).

s t u d i e d t h e m u l t i p l i c a t i o n s on an a b e l i a n

group by g i v i n g them a g r o u p s t r u c t u r e . (e) " M u l t i p l i c a t i o n s " t u r n i n g a r i n g i n t o a composition r i n g a r e s t u d i e d i n A d l e r (1). See a l s o t h e l i n e s o n l a m i n e t e d n e a r - r i n g s

i n 9.277.

325

9d Near-rings on given groups

2 . ) N E A R - R I N G S O N SIMPLE AM0 ON C Y C L I C G R O U PS

9.101 PROPOSITION (e.q.

(r,t,.)

g r o u p and

(a)

Heatherly ( 2 ) ) .

=:r

( r , t ) be r . Then

Let

a n e a r - r i n g on

a simple

r~?'!~v

(b)

v

(c)

r

: ($;

yEII

i s a monomorphism).

a ) v (4;

==>I y e r : (4; = a)

finite

-Proof.

=

v ($;EAut(r,+)).

(a) f o l l o w s from t h e f a c t t h a t

(To,+)

( b ) and ( c ) r e s u l t from c o n s i d e r i n o 9 . 1 0 2 THEOREM ( H e a t h e r l y ( 1 ) , ( 2 ) ) . on the f i n i t e simple qroup

r

Let (r,t).

11 ( r , t ) .

Ker b;.

= (r,t,.)

Then

r

be a n r . f a l l s i n t o one

o f the following d i s j o i n t classes: i s the "zero m u l t i p l i -

( i n t h i s case,

( a ) v y c r : 4; cation").

= 6

( b ) I'd = { o l

a n d I' h a s a r i g h t i d e n t i t y . and r has an i d e n t i t y .

(c) r d

=+

Io)

P r o o f . Suppose t h a t i s not the zero multiplication. 9.101(c), 3 6 e r : @:: l' + r E A u t ( r , + ) . NOW

3

* k k e n : (4,) = idr

,

= ya If

and

rd 8 I o ) ,

Since

ak

Y

.\r

+

Ya

By

* k

(0,) (Y)

yer : y = i d ( y ) =

=

i s a right identity. take

6cri.

Consider

,Ilr

i s not the zero multiplication,

(o:r) 4 (I-,+-),

:r+r EEnd(r,t). y+ay

(o:r) = {ol

6 Q $. 6 a n d ( a s a b o v e ) & $ ~ A u t ( r , t ) a n d some p o w e r o f 6 i s a l e f t i d e n t i t y , hence the identity. for

Observe t h a t 9.109

Hence

i m p l i e s t h a t i n c a s e ( c ) o f 9.102,I'

be a f i n i t e prime f i e l d . We now t u r n t o c y c l i c g r o u p s .

has t o

89 MORE CLASSES OF NEAR-RINGS

326

9 . 1 0 3 THEOREM ( H e a t h e r l y ( 1 ) a n d o t h e r s ) .

B with a generator

Bn . o r ring.

then N i s a commutative

gEPid

N i s an a b e l i a n dpnr., hence a r i n q .

I n t h i s case,

Proof.

L e t N be a n r . on

E v e r y r i n g on a c y c l i c g r o u p i s c o m m u t a t i v e ( s e e (Beaumont)). 9 . 1 0 4 COROLLARY ( H e a t h e r l y ( 1 ) ) . I f N i s a n r . or Z

with

(01

Nd

t h e r e i s some

Pp

(usual product i n Proof.

If

with

xEN

dcNd

on

zp

(PEP

1

t h e n N i s a commutative r i n q and or

$. 0 ,

i s

shows t h a t 1 i s a l s o

n,n'EN

: nn'

= n-n'*,x

Z ). a s h o r t c a l c u l a t i o n (cancel d!)

EN^;

--

now we may a p p l y 9 . 1 0 3

t o get the f i r s t assertion. Let

1.1 = : x

and

n n ' = (l+ , . . +l ) ( l t n-summands 9.105

n,n'EN.

. . .t l )

1cNd.

= n*n'*x.

n'-summands

R E M A R K T h e same r e s u l t a s i n 9 . 1 0 4

if

Then

= n.n'*(l.l)

holds i n every

Zn On t h e o t h e r h a n d , H e a t h e r l y ( 1 ) q i v e s a n

E4 which i s n o t a r i n n ( i n f a c t , 1 and 3 a r e n o t d i s t r i b u t i v e ) .

example o f a n r . o n

For t h e next r e s u l t , l e t C(k,j) be t h e number o f c o m b i n a t i o n s o f k e l e m e n t s t o t h e c l a s s j ) . N i t h o u t p r o o f we s t a t e 9 . 1 0 6 THEOREM ( R . J a c o b s o n ( 1 ) ) . T h e n u m b e r o f d i f f e r e n t d e f i n a b l e on

(Zp.+)

nr.'s

( P E P ) i s given by

Rore i n f o r m a t i o n s can be f o u n d i n A d l e r ( l ) , F e i g e l s t o c k ( 2 ) and Heatherly (2).

9d Near-rings on given groups

327

3 . ) NEAR-RINGS WITH IDENTITIES O H G I V E ? { GROUPS -

ble s t a r t w i t h

9 . 1 0 7 PRO-P O S I T I O N ( C l a y ( 4 ) ) . L e t N = ( r , + , * ) be a n r . on m Then ~ E N i s a n i d e n t i t y o f N i f f m l = i d arid

w

YET:

OY(U

r,.

= Y.

The p r o o f i s o b v i o u s .

O u t o f 9 . 1 0 3 a n d 9 . 1 0 4 we q e t ( o b s e r v e t h a t u n d e r t h e a i v e n assumptions, x o f 9.104 i s i n v e r t i b l e i n :I): 9 . 1 0 8 C O R O L L A R Y ( C l a y - M a l o n e ( 1 ) ) . I f N i s a n r . E??, on t h e c y c l i c q r o u p (N,+) t h e n 11 i s a c o m m u t a t i v e r i n ? . A l l n r . ' s on ( N , t ) a r e i s o m o r p h i c . T h e r e a r e o ( n ) ( @ t h e E u l e r f u n c t i o n ) o n e s on ( Z n , t ) a n d 2 on Z.

9.109

C O R O L L A__ R Y There a r e e x a c t l y p - 1 n r . ' s with i d e n t i t y d e f i n a b l e on ( h p , t ) ; a l l o f t h e m a r e i s o m o r p h i c t o t h e f i e l d Zp and hence a l l a r e f i n i t e prime f i e l d s .

T h i s r e s u l t was o b t a i n e d by M a l o n e , C l a y , Maxson a n d H e a t h e r l y under d i f f e r e n t circumstances. O b s e r v e t h a t i f i n ( r , t , - ) ~ ? ' ?( ~r , t ) i s a b e l i a n w i t h e x a c t l y ?r o n e p r o p e r s u b o r o u p t h e n ( r , + ) = Zp2 a n d (I-,+,.) is a c o m m u t a t i v e r i n r : by 9 . 1 0 8 ( L i q h ( 9 ) ) . B u t t h e r e do e x i s t n o n r i n g s w i t h i d e n t i t y on rrroups o f o r d e r p 2 ( c f . a l s o 9 . 1 1 5 ( c ) ) : 9 . 1 1 0 PROPOSITION (Maxson ( 1 ) ) . F o r e a c h P E P t h e r e e x i s t s a g r o u p r o f o r d e r p 2 a n d a non-rice w i t h i d e n t i t y on r .

T h e p r o o f i s e s t a b l i s h e d by d e f i n i n g a m u l t i p l i c a t i o n on (I-,+): = ( H p , + ) B ( Z p , t ) i n an a p p r o p r i a t e manner ( s e e Mzxson ( 1 ) f o r d e t a i l s . )

328

$ 9 MORE CLASSES OF NEAR-RINGS

Now we s t u d y n r . ' s o f s q u a r e - f r e e o r d e r . F i r s t we need 9 . 1 1 1 P R O P O S I T I O N ( C l a y - M a l o n e ( l ) , Maxson ( 1 ) ) . L e t (I-,+,*) be a n r . w i t h i d e n t i t y 1 on t h e f i n i t e q r o u p r. Let o r d ( y ) be t h e o r d e r o f Y E r . T h e n o r d ( 1 ) = l . c . m . I o r d ( y ) l y ~ r ) = : II. Proof. I f yEr, o = oy = ( o r d ( l ) . l ) y = o r d ( l ) . y , SO o r d ( y ) / o r d ( l ) . Hence f . / o r d ( l ) . B u t l c r , h e n c e o r d ( l ) / I I whence o r d ( 1 ) = II. 9 . i 1 2 T H E O R E M- ( F l a x s o n ( 1 ) , ( 2 ) ) . s q u a r e - f r e e o r d e r . Then i s a c o m m u t a t i v e rincr. ,-

(I-,+,* j E n 1 have f i n i t e ( r , + ) i s c y c l i c , a n d (I-,+,.) Let

P r o o f . Let = plp 2...pr, where p l , . . . , p r are d i s t i n c t primes. U s i n o t h e Sylow t h e o r y we p e t f o r Hence e a c h i E { l , ..., r ) some y i € r o f o r d e r p i . [GI t ord(1) = l . c . m . { o r d ( y ) l y ~ r l t 2 l.c.m.rord(y,) So o r d ( 1 ) = I G I

,..., o r d ( y r ) )

= (GI.

and G i s c y c l i c . N o w u s e 9 . 1 0 3 .

Several groups cannot b e a r a nr. w i t h i d e n t i t y ( c a l l a s u b s e t P o f a p a r t i a l l y o r d e r e d s e t an a n t i c h a i n i f no d i s t i n c t e l e m e n t s a r e comparable): R E M (Krimmel ( 1 ) , ( 2 ) ) . Let (r,+) be a g r o u p h a v i n r l 9 . 1 1 3 T H E Oe l e m e n t s y l , . . ,y, of d i s t i n c t prime o r d e r s p l , pr ( r 1 2 ) . I f e v e r y a n t i c h a i n i n t h e l a t t i c e of normal subgroups o f r has c a r d i n a l i t y < r then (r,t) c a n n o t be t h e a d d i t i v e q r o u p o f a n r . w i t h i d e n t i t y .

...,

.

P r o o f . S u p p o s e t h a t ( r , + , * ) i s a nr. w i t h i d e n t i t y 1. I f there are i,jc{l, r l w i t h ( o : y i ) F ( o : y J. ) then h: ( r y i , + ) * ( r y j , + ) i s a well-defined aroupYYi * YYi e

...,

homomorphism. B u t

h ( y i ) = h ( l y i ) = l y j = yj,

whence

329

9d Near-rings on given groups pj = ord(yj)/ord(yi)

{ ( o : y l ) ,...,( o : y r ) l a contradiction.

pi, so i = j. Hence i s an a n t i c h a i n w i t h r e l e m e n t s ,

O b s e r v e t h a t we d i d n ' t u s e a s s o c i a t i v i t y o f only a l e f t i d e n t i t y .

0

;

1 c o u l d have been

9 . 1 1 4 C O R O L L A R Y ( C l a y - M a l o n e ( 1 ) ) . A nr. w i t h i d e n t i t y on a f i n i t e s i m p l e rjroup r i s a f i n i t e p r i m e f i e l d . Proof.

r c a n n o t h a v e a c o m p o s i t e o r d e r by

9 . 1 1 3 . Hence r i s a simple p-qroup, t h u s c o i n c i d i n a with i t ' s a n d we c a n a p p l y c e n t e r . So r i s isomorphic t o H P 9.109.

9 . 1 1 5 COROLLARIES (Krimmel ( 2 ) ,

Clay-Halone

( l ) , Clay-Doi ( l ) ,

Ligh ( 9 ) ) . The f o l l o w i n g g r o u p s T c a n n o t be t h e a d d i t i v e a r o r r p s o f near-rings w i t h identity: ( a ) groups o f composite o r d e r i n wh;ch t h e l a t t i c e of n o r m a l s u b q r o u p s i s l i n e a r l y o r d e r e d (e.cj. S n ( n > 3 ) ) , ( b ) simple groups of composite o r d e r (e.9.

An (nr4)),

( c ) f i n i t e non-abel i a n o r o u p s w i t h e x a c t l y one p r o p e r normal non-zero s u b q r o u p , ( d ) n o n - c y c l i c clroups o f s q u a r e - f r e e o r d e r .

P r o o f . E v i d e n t l y 9 . 1 1 4 =z ( a ) =z ( b ) a n d 9 . 1 1 2 -> ( d ) . I n ( c ) , r must be o f c o m p o s i t e o r d e r s i n c e o t h e r w i s e r i s a n o n - a b e l i a n p-qroup, hence o f o r d e r pk w i t h kr3. I n t h i s c a s e , r h a s a t l e a s t two n o n - t r i v i a l n o r m a l s u b g r o u p s ( s e e e . g . ( R o t m a n ) , C o r . 5.5 a n d Ex. 5.2). We now m e n t i o n w i t h o u t p r o o f some more r e s u l t s o n t h i s s u b j e c t .

If N i s f i n i t e such t h a t t h e i n v a r i a n t subgroups o f ( N , + ) f o r m a c h a i n t h e n 1.1 i s i s o m o r p h i c t o a r i n g Z Pn*

$9 MORE CLASSES OF NEAR-RINGS

330

9.116 THEOREM

( a ; ( L i g h ( 9 ) ) . T h e r e i s no n r . w i t h i d e n t i t y d e f i n a b l e on a t o r s i o n d i v i s i b l e g r o u p . ( b ) ( C l a y - D o i ( 1 ) ) . The same h o l d s f o r and

Am: =

u An. n c IN

Sm:

=

U IN 'n

nE

(1)). There a r e a l s o no n r . ' s d e f i n a b l e on q e n e r a l i z e d q u a t e r n i o n F r o u p s .

( c ) (Clay-Maxson

( d ) ( L i g h ( 1 3 ) ) . T h e r e do e x i s t n r . ' s w i t h i d e n t i t y on p e r f e c t groups ( t h a t a r e qroups coincidincl w i t h i t s commutator s u b g r o u p ) ( c f . Liqh ( 9 ) ) .

( e ) See Johnson ( 4 ) f o r the n r . ' s on the d i h e d r a l nroups D p n o f o r d e r 2 n . T h e r e a r e n o n r . ' s EV, on D Z n f o r odd t i ( t h i s f o l l o w s f r o m 9.111), f o r t h e o n l y o n e s e x i s t on D 4 p ( P E P) . They a r e z e r o - s y m m e t r i c a n d normal N - s u b g r o u p s a n d l e f t i d e a l s c o i n c i d e ( a n d a l l l e f t ideals are annihilator l e f t ideals). There a r e ( u p t o isomorphism) 7 n r . ' s w i t h i d e n t i t y on D 8 ( p . 4 1 8 ) and ( a q a i n u p t o isomorphism) j u s t one o n D ( p ~ I l ' \ { Z l ) . T h e r e i s j u s t one s u c h 4P n r . on t h e i n f i n i t e d i h e d r a l g r o u p ( L o c k h a r t (1),(3)). ( f ) (Clay-Maxson ( 1 ) ) . A l l n r . ' s w i t h i d e n t i t y d e f i n a b l e on p-groups w i t h e x a c t l y one s u b c r o u p o f o r d e r p a r e commuta t i ve r i n o s . ( T h i s f o l l o w s from 9.108 and ( c ) s i n c e a group a s d e s c r i b e d above i s e i t h e r c y c l i c o r a o e n e r a l i z e d q u a t e r n i o n group.)

4 . ) N E A R - R I N G S W I T H O T H E R P P q P E R T I E S OFJ G I V E N G R O U P S

Now we b r i e f l y s t u d y n r . ' s w i t h s p e c i a l p r o p e r t i e s ( o t h e r t h a n h a v i n r l an i d e n t i t y ) c n some q r o u p ( r , t ) . We w i l l o n l y c i t e t h e r e s u l t s o r e v e n o n l v t b e n a o e r s w h i c b a r e concerned w i t h t h e s e t o p i c s . See a l s o t h e c h a p t e r s concerning t h e types o F n e 3 r - r i n g s i n dis::ission. For ex?nple,

33 1

9d Near-rings on given groups

t h e r e a r e no n e a r - f i e l d s d e f i n a b l e on n o n - a b e l i a n n r o u p s ( 8 . 1 1 ) , a n d so o n .

Ide s t a r t w i t h n r . ' s w i t h c h a i n c o n d i t i o n s . G e n e r a l i z i n n 9 . 1 0 2 one q e t s

9 . 1 1 7 THEqREFI ( L i n h ( 3 ) ) . L e t N b e a n r . w i t h DCC on m o n o n e n i c N - s u b n r o u p s o n t h e s i m p l e nrOlJp ( P i , + ) such t h a t N d $. { O I . Then t i i s e i t h e r t h e z e r o - n r . o r a f i e l d . 9.118 R E M A R K For a d e t a i l e d s t u d v o f n r . ' s N o n a nroup which f u l f i l l t h e D C C o n m o n o n c n i c M - s u b n r o u p s a n d t h e " A C C on p r i n c i p a l a n n i h i l a t o r l e f t i d e a l s " ( i . e . each ( 0 : x ) c c ( 0 : x 2 ) s ( o : x 3 )C terminates) s e e Lioh-RarakotaiahReddy ( 1 ) .

...

9 . 1 1 9 THEQPEE1 (Timn ( 3 ) ) . (T',+) i s t h e a d d i t i v e nroup of a ( n o t n e c e s s a r i l y a s s o c i a t i v e ( ! ) ) near-rirrrr i n which every non-zero element h a s a r i o h t i n v e r s e i f f r i s invariantly simple a n d e v e r y y t r h a s ( t h e s a m e ) p r i m e o r d e r . The q u e s t i o n c o n c e r n i n n t h e a d d i t i v e c r o u p o f n e a r - f i e l d s i s s e t t l e d by t h e f o l l o w i n n t h e o r e m .

9 . 1 2 0 T H E O P E V (Timm ( 3 ) ) . The f o l l o w i n g c o n d i t i o n s o n (r,+) are equivalent:

a vroup

r r

i s t h e a d d i t i v e nroup of a n e a r - f i e l d . i s a b e l i a n and t h e a d d i t i v e nroup of a n r . w i t h r i o h t c a n c e l l a t i o n law. ( c ) 'I i s t h e a d d i t i v e n r o u p o f a v e c t o r s p a c e o v e r some field. ( d ) r i s t h e a d d i t i v e nroup of a conmutative f i e l d . ( e ) r i s the a d d i t i v e nroup o f an a l t e r n a t i v e f i e l d . ( f ) T h e r e i s Some P E P s u c h t h a t r i s t h e d i r e c t sum o f t h e nroups ( Z P , t ) o r r i s a d i r e c t sum o c c o p i e s o f

(a) (b)

(@,+). ( g ) r i s a b e l i a n a n d e i t h e r e a c h e l e m e n t h a s t h e same prim.; 91.d.r o r r i s ~ s r : i o n ~ r - e ed i v j s i b l e .

332

$ 9 MORE CLASSES OF NEAR-RINGS

F i n a l l y , we c o n s i d e r t h e a d d i t i v e g r o u p o f d g n r . ' s a n d o f i n t e gral n r . ' s . 9.121 REMARKS ( a ) (Ligh ( 1 0 ) ) . There a r e just 3 non-isomorphic d g n r . ' s

o n S6 and a t l e a s t 3 on S n ( n

2 5 ,

n j 6).

There a r e p r e c i s e l y 3 non-isomorphic d g n r . ' s d e f i n a b l e ( p E IP) , b u t none on t h e i n f i n i t e d i h e d r a l g r o u p on D 2P Dm ( L o c k h a r t ( 1 ) , ( 3 ) ) . ( b ) ( L i g h ( 1 3 ) ) . The a d d i t i v e g r o u p o f a s i m p l e d g n r .

is

perfect. ( c ) D g n r . ' s w i t h i d e n t i t y on f r e e g r o u p s a r e e x t e n s i v e l y s t u d i e d i n Zeamer ( 2 ) . ( d ) D g n r . ' ~o n g r o u p s

group

r'

r,

i n which t h e index of t h e d e r i v e d

i s p r i m e , a r e c o n s i d e r e d i n Chandy ( 3 ) .

( e ) ( M a l o n e ( 7 ) ) . T h e r e a r e e x a c t l y 16 d g n r . ' s on a g e n e r a l i z e d q u a t e r n i o n g r o u p . A l l o f them a r e d i s t r i b u t i v e . ( f ) More on d i s t r i b u t i v e n e a r - r i n g s

on g i v e n g r o u p s c a n be

found i n Jones ( 1 ) and W i l l h i t e ( 1 ) . ( 9 ) From 9 . 5 1 we k n o w t h a t i n t e g r a l n e a r - r i n g s groups

r

force

r

t o be n i l p o t e n t . I f

r

on f i n i t e

i s non-abelian

o f o r d e r p 3 ( P E I P ) w i t h p 3 = 2 ' t l t h e n t h e r e a r e no i n t e g r a l n r . ' s N d e f i n a b l e on r', s u c h t h a t N h a s a t l e a s t o n e r i g h t i d e n t i t y -1 0 . I f Irl = p t l , P E I P , p f 2 t h e n e i t h e r a g a i n no s u c h N e x i s t s o r p t l i s a power p t l = 2" o f 2 a n d N i s a G a l o i s - f i e l d ( O l i v i e r (2), H e a t h e r l y - O l i v i e r ( 2 ) ) . (h)

(Lawver ( 3 ) ) . A l l n e a r - r i n g s o n Zm a r e p l a n a r . There P a r e no i n t e g r a l p l a n a r n r . ' s d e f i n a b l e o n Z y , b u t t h e r e a r e some on Z F ( w i t h c h a r a c t e r i s t i c f 5 ! ) .

( i ) "H-monogenic" n e a r - r i n g s ( s e e 9 . 2 7 5 ) a r e g e n e r a l i z a t i o n s o f i n t e g r a l n e a r - r i n g s . A d d i t i v e g r o u p s o f H-monogenic near-rings a r e studied in O l i v i e r ( 2 ) a n d HeatherlyOlivier (3).

9e

Ordered near-rings

333

I f a g r o u p r i s g i v e n by a p r e s e n t a t i o n , i t i s a h i g h l y nont r i v i a l m a t t e r t o c h a r a c t e r i z e a l l n e a r - r i n g s on r . F i r s t studies i n t h i s directions (including "pre-near-rings" ( = m u l t i p l i c a t i v e l y n o n - a s s o c i a t i v e n e a r - r i n g s ) ) can b e found i n L o c k h a r t ( 1 ) and L a x t o n - L o c k h a r t ( 1 ) .

9.122 DEFINITION A nr. N i s c a l l e d p a r t i a l l y ( f u l l y ) ordered by 5 i f

(a) (b)

I makes

i n t o a p a r t i a l l y ( f u l l y ) ordered nroilo.

(N,t)

n,n'EN:

(nrO

A

n'rO

=>

nn'rO).

" O r d e r e d " means " p a r t i a l l y o r d e r e d " .

9.123 REVARKS

( a ) T h u s an o r d e r e d n e a r - r i n n i s a n r . w h e r e (PI,<) is a n o r d e r e d s e t such t h a t n29, n ' 2 0 i m p l i e s n t n ' > ? nn'2O.

and

( b ) The s t a n d a r d w o r k on o r d e r e d a l a e b r a i c s y s t e m s ( s e n i -

g r o u p s , clroups, r i n n s a n d f i e l d s ) i s ( F u c h s ) .

( c ) F o r a n o r d e r e d n e a r - r i n g we w i l l w r i t e simply

(N,+,*,s)

or

(3,s).

( d ) P a r t s of o u r d i s c u s s i o n i s i m p l i c i t in ( G a b o v i c h ) .

O f course,

nl

S

n 2 and 0 s n i m p l i e s

nlnS

n 2 n i n an o r d e r e d n r . N .

Some a u t h o r s ( K . B . P . Rao ( l ) , f o r i n s t a n c e ) r e q u i r e t h a t a l s o n n l s n n 2 follows. Cf. 9 . 1 5 2 ( b ) , ( d ) .

59 MORE CLASSES OF NEAR-RINGS

334

9 . 1 2 4 NOTATION I4e a d o p t t h e u s u a l c o n v e n t i o n s t o w r i t e n < n ' , n ? n ' , n > n ' , nl1 n ' ( n a n d n ' a r e incomparable, i . e . neither nrn' nor n ' s n holds). " P a r t i a l l y o r d e r e d " w i l l be a b b r e v i a t e d by " P . o . " , " f u l l y o r d e r e d " by " f . o . " . J u s t a s i n t h e t h e o r y o f o r d e r e d n r o u p s o r r i n n s , i t i s more c o n v e n i e n t t o work w i t h t h e s e t o f " p o s i t i v e " e l e m e n t s i n s t e a d of the o r d e r r e l a t i o n i t s e l f : 9 . 1 2 5 THEOREkt ( a ) Let ( E l , + , * , < ) be a p . 0 . n r . ; t h e n t h e "-___-positive cone" P : = P: = { n E l ! l n r o l fulfills 5

(a) PtP P. ( R ) P n ( - " ) = (01, where, as u s u a l , ( y ) d nEpI: n t P = P t n . ( 6 ) P * P c P.

-P:

= {nlnir)).

( b ) C o n v e r s e l y , f o r e v e r y s u b s e t P o f a n r . rl f u l f i l l i n i

( a ) - ( 6 ) we clet a n o r d e r e d n r . n sP n ' : <=> n ' - n ~ P .

(N,sp)

via

( c ) T h i s c o r r e s p o n d e n c e between o r d e r r e l a t i o n s a n d subsets w i t h ( a )

-

(6) is

t h a t means t h a t

1-1,

The p r o o f o f 9 . 1 2 5 i s e a s y a n d l e f t t o t h e r e a d e r . 9 . 1 2 5 e n a b l e s u s t o s a y t h a t " t h e n r . PI i s o r d e r e d by P " . The f o l l o w i n c l r e s u l t i s o b v i o u s . 9 . 1 2 6 PROPOSITI3PI Let FI be o r d e r e d by P.

( a ) sP (b)

i s a f u l l o r d e r <->

sP i s t r i v i a l ( i . e .

P w(-P)

n I, n ' < - >

=

N. n = n ' ) <->

There i s no p l a c e f o r f i n i t e near-rirlns i n t h i s s e c t i o n :

P

=

{Ol.

335

9e Ordered near-rings

9 . 1 2 7 PROPOSITION Every n o n - t r i v i a l l y o r d e r e d n e a r - r i n o i s infinite. Proof.

3

nErl:

n>O.

B u t then

... .

n < n t n = 2n<3n<

9 .1 2 8 DEFINITION Let N , N ' be n r . ' s o r d e r e d b y P,P', respectively. f: N + N' i s o r d e r - p r e s e r v i n f l : <=+ <=> f ( P ) E P ' . I f t h e r e i s an o r d e r - p r e s e r v i n a mono- (iso-)mor phism f : rI + N ' we w r i t e N % N I (N2 ria, respectively) 1 ( f o r i s o m orphisms f we a l s o w a n t t o be o r d e r p r e s e r v i n q s i n c e t h e c a t e r l o r y of o r d e r e d n e a r - r i n o s a n d o r d e r - p r e s e r v i n n homomorphisms i s n o t b a l a n c e d ) .

;-

9 . 1 2 9 D E F I N I T I O N A s u h s e t T of a n o r d e r e d n r . FI i s c a l l e d convex i f

I _ -

tl,t2ET

nEN

: ( t l < n s t 2-=> n ~ f ) .

S i m i l a r t o r i n a t h e o r y ( s e e ( F u c h s ) ) one can e a s i l y prove t h e f o l l o w i n o two r e s u l t s . A o a i n , t h e y a r e c o r o l l a r i e s of theorer is of ( G a b o v i c h ) . 9 . 1 3 9 TtlEQ?EF-1A s u b s e t I of a n o r d e r e d n r . N i s t h e k e r n e l o f a n o r d e r - p r e s e r v i n q nr.-homomorphism from \I t o some o r d e r e d n r . N ' i f f I i s a convex i d e a l . be n r . ' s , o r d e r e d b y P , P ' . Lei 9.131 THEOREPI Let N , N ' h : N -H. N ' be a n o r d e r - p r e s e r v i n n epimorphism ( i . e . h(N) = N ' and h(P) = P I ) . -1 If I ' A N' i s convex t h e n h ( 1 ' ) = : I i s a convex % i d e a l o f N a n d N / I -o N ' / I l . C o n v e x i t y i s q u i t e t r i v i a l , s o we w o n ' t p r o v e i t . 9 . 1 3 2 P R O P Q F I T I O N Every o r d e r ? i n N can be e x t e n d e d t o a maximal o r d e r P. Tbe p r o o f i s a c c o a p l i s h e d b y a n a p p l i c a t f o n o f 7 i ) r n ' S Lemnra.

09 MORE CLASSES OF NEAR-RINGS

336

9 . 1 3 3 PROPOSITI9N To e v e r y a b e l i a n o r d e r e d n r . r l t h e r e e x i s t s a n ( a b e l i a n ) o r d e r e d n r . fl w i t h i d e n t i t y s u c h t h a t FI

"b fi.

P r o o f . By 1 . 8 6 , we c a n f i n d a g r o u p r w i t h N q L f ( r ) ( s a y by h ) . I f N i s o r d e r e d by P , t a k e ?: = h ( P ) . Then a ) , P ) , 6 ) a n d ( s i n c e FJ i s a b e l i a n ) a l s o y ) o f 9 . 1 2 5 a r e c l e a r l y f u l f i l l e d , h e n c e fi: = M(r) i s o r d e r e d by 6 a n d h i s a n o r d e r - p r e s e r v i n g monomorphism. 9.134

PEMARKS

( a ) Plot e v e r y o r d e r e d n r . c a n be errbedded i n t o a n r . w i t h i d e n t i t y ( t h i s f o l l o w s from 9 . 1 3 7 ) .

f.0,

( b ) I f N c o n t a i n s a n i d e n t i t y 1 t h e n 120 o r 111 0 , f o r 1<0 implies ( - 1 ) > O , hence ( - 1 ) ( - 1 ) 2 9 , w h e n c e 120, a c o n t r a d i c t i o n t o t h e a s s u m p t i o n !<@. -

I n some i n s t a n c e s we c a n d e s c r i b e p . o f 9 . 1 3 2 e x p l i c i t p l y : i s ordered 5y 9 . 1 3 5 PROPQSITIOA L e t N be a n r . s u c h t h a t PIc Pc and such t h a t Pc forms a b a s e ( 1 . 9 1 ) . Then P : = { n E N 1 v p c P c : n p 2 O l i s t h e u n i q u e maximal extension of Pc t o an o r d e r o f P I .

Uniqueness i s c l e a r f o r p = ( P c : P c ) : and a ) - 6 ) o f 9.125 a r e e a s i l y v e r i f i e d . So t h e proof i s easy. Nevertheless, t h i s p r o p o s i t i o n has heavy c o n s e q u e n c e s , e.g. t h a t i n n e n e r a l P ( 9 ) c a n n o t be f u l l y o r d e r e d : e a c h p E P ( R ) w o u l d h a v e t o h a v e o n l y positive or only nenative values a t { r E R l r Z O l ! I f N i s complete ( i . e . i f nEN \I k E N c 3 n c ? t C E H ~ : nc = = n(c+k)) t h e n 9.135 i m p l i e s t h a t one c a n n a t g e t f u l l o r d e r s e x c e p t N = N o o r N = N, ( s e e P i l t (1),(3),(6)). Fundamental f o r t h e f o l l o w i n n i s

9 . 1 3 6 TFEQPEb4 L e t N be nEN

\d

f.5.

(by P ) a n d

c ~ P ~ : = P ~ r l nc ,:

=

no.

N c A ('31.

Thrn

337

9e Ordered near-rings

P r o o f . S u p p o s e t h e r e a r e some ncN a n d c c P C w i t h nc n O . 1 . l . l . o . g . we may a s s u m e t h a t n O 2 O ( o t h e r w i s e chanqe t o - n ) , Now n o t (n-nO)c = nc-n0 0. Hence n o 0. I f n o w o u l d be > O t h e n Osnoc 0, whence n o c > r ) II f u l f i l l s IIc = n o c > O , C o n s i d e r R: = n-nctn,; w h e n c e R > O . O n t h e o t h e r h a n d , 9,r) = n 9 - n c = = -n,c
+

9.137 C O R O L L A R V I f H has a

then

+

%

left

i d e n t i t y and i s f u l l y o r d e r e d

YE^,.

9 . 1 3 8 R E U-A R K I f under t h e c i r c u m s t a n c e s o f 9 . 1 3 6 PIc forms a base and N i s considered a s subnear-rinn o f Y ( N C ) ( 1 . 9 6 ) t h e n 9 . 1 3 7 t e l l s us t h a t e a c h n E N i s c o n s t a n t Me w i l l s e e t h a t i n on a l l p o s i t i v e e l e m e n t s o f k4 C ' f a c t f.0. n r . ' s hre closely related t o constant nearrinas (9.141(a)).

9 . 1 3 9 DEFINITIQN a r ch im ed ian ordered - An o r d e r e d nr. N i s c a l l e d i f (N,+) h a p p e n s t o be t h i s ( s e e e . g . ( F u c h s ) ) .

If a f.0.

n.r.

N i s archimedian,

( N , t ) C > o (IR , + ) .

(N,+)

i s a b e l i a n and

I f FI i s n o t a r c h i m e d i a n t h e n t h e r e e x i s t o f N with k . l a \ = l a l + l a l t ...+I a l < ! b l for a l l

pairs

2

(a,b) (where \ a \ h a s t h e u s u a l meaning - s e e ( F u c h s ) ) . I n t h i s c a s e we c a l l a " s m a l l w . r . t .b " a n d w r i t e a<< b . If AcN, " A < < b " a n d s i m i l a r n o t a t i o n s a r e t h e n c l e a r . ksIN

9 . 1 4 0 D E F I N I T I O N L e t N be o r d e r e d and

nE!l.

n i s n e a r l y c o n s t a n t : <=> n 0 << Pi:. 9 . 1 4 1 T H E n R E ' 1 L e t N be f u l l y o r d e r e d a n d L

( a ) E v e r y nc.N ( b ) H, =k !O; =>

N c I {ti}.

i s nearly consrant.

the order i s non-srrbfrnelian.

Then

$9 M O R E C L A S S E S O F N E A R - R I N G S

338

A, { Q l , t a k e a n y nEN w i t h ak: k.(-nc+n)-nc, bk: = (k-l)*no+n.

Proof.

N,

(a) I f

For

akO = - n C < O ,

Then

bkO = n c > r ) , Thus

hence

whence

-nc+n-n

C

+...+

0

ak
kEIN:

. . .- n c t n .

$0

From t h e f i r s t i n e a u a l i t y Hence

0

C

and

so

n - n C
ken -nc
-n <-k.n

so

ak
bk>O,

-nc+k.no
nc>O.

let

k EIN ,

<-n t n = n 0

k*no-nc
C

k.lnol
C

for all

kEIN. Now l e t Let

and

XCV

m:

= xo+c.

c~P.1;

be a r b i t r a r y ,

Yince

mc = c>n,

c o n s i d e r a t i o n s above t o m and o e t Since

mo

=

xo'

x o <<

we s e e t h a t

but

CZO.

we c a n a p p l y o u r

~ E I N :k.'fiol<mc.

Nr.

(b) f o l l o w s a t once from ( a ) . I t i s h i q h t i m e f o r examples.

9 . 142 EXAMPLES (a) Non-zero near-rincrs o f t h e k i n d (1.4(a)),

R ~ x ! and

Yaff(V)

V(r),

'lcont(l')

c a n n o t he f u l l y o r d e r e d

s i n c e t h e y c o n t a i n an i d e n t i t y arid h a v e n o n - z e r o constants.

(b) L e t R be a f u l l y ordered comnutative n r . w i t h i d e n t i t y . Then

Ro[x2

k

P:

with

{

1 aixilak>rll Y C O I

("lexico-

i= r )

qraphic order")

P: = i

aixi

i2 k

and

lak>r))

Ro[[x]]:

uia)

= (R[[x]]),

with

('Iantilexiconraphic order")

s u p p l y n o n - t r i v i a1 e x a m p l e s o f o r d e r e d n e a r - r i n a s

4-

I n Ro[x], f o r e x a m p l e , we n e t t h e f o l l o w i n n "archimedian classes" {pldeg p 11 << { p ' d e g p = CL!

<<

... .

<<

339

9 e Ordered near-rings

( y o u may m u l t i p l y a l i n e a r p o l y n o m i a l by a n y n a t u r a l number you w a n t a n d y o u w i l l n e v e r a r r i v e a t a quadratic one). ( c ) L e t R be a r i n r l , f . 0 . by P. T a k e a n a r b i t r a r y , b u t fixed subset Qy-P*, and form t h e n e a r - r i n n

Nq: of

= I r l + r 2 ' x n l r l , r 2 ~ R R (xr, )

the i n d i c a t o r function

a).

Orderi no NQ l e x i c o q r a p h i c a l l y nives a f . 0 . n.r. w i t h {'I} -1. N o , i n which a l l e l e m e n t s a r e nearly constant.

+

9 . 1 4 3 PEKnRY, 9 . 1 4 1 shows t h a t o n e c a n n o t e x p e c t f o r " r e a l ~

'I

n e a r - r i n a s t o q e t any " n e a t " f u l l o r d e r . T h i s i s n o t v e r y s u r p r i s i n n : o n e h a s some s u b n e a r - r i n n o f some V(r) i n h a n d , w h o s e c a r d i n a l i t y i s , i n o e n e r a l , "much b i n q e r " t h a n t h o s e o f r . P n y b o d y , who e v e r h a d t o b r i n a o r d e r i n a l a r a e s t o r e h o u s e knows: t h e l a r n e r t h e s e t , the harder i s i t t o net a f u l l order. Also, one m i n h t e x p e c t t h a t b/ell-ot*dered n e a r - r i n n s a r e quite cpecial: be we1 1 - o r d e r e d . Then 9 . 1 4 4 T H F O D Eb f (Maxson ( 1 1 ) ) . L e t Nc?'$ N 2 0 (Z,+,-,<) ( I the usual o r d e r i n Z ) . P r o o f . A o a i n , l e t P be t h e p o s i t i v e c o n e o f P I . L e t a be t h e s m a l l e s t e l e v e n t o f P * and 4 t h e c y c l i c subq r o u p o f (PI,+) o e n e r a t e d by a . Suppose t h a t A FI. L e t b be t h e s m a l l e s t p o s i t i v e element of U: = ?&A. Consider a-b. I f a-b>'I t h e n a - b = a , a n o n s e n s e . Hence b - a > r ) . b - a ~ U - > b - a = b , w h i c h i s a c o n t r a diction, too. Hence U = fl and N i s c y c l i c a n d i n f i n i t e ( 9 . 1 2 7 ) . The map h : N = A + Z i s a n o r d e r - i s o m o r p h i s m z za between t h e a d d i t i v e groii?s.

-.

340

0 9 MORE CLASSES OF NEAR-RINGS

By 9.137, (usual

= -z*z'.

z*z':

z

+

2 a n d Z'

I n an o r d e r e d n r .

N one can ask, I n oeneral

Rrx]

are order-isomorphic

PI =o H %

Hence i n any c a s e

-2.

m i q h t be r e l a t e d . but for

H = (H,+,*) h': = (Z,+,*)w i t h

N i s isomorphic t o t h e r i n n

multiplication) or

how

Inn'\

via

.

and

In1 I n ' '

t h e r e i s no d i r e c t r e l a t i o n s h i p ,

o f 9.142(b) we n e t t h e f o l l o w i n n r e s u l t w h i c h

we s t a t e w i t h o u t p r o o f .

9.145

T H E O R EY (Pi12 (4)).

In

we h a v e f o r a l l

(R[x],t,o)

P,qER[x]: (a)

l p o q l = / p l o j q l c->

( p c o n t a i n s o n l y e ~ c no r

(420) v

o n l y odd d e q r e e s ) .

(b)

l p o q l 5 l p l o l q l <=>

(qcr)) A ( t h e c o e f f i c i e n t s o f t h e o r e a t e s t e v e n and n r e a t e s t odd d e n r e e o f p h a v e t h e same s i n n ) .

(c)

l p o q / 2 I p l o ' q j <=>

( q < O ) A ( t h e c o e f f i c i e n t s of t h e q r e a t e s t even and o r e a t e s t odd p h a v e o p p o s i t e sinn).

deqree of

J . Zemmer h a s s h o w n t h a t a d i r e c t s u m o f

f.0.

r i n n s can be

f . o r d e r e d i f f a l l b u t a t m o s t o n e o f t h e summands a r e z e r o r j n n s . Me now o b t a i n a s i m i l a r r e s u l t f o r n r . ' s ments on t h e s t r u c t u r e o f

9.146 T H E O R E M l f one of

the

i m p l y i n n some s t a t e -

nr's.

f.0.

S

N =

@

i s

Ni i-1

Nils

then i n a l l but a t most

f.0.

a l l p o s i t i v e elements ( i n the order

i n d u c e d b y N v i a t h e p r o j e c t i o n R a p s ) a n n i h i l a t e Fli from the r i q h t . Proof.

3

Assume t h a t

3

i,jE{l,

-

3

..., sl,

i I- j 101 A N . n ~j a n d ns;cNj,

O
positive,

If

n;
;uch then

that

J a contradiction.

nin.>r) 1

1

,

r ) < n i n i.-< n ! n

n;>o;

a n d we h a v e n o f u l i

ar,.: = r)

CO). both

n!n.>l. J J (2.27 a n d 2 . 6 ( b ) ) ,

y i s l f s tbe szrn~.

order.

50

n;

11 n j ,

341

9e Ordered near-rings S

9.147 C O R O L L A R Y Let

PI =

@ Ni

be f u l l y o r d e r e d a n d a l l

i=l Ni (0). If P i i s e i t h e r s t r i c t l y ordered ( i . e . n > 0 , n ' > n -> ==> n n ' > O -R[x! of 9.142(b) i s s t r i c t l y ordered i f R i s i n t e q r a l ) o r i f PI c o n t a i n s a l e f t i d e n t i t v t h e n s = 1 .

P r o o f . F o r s t r i c t o r d e r s t h i s i s i m m e d i a t e from 9 . 1 4 6 . S

1 ei i=l eiE?i As i n 3 . 4 3 , e i i s a l e f t identity i' a n d a q a i n w e can employ 9 . 1 4 6 t o n e t s = 1 .

I f ri c o n t a i n s a l e f t i d e n t i t y e , l e t with i n :Ii

9.148 C r l R O L L A R V Every 1 - s e m i s i m p l e is simple.

f.0.

e =

!ienl

nr.

with DCCL

Proof: by 9.137, 5.31 a n d 9.147. 9 . 1 4 9 REM4RK rlne c a n n o t i m n r o v e 9 . 1 4 6 t o n e t t h e e x a c t a n a l o n u e o f Zemner's r e s u l t : t a k e f o r N l , F 1 2 any f . 0 . c o n s t a n t non-zero near-rinns a n d use the lexicoaraphic order. Exanininn a b s t r a c t affirle near-rinas oives a stranoe r e s u l t w h i c h shows t h a t " n e a r l y n o " a . a . n . r . c a n b e f u l l y o r d e r e d : L e t PI be a n a . a . n . r . s u c h t h a t o r d e r e d . Then I! c a n be f . 0 . <==> FI o H c = v No i s a z e r o r i n c l ) .

rJo,Y,

9 . 1 5 0 TIIE3PE'f -___-

(Q)

A

are fully ( l l c = {'I)v

P r o o f . =>: ( a ) F i r s t we show t h a t ( N , + ) = ( N C , + ) G ( N , , + ) ( 9 . 7 3 ( a ) ) must have t h e l e x i c o o r a p h i c o r d e r . I f n r O t h e n n c = n9lc); l i k e w i s e n s ' l i f n p l f e s

n C SO. I f nc = 0

then

norO),

n

n 0 >O.

n E N : n20 <=+ (nc>9) v (nc i . e . the lexicographic order.

So He q e t f o r A

n > 0 <=>

r) A

342

S 9 MORE CLASSES OF NEAR-RINGS

( b ) Assume n o w t h a t N O H c {Ol. q i n c e N o = N d , we c a n f i n d n o E N 0 and PtccN, w i t h n c > 9 and nonc<9. T h e n n : 2no-nonc>r) bv ( a ) . B u t nn, = nonc O nonAn. L e t n : = no>O and n ' : = n j t n c > O . Then n n ' = n o n ; t t n o n c = nonh
A n;
{Ol. A

A

The m u l t i p l i c a t i o n r u l e i n Ii i s n n ' = ( n O t n C j ( n o' + nc ' ) nonA+nc. I f N, = { O l we n e t N = N 0 a s a f . 0 . r i n g . I f N, =I= I 0 1 t h e n N o2 = I 0 1 a n d a i l n n ' = n C In t h i s c a s e i t i s e a s i l y v e r i f i e d t h a t t h e l e x i c o g r a p h i c o r d e r i n N c t N o = PI makes PI i n t o a f . o . n r .

<=:

-

5 . 1 5 1 C O R O L L A R Y No a . a . n . r . N 4 N, i n which b a s e ( t h i s happens e . n . i n kfaff(V) a n d be f u l l y o r d e r e d .

N,

forms a

V a ( r ) ) can

P r o o f . I f N i s f . o . , t h e same c a n b e s a i d a b o u t N,. NoNc = {nl i m p l i e s N o = { 9 l , hence

No and N = PIc.

9.152 REVARKS ( a ) S e e Kerby ( 1 ) , ( 3 ) , ( 5 ) f o r a t h e o r v o f o r d e r e d n e a r f i e l d s w i t h some o e o m e t r i c i n t e r p r e t a t i o n s .

( b ) (Pilz ( 1 ) , ( 4 ) ) . nEN i s c a l l e d even (odd) i f n'EN: n ( - n ' ) = nn' (n(-n') = -nn', respectively). For i n s t a n c e , fEY(1R) i s even ( o d d ) i f f f i s an even ( o d d ) f u n c t i o n . A n f . c o n t a i n s o n l y odd e l e m e n t s ( 8 . 1 O ( b ) ) ; t h e same a p p l i e s t o r i n a s . N i s s a i d t o be c l e a v a b le i f e a c h neN i s t h e s u m o f an e v e n a n d an a d d e l e m e n t . R ~ x ? , R [ c x ? ! , P0[x! and the s u h n r . N n e n e r a t e d b y i d , s i n a n d c o s i n M(IR ) a r e e x a m p l e s o f c l e z v a b l e n e a r - r i n o s . 9 . 1 3 5 can b e e x t e n d e d t o f . c ; . c l ~ a v a b l pn e z r - r i n g s .

9e Ordered near-rings

343

with the antilexicoqraphic order ( c ) Ro[x] k r ( a k x t . . . t a r x > O : <=> a I:>0). i s n o t a f . 0 . n r . ( a l t h o u o h a f . 0 . r i n o when we u s e m u l t i p l i c a t i o n instead of composition ( F u c h s ) ) , f o r e.o. in t h i s ordering -xtlzr), x+2>Q, b u t ( - x + l ) d ( x t 2 ) = -x-l n n ' i n n " ( n n l > n n " ) .

n i s p o s i t i v e d e f i n i t e : <===>V n'EN : n n ' r 0 .

See P i l z ( 1 ) f o r r e s u l t s on t h e s e concepts.

( e ) ( P i l z ( 8 ) ) . L e t N be a n r . w i t h ( N , t ) = ( N o , t ) ~ ( ! 4 c , t ) ( c f , 1 . 1 3 ) , where N o a n d Pic a r e f . 0 . n . r . ' s (by P o + . T h e n t h e f . 0 . of No a n d N c can be e x t e n d e d t o a f . 0 . on N i f f po~:Po 1 pC€PC nocNo: p o ~ ( n 0 o + p c ) ~ P o . In t h i s c a s e t h e o r d e r i s the " l e x i c o o r a p h i c " one d e t e r m i n e d by n o t n C -z 0 <=> ( n C > O ) v ( n C = r) A n o 2 0 ) (see 9.150). ( f ) I t i s hard t o q e t f u l l orders in "non-deaenerated"

n e a r - r i n o s ( 9 . 1 4 1 ) . B u t i t i s very n a t u r a l t o l o o k f o r l a t t i c e - o r d e r s ( i . e . such t h a t (N,s) i s a l a t t i c e ) . F o r i n s t a n c e , M(T), where 7 i s a f . 0 . q r o u p , c a n be a i v e n a l a t t i c e o r d e r by

msm' :

<->

y

~ :r m ( y ) m l ( y ) .

F o r d e t a i l s a n d c o n n e c t i o n s t o " F - n e a r - r i nns" N ( t h e s e a r e s u b n r . ' ~a n d s u b l a t t i c e s o f a d j r e c t IT l i i o f f . 0 . n r . ' s €cllc, l a t t i c e - o r d e r e d product if1

( . . . , n i ,... ) I ( . . . , n ; ,... ) : c-> i c I : n i s n !1) and t o v e c t o r - n e a r - r i n q s ( F - n r . ' s , where is 3 s u b d i r e c t p r o d u c t o f t h e N i l s ) s e e O i l z (I), B b a n d a r i R a d h a k r i s h n a (I) a n d P a d h a k r i s h n a ( 1 ) . by

(9)

Kerby ( 1 ) , ( 3 ) , ( 5 ) a n d G r o g e r ( l ) ,( 2 ) s t u d i e d o r d e r e d n e a r - f i e l d s . A n f . F i s f o r m a l l v r e a l i f -1 i s n o t

344

89 M O R E CLASSES OF N E A R - R I N G S t h e sum of p r o d u c t s of s q u a r e s . F can be f u l l y o r d e r e d i f f F i s formally real (Groger ( 1 ) ) . ( h ) Extensions o f p a r t i a l orders t o f u l l orders a r e studied i n K . B . P . Rao ( 1 ) . (i) N a t a r a j a n ( 3 ) and K . B . P .

o r d e r e d N-groups. ( j ) See a l s o Kiisel

(1).

Rao ( l ) , ( 2 ) a l s o c o n s i d e r e d

345

9f Regular near-rings

f ) REGULAR N E A R - R I N G S Von Neumann r e g u l a r r i n g s p l a y a n i m p o r t a n t r 8 1 e i n r i n g t h e o r y . T h e y g e n e r a l i z e some p r o p e r t i e s o f n e a r - f i e l d s t o a m u c h w i d e r class of rings.

This concept n o t only transfers t o near-rings,

i t i s a l s o m o t i v a t e d by t h e f a c t ,

types o f near-rings

t h a t some o f t h e m o s t i m p o r t a n t

are r e g u l a r (see 9.154).

9.153 DEFINITION A n e a r - r i n g i s c a l l e d r e g u l a r i f Vn E N 9.154

3x

E

N:nxn

=

n

EXAMPLES R e g u l a r n . r . I s a r e o b v i o u s l y : ( a ) M ( 7 ) and M o ( r )

(Beidleman ( 1 0 ) ) .

( b ) Constant near-rings.

( c ) D i r e c t sums a n d p r o d u c t s o f n e a r - f i e l d s . (d) Integral planar near-rings

N ( s i n c e f o r n E N we c a n

f i n d X E N w i t h n = xn2 by 8.88 =

(el (N,t,*)

I n 9.153,

(b);

now ( n - n x n ) n

=

n 2 - n 2 = 0 g i v e s t h e r e s u l t (Mason ( 5 ) ) . f o r any group

( N , + ) a n d n*m:

n i f m f O O i f m = O

xn can be c o n s i d e r e d as a " p r i v a t e r i g h t i d e n t i t y "

and nx as a " p r i v a t e l e f t i d e n t i t y " f o r n.

n has

I f N E ~ , and

an i n v e r s e x t h e n n x n = n , o f c o u r s e .

9.155 REMARKS ( a ) I n 9.153,

n x and xn a r e i d e m p o t e n t .

( b ) By 9 . 1 5 4 ,

regular near-rings are not necessarily

abel ian. ( c ) Homomorphic images,

d i r e c t sums a n d d i r e c t p r o d u c t s

o f regular near-rings are regular.

7.33,

1.86 and 1.88,

By 9 . 1 5 4

every (zero-symmetric)

r i n g c a n b e embedded i n a ( z e r o - s y m m e t r i c ) regular near-ring.

nearsimple

9 . 1 5 4 ( a ) a l s o shows t h a t i n

general a regular nr.

(P,)-(P4)

(a),

has n e i t h e r t h e IFP n o r

(see 9.1 and 9 . 4 )

09 M O R E CLASSES OF N E A R - R I N G S

346

( d ) By 9 . 1 5 4 ( a ) , s u b n e a r - r i n g s o f r e g u l a r n r . ' s a r e n o t regular in general. N e v e r t h e l e s s , s e v e r a l c o n n e c t i o n s t o I F P - n r . ' s and t h e i r p r o p e r t i e s w i l l s h o w u p . We now c h a r a c t e r i z e r e g u l a r n e a r - r i n g s a n d d i s p l a y some o f t h e i r p r o p e r t i e s a f t e r w a r d s . 9.156 T H E O R E M (Beidleman ( l o ) , Ligh ( 7 ) ) . Let N < = > v n E N 3 e = e 2 E N : Nn = Ne.

EN,.

N i s regular

P r o o f . =+ : T a k e X F N w i t h n x n = n . T h e n Nn = N ( x n ) d o e s t h e j o b by 9 . 1 5 5 ( a ) . + = : T a k e n c N . Then N n = Ne

Ne, t h e r e i s some x E N w i t h x n = e . S i n c e N ' f l , , n E Nn = Ne, h e n c e n = y e f o r some y t N , a n d we g e t n = y e = y e e = = yexn = n x n .

f o r some i d e m p o t e n t e .

Since e

F

9.157 COROLLARY (Beidleman ( 1 0 ) ) . A r e g u l a r n e a r - r i n g with i d e n t i t y c o n t a i n s no n o n - z e r o n i l N - s u b g r o u p s . H e n c e we m i g h t l o o k a t r e g u l a r n e a r - r i n g s w i t h o u t n i l p o t e n t elements. 9 . 1 5 8 T H E O R E M ( L i g h ( 7 1 , Chao ( I ) , O s w a l d ( 9 ) ) . L e t N {Ol be a r e g u l a r n e a r - r i n g w i t h i d e n t i t y . E q u i v a l e n t a r e :

(a) N

h a s no n o n - z e r o n i l p o t e n t e l e m e n t s . ( b ) A l l i d e m p o t e n t s of N a r e c e n t r a l . ( c ) N i s a s u b d i r e c t product o f n e a r - f i e l d s . P ro o f . ( a ) * ( b ) h o l d s by 9 . 4 3 ( 6 ) . ( b ) + ( c ) : By 1 . 6 2 , N i s isomorphic t o a s u b d i r e c t product of s u b d i r e c t l y i r r e d u c i b l e n r . ' s . N i ( i E I ) . These Nils a r e r e g u l a r by 9 . 1 5 5 ( c ) , EY, a n d f u l f i l l t h e c o n d i t i o n ( b ) , = No

t o o . Let A : = n ( O : e ) , w h e r e e r u n s o v e r a l l i d e m p o t e n t s f 0 and 1 in N i . Since each e i s c e n t r a l ( 0 : e ) (and hence A ) a r e i d e a l s . I f ( 0 : e ) = I01 t h e n e = 1 , a c o n t r a d i c t i o n . By 1 . 6 0 , A f {0}

+

347

9f Regular near-rings

T a k e a E A , a f 0 . Now a x a = a f o r some x c N .1 . I f e 2 = e E N. t h e n a e = 0 , h e n c e e a = 0 , w h e n c e 1 e c ( 0 : a ) . ax i s idempotent by 9.155 ( a ) . I f ( 0 : a x ) = {O}, a x = 1 a n d e = e ( a x ) = ( e a ) x = Ox

=

If (0:ax) { a } , we g e t a E ( O : a x ) , hence a = ( a x l a = a ( a x ) = 0 , again a c o n t r a d i c t i o n . T h i s shows t h a t 0 and 1 a r e t h e o n l y 0 and n = n x n idempotents in N i . If n c N i i s =

0, a contradiction.

+

t h e n nx a n d x n a r e 0 and hence = 1 (by 9.155 ( a ) ) . Hence N i i s a n e a r - f i e l d . ( c ) * ( a ) i s t r i v i a l . The e q u i v a l e n c e ( b ) o ( c ) i s t r u e w i t h o u t t h e a s s u m p t i o n M E ? " , . T h i s r e s u l t ( and i t s proof

) show

9 . 1 5 9 COROLLARIES ( B e i d l e m a n ( l o ) , L i g h

( 7 ) , Heatherly ( 8 ) ,

Chao ( I ) , M a r i n ( I ) ) . ( a ) A r e g u l a r n e a r - r i n g whose i d e m p o t e n t s a r e c e n t r a l i s a b e l i a n , 2 - s e m i s i m p l e a n d an I F P - n e a r - r i n g ( 9 . 3 7 ) . ( b ) A r e g u l a r d g n r . whose i d e m p o t e n t s a r e c e n t r a l i s a semisimple r i n g . ( c ) A regular near-rinq N with i d e n t i t y 1 4 0 i s a nearf i e l d i f f 0 a n d 1 a r e t h e o n l y i d e m p o t e n t s i n N. ( d ) A r e g u l a r nr. w i t h D C C I whose i d e m p o t e n t s a r e c e n t r a l

is a finite direct

sun1

of n e a r - f i e l d s .

( e ) I n a r e g u l a r nr. w i t h i d e n t i t y whose i d e m p o t e n t s a r e c e n t r a l , e v e r y N-subgroup i s a l e f t i d e a l . ( f ) A r e g u l a r nr. w i t h i d e n t i t y i s i n t e g r a l i f f i t i s a near-f iel d. This gives another c h a r a c t e r i z a t i o n of r e g u l a r n e a r - r i n g s 9 . 1 6 0 T H E O R E M (Chao ( I ) ) Suppose N

= PIo nilpotent elements. N i s regular-

h a s no n o n - z e r o Na i s a d i r e c t summand

of N f o r each a t N . P r o o f . =, : By 9 . 1 5 9 ( a ) a n d ( e ) , N i s a b e l i a n a n d e a c h -__ Na s R N . B u t Na = Ne f o r some i d e m p o t e n t e , w h e n c e ( N , + ) = Ne i ( 0 : e ) by 1 . 3 3 .

09

348 e :

MORE CLASSES OF NEAR-RINGS

l e t L b e a l e f t i d e a l of N w i t h ( N , + )

For n E N ,

=

NnTL.

n

=

n.1

There a r e m E N a n d 1 E L w i t h 1 =

n m n + n l by 2 . 2 9 .

B u t nl

=

=

m n + l , whence

=

-nmn+n

E

NnnL

=

{O)

b v 1 . 3 4 . So n = n m n .

In Ligh - Utumi elements then N We a l s o mention with I N \ > 1 . F o r

( 1 ) i t i s shown t h a t i f N E ? ? ~ has i s r e g u l a r i f f nN = nNn holds f o r a n o t h e r r e s u l t o f Ligh ( 2 ) : L e t N e a c h n E N t h e r e i s e x a c t l y one x E

no nilpotent a l l n r N. be a d g n r . N with n x n = n

iff N is a near-field. R e g u l a r n r . ’ s w i t h one ( a n d hence a l l ) of t h e t h r e e c o n d i t i o n s s t u d i e d i n 9 . 1 5 8 a r e o b v i o u s l y of p a r t i c u l a r i m p o r t a n c e . They deserve a s p e c i a l n o t a t i o n . 9.161 DEFINITION A regular near-ring N i s called strongly N EN, a n d i f N f u l f i l l s t h e c o n d i t i o n s regular if { O } in 9.158.

+

9 . 1 6 2 T H E O R E M (Marin ( I ) ) . N ~ f l ~ ~i sm s ,t r o n g l y r e g u l a r i f f V n E N

~ x E N :n

P-r o o f . +: Take -

n

E

2 xn . N. Then n =

=

n x n f o r some x

E

N.

Hence

x n i s i d e m p o t e n t , hence c e n t r a l .

.

2 n = n x n = xnn = xn = : Let n E N. Then n = x n 2 f o r some x r N . Hence n 2 = x n 3 , a n d s o o n . Thus t h e r e c a n n o t be some k E N k-I f 0 , and N i s shown t o b e a with n k = 0 , b u t n n r . w i t h o u t n i l p o t e n t e l e m e n t s . Now n 2 = n x n 2 , whence ( n - n x n ) n = 0 , hence n ( n - n x n ) = 0 . So

1/Je g e t ( n - n x n ) ’ consequently n =

=

n(n-nxn) nxn.

- nxn(n-nxn)

=

0-0

=

0,

349

9f Regular near-rings

We r e m a r k t h a t M ( T ) a n d M o ( r ) f o r m e x a m p l e s of r e g u l a r , b u t n o t strongly regular near-rings. Integral p l a n a r near-rings are e x a m p l e s o f s t r o n g l y r e g u l a r n e a r - r i n g s . Many r e s u l t s on s t r o n g l y r e g u l a r n e a r - r i n g s c a n be f o u n d i n Mason (5). We p r e s e n t some o f t h e s e r e s u l t s : 9 . 1 6 3 T H E O R E M L e t N be s t r o n g l y r e g u l a r . ( a ) E v e r y p r i m e i d e a l of N i s maximal ( c f . 2 . 7 2 ) . (b)

\In

c N

3

X E

N : n = xn2

A

x is invertible.

( c ) Every N-subgroup o f N i s a ( t w o - s i d e d ) i d e a l . 2 ( d ) Every i d e a l I of N f u l f i l l s I = I . P r o o f . ( a ) L e t P be a p r i m e i d e a l a n d s u p p o s e t h a t P c M c N , M S N . I f m E M \ P t h e r e i s some X E M w i t h n

0 = m - x m L = ( I - x m ) m . By 2 . 6 1 a n d t h e I F P we g e t e i t h e r m E P ( a c o n t r a d i c t i o n ) o r 1 - x m E P , whence 1 E M , again a contradiction. ( b ) n = a n 2 f o r some a E N and a = z a 2 f o r some z . L e t x : = I - z a + a . Then x n 2 = ( I - z a + a ) n 2 = n 2 - z a n 2 + a n 2 = 2 = n2-za(an ) n + n = n2-n2+n = n , a n d xa = ( I - z a + a ) a = 2 = a 2 . I f x i s c o n t a i n e d i n a maximal i d e a l = a-a+a M, a 2 E M , b y 2 . 7 2 hence a E M . SO 1 E M , a c o n t r a d i c t i o n . Hence x i s a u n i t . ( c ) Lde k n o w a l r e a d y ( 9 . 1 5 9 ( e ) ) t h a t e v e r y N-subgroup S of N i s a l e f t i d e a l . I f s E S and n E N then s

E

Ns

Ne f o r some i d e m p o t e n t

=

e , hence s

=

ne.

N o w e i s c e n t r a l . Hence s n = Ken = n n e E Ne = N s s S . 2 ( d ) Of c o u r s e , I c I . I f i E I t h e n t h e r e i s some X E

N with i

=

xi2

=

(xi)i

E

I

2

.

I n f o r m a t i o n c o n c e r n i n g t h e r a d i c a l s o f a r e g u l a r n r . was o b t a i n e d i n J o h n s o n ( 6 1 , which we s t a t e w i t h o u t p r o o f . 9.164

T H E O R E M Let N

( a ) %,,2(~)

=

be r e g u l a r . Then {OI

89 MORE CLASSES OF N E A R - R I N G S

350

#

( b ) Every minimal l e f t i d e a l

{O}

i n N i s a minimal

N - s u b g r o u p. ( c ) I f N has t h e DCCN t h e n N i s r e g u l a r i f f N i s 2-semis i m p l e w i t h n EN^ f o r a l l n E N . ( d ) I f N has t h e DCCN t h e n maximal i d e a l s c o i n c i d e with orirnitive ideals. S t i l l more

i n f o r m a t i o n c a n be found

( 1 ) and Ramakotaiah ( 3 ) .

i n Choudari-Goyal

We s h a l l c o n s i d e r r e g u l a r n e a r - r i n g s

o f t h e t y p e M (1') i n 9 9 ( h ) . S

a ) TAME NEAR-RINGS

I n t h i s c h a p t e r we i n v e s t i g a t e a c l a s s o f n r . ' s

which i s c l o s e l y

r e l a t e d t o c o m p a t i b l e n e a r - r i n g s as d e f i n e d i n 7 . 1 3 7 . o f t h e r e s u l t s i n t h i s t h e o r y a r e due t o S.D.

D E F I N I T I O N An N - g r o u p

r

i s c a l l e d __ tame i f e v e r y No-

subgroup o f NT i s an i d e a l . a faithful

tame N-group

Hence i n t a m e N - g r o u p s ,

Scott. For the

1.34.

following definition cf. 9.165

Most

r

A n e a r - r i n g N i s ___ tame i f N has

( t h e n N i s c a l l e d t a m e o n Ni'-).

i d e a l s and No-subgroups c o i n c i d e .

There a r e several examples which work f o r d i f f e r e n t reasons. 9 . 1 6 6 EXAMPLES (a) I f N i s 2-primitive (since

,,,r

t h e n N i s tame on

on

has no n o n - t r i v i a l

l e t S ( r ) be the nr. a d d i t i v e l y

(b) I f Inn(r)sSsEnd(r), generated by S.

I f S = I n n ( r ) then

S = E n d ( 1 ' ) t h e n S(T)

=

r,

S(T)

E(1'), and so on.

hence z e r o - s y m m e t r i c and group

Nr

No-subgroups i n t h i s case). =

I(T), if

S ( r ) i s d.g.,

S ( r ) i s tame on t h e S ( T ) -

since every S(r)-subgroup

A i s normal

(because A i s i n v a r i a n t under a l l i n n e r alAtomorphisms), hence an i d e a l by 6.6.

351

9 g T a m e near-rings

( c ) L e t l? be a v a r i e t y o f ( 2 - g r o u p s a n d A EU. Then ( A , + ) 21 i s a tame A and P ( A ) - g r o u p , and P ( A ) i s tame

[XI-

on A .

This holds since B 5

p(b)E B for all

P F

A implies t h a t Po(A) P o ( A ) a n d b E B . Hence a l l f i n i t e

s u m s o f t h e s e e l e m e n t s a r e i n B y w h e n c e B !A by 7 . 1 2 3 . B u t t h e e l e m e n t s o f P ( A ) a r e C o m p a t i b l e by 7 . 1 2 2 ; c c n s e q u e n t l y B i s an i d e a l of p ( A ) A . ( T h e same a r g u m e n t s a r e a p p l i c a b l e f o r A*[x] i K s t e a d o f P ( A ) , w i t h t h e o n l y e x c e p t i o n t h a t Al'lx] a c t s n o t n e c e s s a r i l y i n a f a i t h f u l m a n n e r on A . ( d ) More g e n e r a l l y , e v e r y n e a r - r i n g N b e t w e e n P ( A ) a n d C ( A ) i s tame on A ( c f . a l s o 7.140 and 9 . 1 6 8 ! ) .

( e ) Every ring-module i s tame. Every r i n g w i t h i d e n t i t y i s tame. ( f ) Many m o r e e x a m p l e s w i l l come u p b y 9 . 1 6 8 a n d 7 . 1 3 7 ! S c o t t r e m a r k e d t h a t S(T

i n 9 . 1 6 6 ( b ) i s a l s o ( b y 9 . 1 6 8 we w i l l s a y : " m o r e o v e r " ) c o m p a t b l e on r : I f Y E r a n d i f s E S o r - s c S t h e n e i t h e r s(y+6 - s y = s y + s G - s y = s ( y + i s - y ) o r = s t ~ " o r a l l S t r . N o w : n n ( r c _ S , h e n c e t h e r e i s some n E S(r) w i t h i t r . T h i s e x t e n d s t o a l l f i n i t e is(),+fi) - s y = n 6 f o r a1 s u m s o f e l e m e n t s o f S . H e n c e S ( 1 ' ) i s c o m p a t i b l e on r . T h e s i m i l a r i t y between the c o n c e p t s "tame" and " c o m p a t i b l e " i s r e v e a l e d by

~v~

and NT i s u n i t a r y t h e n 9 . 1 6 7 PROPOSITION ( S c o t t ( 1 7 ) . I f N Nr i s t a m e i f f f o r a l l y , 6 E r a n d n E N o t h e r e i s some m E No w i t h n(y.6) - ny = m 6 . Pr o o f . I f Nr i s u n i t a r y a n d t a m e t h e n e a c h N o & i s an i d e a l o f N T c o n t a i n i n g 6 . Hence n(y.6) - ny E N o & . C o n v e r s e l y , s u p p o s e t h a t n 5 r . I f y E r and ~ E t hA e n y + 6 - y = l ( y + 6 ) - l yN O = my E n ( f o r some Hence A i s n o r m a l . I f y E then n(y.6) - n y = n C ( y + 6 ) - noy a E No).

m E

No).

r, =

6 E A and n E N a 6 E A ( f o r some

89

352

MORE CLASSES OF NEAR-RINGS

T h i s i s shown by t h e f o l l o w i n g p i c t u r e ( c f . t h e d i a g r a m a f t e r 7.136 !)

tr

9.168

C O R O L L A R Y Every u n i t a r y c o m p a t i b l e N-group i s t a m e .

( 2 0 ) a n d (21), S.D. S c o t t g o e s o n t o t h e s t u d y o f a t y p e o f n e a r - r i n g s between t a m e a n d c o m p a t i b l e n e a r - r i n g s : In ( 1 7 ) ,

9 . 1 6 9 DEFINITION - L e t k b e a c a r d i n a l number. A n N - g r o u p r i s k - t a m e on m~

Nri f

for all n

No w i t h n ( y + S i ) - n ( y )

of a t most k e l e m e n t s S i

=

t

No and

m(si)

Y E

r

t h e r e i s some

f o r any c o l l e c t i o n

i n I'.

H e n c e we g e t f o r u n i t a r y N - g r o u p s : + k - t a m e + ... =. 2 - t a m e + I - t a m e c o m p a t i b l e =r

...

=

tame.

We c i t e some r e s u l t s on 2 - t a m e n e s s w i t h o u t p r o o f .

-9 . 1 7 0 T H E O R E M ( S c o t t ( I ) ,

(81,

(20)). L e t NE%,,,-,~,

be 2-tame

on t h e u n i t a r y N - g r o u p r. ( a ) I f h i s an N-endomorphism o f r then i d - h i s an N - endorno r p h ism , t o o ( b ) I f h i s an N-automorphism of r and i d - h i s a n N a u t o m o r p h i s m , t o o , t h e n - i d i s an N-automorphism.

.

9g Tame near-rings

353

( c ) If A u t N ( T ) c o n t a i n s a f i x e d - p o i n t - f r e e element t h e n - i d i s an N - a u t o m o r p h i s m . ( d ) I f - i d i s an N-endomorphism t h e n ( E n d ( r ) , + , " ) i s a r i n a and E n d ( r ) = E ( T ) . ( e ) I f - i d i s a n N-endomorphism and i f r i s f a i t h f u l w i t h o u t e l e m e n t s o f o r d e r 2 t h e n PI i s a r i n g a n d NI' i s a n N-module. ( f ) If

N r has

r

DCCI a n d A C C I and i f

=

Ali

...; A r

=

where t h e A i l s a n d E ' s a r e indecomposable j i d e a l s of N r t h e n r = s and t h e r e i s a p e r m u t a t i o n p o f { I ,... , n } w i t h A l 1 E p ( , ) ,... , A r = N E p ( r ) . = E,;

...+

E,,

( "Krull -Schmidt-Theorem"). ( 9 ) I f N i s 2 - p r i m i t i v e on I' a s w e l l t h e n N i s a r i n g o r N i s d e n s e i n Mo(I') ( i . e . G

=

(id} i n 4.60).

( h ) I f n o n o n - z e r o homomorphic image o f N i s a r i n a ( N i s then c a l l e d r i n g - f r e e ) and i f N h a s t h e D C C L then N i s f i n i t e .

We n o w m e n t i o n some e l e m e n t a r y f a c t s a b o u t tame n e a r - r i n g s 9.171

PROPOSITION L e t a l l a p p e a r i n g N - g r o u p s be u n i t a r y ( a ) L e t N be tame o n r a n d A a N r . Then N i s tame o n A (b)

and on rlA. I f N i s tame o n T i

r:

=

( i c I ) t h e n N i s tame on

Ori.

i €1 ( c ) I f N i i s tame o n T i

(i

E

I ) then

II N i

i s tame o n

i €1 @Ti.

iEI ( d ) I f N i s 2-semisimple t!ien N i s tame. P r o o f . ( a ) f o l l o w s from 1 . 3 0 , ( b ) i s s t r a i g h t f o r w a r d since A 5 r implies ( . . . 0 , ~ , ~ , . . . ) ( . . . , 6 i , . . . ~ NO

=

,... ) I (. ..

( . . . , 0 , 6 i ,O

E

A , whence A = @ A i

i €1

=

for

A} ( c ) f o l l o w s f r o m t h e f a c t t h a t i f N i i s tame o n r i t h e n N i s tame on I' ( b y ( . . . , n i ,... ) y 1. : = n1 . y1 . ) , t o o .

Ai = { S i

E

Ti

Now a p p l y ( b ) .

,9,.. .)

E

354

89 MORE CLASSES OF NEAR-RINGS ( d ) I f T i ( i E I ) r e p r e s e n t a l l non-N-isomorphic N - g r o u p s o f t y p e 2 t h e n N i s tame o n t h e i r d i r e c t sum ( S . D . S c o t t ) .

A s p l i t t i n g of N d o e s n o t i n d u c e a s p l i t t i n g of

Nr

i n general.

B u t i t d o e s f o r tame n r . ' s . 9 . 1 7 2 THEOREM ( S c o t t ( 1 7 ) . Let N f be tame, u n i t a r y a n d f a i t h f u l .

If N f = A;B

i s t h e d i r e c t sum o f t h e i d e a l s I a n d J t h e n w i t h (0:A) = J a n d ( 0 : B ) = I .

P r o o f . In N = I + J , 1 decomposes as 1 = e l + e 2 , where e 1 ' e 2 a r e c e n t r a l o r t h o g o n a l i d e m p o t e n t s . Let A: = e l l ' a n d B : = e 2 r . N o w J = ( o . A ) , s i n c e JA = = J e l i ' = { o } a n d i f n E ( o : A ) , n = ne + n e 1 2' thep ne:y = 0 h o l d s f o r a l l y F r , b e n c e n e , = 0 a n d n = n e 2 c J. S i m i l a r l y , I = ( o : B ) . Next we show t h a t A 5N I' ( t h e n A g N 1 ' ) . N A = N e l r

=

0

A f o r a l l 6 E A we show t h a t A i s c l o s e d u n d e r a d d i t i o n . L e t e l y l , e l r 2 E A . Then e 2 ( e l y l + e l y 2 ) - e 2 e l y 1 = m e l y 2 f o r some m t N. S i n c e e 2 e l y 1 = 0 we g e t e 2 ( e l ~ l + e l y 2 =) = mely2. M u l t i p l i c a t i o n by e 2 g i v e s e 2 ( e l y , + e , y 2 ) = = e2rnely2 = e e m y 2 = 0 , whence e , y l + e l y 2 E A. 2 1 C o n s e q u e n t l y 5 --N 1', a n d s i m i l a r l y 6 c N r . O b v i o u s l y =

elNrc_elr = A.

r

=

=

ele2y2

Since ( - I ) &

E

0

A+B.

If e l y l =

=

0 . Hence

0

e2y2

r

=

E

A n B then

elyl

=

s

elyl

=

A+B.

W i t h o u t p r o o f we m e n t i o n

n, h a v e

M ( S c o t t ( 1 7 ) . Let N c N O n 9.173 T H E O R El e t N r be u n i t a r y , tame a n d f . g . ;

is f.g.,

the ACCL, and then a n y i d e a l of N T

too.

Next we l o o k , h o w f o r S c - N t h e l e f t i d e a l 'looks l i k e .

tS>

v. g e n e r a t e d

by S

355

9g Tame near-rings

9.174

PROPOSITION ( S c o t t ( 1 7 ) .

L e t N T be tame and M

Then

N. NO

II.

<M>Ry = M Y f o r a l l y If y

Proof.

E

r

I ,

hence


My

~

Thus

NO

r

cE

(My:y)

MY

then

Since Ms(My:y),

<MzK. C ( M y : y ) ,

whence

C M > 'My. ~ ~

9.175

PROPOSITION ( S c o t t ( 1 7 ) .

L e t N b e tame and M , ,

N o - s u b g r o u p s o f M w i t h MIM 2 . . . M k

<M1>R<M2>R.. -~ Proo

f.

.R

=

TO}

{Ol. Then

=

L e t k = 2 ( a n d t h e n p r o c e e d by i n d u c t i o n ) . Now b y 9 . 1 7 4 ,

to1 f o r a l l y i s f a i t h f u l , < M , i e , < M 2 > * , = {U}.

9.176

<M,>% M2y = M 1 M 2 y =

COROLLARY ( S c o t t ( 1 7 ) .

9.177

Proof.

n

E

<M2>Ry = I

1'.

Since

The sum L o f a l l n i l p o t e n t l e f t

N we g e t Ln

=

ELi".

i s a.1 i d e a l o f N .

0

E a c h Lin

i s an N - s u b -

g r o u p o f N and i s n i l p o t e n t , s i n c e N L c L . 1

R

i s a n i l p o t e n t l e f t i d e a l by 9.176,

< L 1. n j L C C L ~=

Nr

too.

o f a tame n e a r - r i n g N = N

For

Suppose

N i s n i l p o t e n t and i f N

5

i s nilpotent,

THEOREM ( S c o t t ( 1 7 ) .

i d e a l s Li

If M

N O

i s tame t h e n < M > R

be

a s we1 1 .

t h a t N i s t a m e on N T . =

. . . ,Mk

Hence whence

L. T h e r e f o r e L n S L .

These a r g u m e n t s a l s o show: 9.178

P R O P O S I T I O N (Lyons-Meldrum

N tame and B S A

(2)).

Nr b e u n i t a r y ,

Let

= N ~ .T h e n t h e i d e a l g e n e r a t e d b y B i n A

i s t h e same a s t h e o n e g e n e r a t e d i n

r

and

i s g i v e n by

CNB. BEB Another i n t e r e s t i n g f a c t i s the following. completely non-abelian o f an i d e a l o f over

A.

C a l l an N-group

r

i f each n o n - z e r o homomorphic image A

Nr i s e i t h e r n o n - a b e l i a n o r no n E N d i s t r i b u t e s

$9 MORE CLASSES OF NEAR-RINGS

356

I t f o l l o w s f r o m 2.23

distributive.

If

r

t h a t t h e i d e a l l a t t i c e o f NT i s t h e n

r

i s non-abelian and simple then

i s an

example o f a c o m p l e t e l y non-abel i a n I ( r ) - g r o u p . 9.179

I f ,,,T i s t a m e a n d c o m p l e t e l y

PROPOSITION ( K a a r l i ( 5 ) ) .

9QN

n o n - a b e l i a n t h e n L 1 y n L 2 y = ( L 1 L~2 ) y f o r a l l L , , L 2 and y

E

Proof.

I'. ( L 1 " L 2 ) y ~ L 1 y n L 2 yi s t r i v i a l .

Conversely,

if

a,B E L 1 y n L 2 y a n d n E N t h e n a + B - a - B a n d n ( a t g ) - n g - n a a r e i n ( L 1 ~ L 2 ) y ,s i n c e ( L 1 n L 2 ) y i s an No-subgroup,

consequently an i d e a l

o f N T , The l e f t i d e a l g e n e r a t e d b y a l l a+B-a-B a n d n ( a + B ) - nR - n n w i t h ~ , B E L , ~ A L a~n dY EN

Nr i s c o m p l e t e l y

coincides w i t h L 1 y n Lzy, since non-abelian. Now we

take

Hence L l y n L z y c ( L , o L z ) y .

a l o o k a t t h e s t r u c t u r e o f a tame n e a r - r i n g .

O b v i o u s l y we g e t 9.180

P R O P O S I T I O N I f NT i s tame t h e n NT i s 0 - p r i m i t i v e i f f i t

i s 2-primitive.

From t h i s i t l o o k s q u i t e p l a u s i b l e t h a t J o ( N ) f o r a tame n e a r - r i n g N. 9.181

22(N) holds

We w i l l r e t u r n t o t h i s q u e s t i o n l a t e r o n L e t N b e tame on N T , M

PROPOSITION ( S c o t t ( 1 7 ) .

s NN a n d 0

r.

Then M+(O:y) a N. R P r o o f . By 2 . 1 5 , M + ( O : y ) 5 -Y E

=

n,n' E M ,

n(mY+n'y)

-

N. I f

m E M ,

a

E

(O:y),

NO

r,

n n ' y E M ~ , s i n c e My <=

whence

NO My P N r .

Therefore n(m+n') - nn'

n(m+atn') - n n ' = n(m+a+n')

9.182 -

nn'

E

( 0 : ~ )+ M

t

Hence

"(N)

E

FI+(O:y),

n(mtn')

t

and

n(mtn')

-

( 0 : ~ )= M + ( o : ~ ) .

-

THEOREM ( S c o t t ( 1 7 ) , L y o n s L e t N b e t a m e w i t h DCCL.

-

Meldrum ( Z ) , Meldrum ( 1 3 ) ) .

Then J2(N)

= m ( N ) = To(N)

=

...

i s nilpotent.

=$2(N).

357

9g Tame near-rings

Proof.

Suppose & ( N )

i s not nilpotent.

Let L S ~ N ,

L S j 2 ( N ) be m i n i m a l f o r b e i n g n o n - n i l p o t e n t . I f 2 2 = L. < L > < L t h e n L 2 , h e n c e L , i s n i l p o t e n t . So < L R S i n c e LO = ( 0 1 , Z o r n ' lemma p r o v i d e s u s w i t h some L ' 4 N, L =

maximal f o r h a v i n g L L ' = { O } .

L"'

=

a contradiction.

{O},

? h e DCCL g u a r a n t e e s with L'< L"

d,N

t h e e x i s t e n c e o f some L "

I f L ' = N,

,

but

.

w i t h o u t l e f t i d e a l s s t r i c t l y between L ' and L "

L" / L

i s a simple No-group.

w i t h L2/LI

L" /L',

:

If K < L 2 , K

OR

N, then

K+L1 = L 2 and L 2 / L 1

Since

fl

YET

(o:y)

because o t h e r w i s e R 1' K/KnL1, a contradiction t o

Kg L

Suppose t h a t M

5

NO

N i s strictly

Nr b e f a i t h f u l a n d t a m e . { O l , t h e r e i s some y

Let

(o:r)

=

SRN

choose one w i t h m i n i m a l Lz.

g N o

the minimality o f L2. between L 1 and L 2 .

Then

Among a l l p a i r s L 1 , L 2

=

~ w ri t h

( o : y ) n L 2 < L 2 , w h e n c e ( o : y ) n L 2 s L 1 b y w h a t we h a v e j u s t seen b e f o r e .

9.181

Hence L 2 n ( M + ( o : y ) )

t e l l s us M+(o:y)

M + ( o : y ) n L 2 (see p. 4 8 ) i s a

=

B u t t h e n ( o : ~ ) AL 2 2 L 1 <

l e f t i d e a l o f N.

M deN, a c o n t r a d i c t i o n . thus L " /L' and

L 2L " 5 L L ' 2

L =

=

(0:L"

0

too.

/L')

L(L"

.

M and

Hence L z / L 1 i s N o - s i m p l e ,

i s No-simple,

Since L < &(N),

"N.

We g o b a c k t o

L.

t o } , whence L L " 5 L ' 2 C o n s e q u e n t l y L s (0:L" ) , a n d =

) as w e l l .

H e n c e we g e t L L "

=

{0},

i n c o n t r a d i c t i o n t o t h e m a x i m a l i t y o f L ' , and &(N) i s shown t o be n i l p o t e n t . The r e s t f o l l o w s f r o m 5.59

This r e s u l t g e n e r a l i z e s 7.58 and

9.77.

(b),

and 5.60.

f o r instance,

as w e l l as 7.127

P s f a r a s t h e a u t h o r k n o w s , n o t a m e n r . tl i s k n o w n

i n which the Jacobson-type r a d i c a l s partial result in this direction i s

7"

do n o t c o i n c i d e .

A

358

39

MORE CLASSES OF NEAR-RINGS

9.183 P R O P O S I T I O N L e t N T b e t a m e w i t h a t l e a s t o n e

{O}.

that (o:y) =

Ny+A,

such

= J2(N). {O} a n d N A i s o f t y p e 0 t h e n c o n s i d e r

whence J o ( N )

If (o:y) =

h:

r

Then e v e r y N-group o f t y p e 0 i s tame,

hence o f t y p e 2, Proof.

Y E

where 6 i s a g e n e r a t o r o f NA.

ny+n6,

Since (o:y) =

{O},

h i s w e l l - d e f i n e d and an

N-endomorphism such t h a t A:N

N /ker

N - g r o u p i s t a m e b y 9.171 ( a ) .

Now a p p l y 9.180.

h.

The l a t t e r

Tame n e a r - r i n g s h a v e a v e r y c l e a r s i t u a t i o n c o n c e r n i n g c h a i n conditions.

We m e n t i o n w i t h o u t p r o o f a r e s u l t ,

us of

5.50 ( c ) .

9.184

T H E O R E M ( S c o t t (17).

If

t h e n N has DCCN,

DCCL,

I n t h i s case,

NrVo,n,

which reminds

i s tame and has t h e

A C C N a n d ACCL a s w e l l .

c e r t a i n minimal and maximal o b j e c t s e x i s t .

The

f i r s t p a r t o f t h e n e x t r e s u l t i s o b v i o u s b y t a k i n g A = Ny f o r some a a o r o o r i a t e y

E

r;

t h e second p a r t i s mentioned w i t h o u t

proof.

9.185 P R O P O S I T I O N ( L y o n s - M e l d r u m ( 2 ) ) .

Nr

L e t N b e tame on

w i t h DCCL. ( a ) I f Nr' f i n

r

(b) If N

{O}

t h e n t h e r e i s some m i n i m a l N - i d e a l

such t h a t

A

i s o f t y p e 2.

c ? ? o o ~ lt h e n N T h a s s t r i c t l y m a x i m a l N - i d e a l s .

This turns our a t t e n t i o n t o minimal N-ideals.

9.186 D E F I N I T I O N T h e sum o f a l l m i n i m a l i d e a l s o f Nr i s c a l l e d t h e socle o f Nr and d e n o t e d b y soc ( N T ) . The s o c l e o f NN ( = t h e sum o f a l l m i n i m a l

l e f t i d e a l s o f N) i s s i m p l y

denoted by soc(N). It follows

f r o m 9.185 ( a ) t h a t s o c ( N r )

assumptions.

{O} under these

T h e s o c l e o f Nr c a n s o m e t i m e s b e u s e d a s a t e s t f o r

s e m i s i m p l i c i t y o f N ( s e e 9.188 ( b ) ) .

By 2.48 we c a n c o n c l u d e

3 59

9g Tame near-rings

Nv

9.187

PROPOSITION I f N cno and N T ~ t h e n ~ o c ( ~ i1s ) t h e d i r e c t s u m of m i n i m a l i d e a l s o f Nr.

9.188

T H E O R E M ( L y o n s - Meldrunl ( 2 ) , S c o t t ( 1 7 ) . L e t N ~ n , n / h ~

h a v e D C C N a n d be tame on t h e u n i t a r y N - g r o u p

r.

g2(N)~

( a ) If A s r f u l f i l l s = {OI t h e n A c s o c ( N r ) . N r i s f a i t h f u l and R ( N ) a n y o n e o f t h e r a d i c a l s of N mentioned in 9 . 1 8 2 t h e n R ( N ) = { @ } + > S O C ( ~ ~= ) r . = Y 2 ( N ) 6 = {O}. P r o o f . ( a ) I f 6 E A t h e n 6 :i N6 a n d By 3 . 1 4 ( a ) , "5 i s a nionogenic N / 7 2 ( N ) - g r o u p . S i n c e N/'J2(N) i s 2 - s e n i i s i n i p l e , N6 i s t h e sum o f minimal i d e a l s o f N r b y 5 . 3 4 ( a ) . Hence As,' N G s ~ o c ( ~ r ' ) . AcA ( b ) By 9 . 1 8 2 , a l l r a d i c a l s i n N c o i n c i d e i n t h i s case. I n particular, R ( N ) = Y 2 ( N ) . If t h i s i s zero, R ( N ) r = 101, whence r s s o c ( r ) b y ( a ) , h e n c e r = S O C ( ~ ~ )C. o n v e r s e l y , r = S O C ( ~ ~i m) p l i e s t h a t NI' i s t h e sum o f n i n i m a l i d e a l s o f N T . S i n c e N r i s tame a n d f a i t h f u l , R ( N ) = J 2 ( N ) = {O}.

( b ) If

12(N)N6

S c o t t p r o v e s a r e s u l t f o r t h e s o c l e o f a tame n e a r - r i n g . We p r e s e n t t h i s r e s u l t w i t h o u t p r o o f .

S.D.

9.189

PROPOSITION (Scott ( 1 7 ) .

on a unitary N-group soc(N)

=

(A:r).

If N

E$,,V,

h a s D C C L a n d i s tame

t h e n t h e r e e x i s t s some

s N r with

Hence s o c ( N ) g N a n d N / s o c ( N ) i s tame on

rln. F i n a l l y we a r e g o i n g t o i l l u s t r a t e t h e s e r e s u l t s f o r t h e tame ( e v e n c o m p a t i b l e ) n e a r - r i n g s S(r) o f 9 . 1 6 6 ( b ) . The p r o o f s c a n be f o u n d i n f u l l d e t a i l i n Meldrum ( 1 3 ) , c h . 1 0 ( t h e f i r s t r e s u l t c a n be shown a s i n 7 . 6 6 ) . 9 . 1 9 0 T H E O R E M (Lyons-Meldrum ( 3 ) ) . Let

r

be a g r o u p . T h e n

i s n o n - a b e l i a n t h e n S(r) i s d e n s e i n M o ( r ) i f f r i s s i m p l e . In p a r t i c u l a r , S ( r ) = Mo(r) i f r i s f i n i t e , n o n - a b e l i a n and s i m p l e .

(a) If

r

89

360 (b) If A

SNr

MORE CLASSES OF NEAR-RINGS

i s minimal

then e i t h e r A i s a b e l i a n or p e r -

fect. ( c ) I f a i n ( b ) i s a b e l i a n t h e n e i t h e r A i s an e l e m e n t a r y abelian p - g r o u p o r A i s d i v i s i b l e ( c f . the f i r s t 6 l i n e s on p.

242 ! ) .

( d ) I f A i n ( b ) i s p e r f e c t then S ( r ) / ( o : A )

i s isomorphic

t o a dense subnear-ring o f M o ( A ) . ( e ) By ( b ) , s ~ c ( ~ ( ~i s) rt h)e d i r e c t sum o f a n a b e l i a n i d e a l A a n d a p e r f e c t i d e a l II. (f) I f S(T) h a s t h e D C C N t h e n X i s f i n i t e a n d t h e r e e x i s t s an i d e m p o t e n t e of S ( r ) w i t h e r = II a n d en = for a l l VETS.

The r e a d e r w i l l have o b s e r v e d t h a t t h i s a r e a has a l o t t o d o w i t h many o t h e r t o p i c s i n n e a r - r i n g a n d g r o u p t h e o r y , so i t i s a l r e a d y f o r t h i s r e a s o n a very i n t e r e s t i n g s u b j e c t . Much more s h o u l d be a v a i l a b l e soon i n a f o r t h c o m i n g paper of S . D . S c o t t on tame n e a r - r i n g s .

IT

9h Bicentralizer near-rings

36 1

h ) BICENTRALIZER N E A R - R I N G S

We now r e t u r n t o t h o s e n e a r - r i n q s w h o s e c o n s i d e r a t i o n s t a r t e d i n

1.4

(a),

was b r o u o h t i n t c l b l o o m i n 4 . 5 2

and matured i n $7 ( a ) ,

namely n e a r - r i n g s o f t h e M S ( r ) - t y p e . I f N i s a n r . a c t i n g o n an N - g r o u p r i t i s c u s t o m a r y , a n d i n accordance w i t h r i n g theory, t o c a l l EndN(r) the c e n t r a l i z e r (4.10) o f Nr, because i t c o n t a i n s t h o s e endomorphisms h o f r w h i c h "commute" w i t h t h e a c t i o n o f N v i a h ( n y ) = n h ( y ) . I f S c E n d N ( T ) i n 1.4

then M S ( r ) , as defined f

on

r

w h i c h "commute"

( a ) , c o n s i s t s o f t h o s e mappings

w i t h some e l e m e n t s S o f E n d N ( r ) w h i c h

i n t u r n "commute" w i t h N,

i.e.

fos = sof f o r a l l s

E

S.

Hence

MS(T) m i g h t be c a l l e d a "double c e n t r a l i z e r " o r " b i c e n t r a l i z e r " of

Nr,

a n d we f o l l o w a n a d v i c e o f G .

creatures "bicentralizer near-rings"

Betsch t o c a l l these i n s t e a d o f t h e commonly

used " c e n t r a l i z e r near-rings",

I f we s t a r t w i t h some g r o u p G o f a u t o m o r p h i s m s o f a g r o u p

r

and

MG(r) t h e n we m i g h t f o r m i t s c e n t r a l i z e r F : = A u t M ( r ) I f Nr i s m o n o q e n i c we g e t ( s e e 9 . 2 2 6 ) MG(r) = Mc(r) i n t h i s c a s e ( " c l o s u r e p r o p e r t y " ) . T h i s f o r m N :=

and compare M G ( r ) w i t h M c ( r ) . also motivates the "bi"

i n t h e t i t l e o f t h i s c h a p t e r : e v e n i f we

s t a r t w i t h some G G A u t ( r ) we g e t M G ( r ) a s a b i c e n t r a l i z e r nearring. The k n o w l e d g e o f M S ( T ) , f r e e automornhisms of

w h e r e S i s __ n o t a group o f f i x e d - p o i n t N f ( s e e 5 7 a ) f o r t h i s c a s e ) was p u s h e d

f o r w a r d i n t h e l a s t few y e a r s m a i n l y b y p a p e r s o f B e t s c h , Maxson-Smith,

Meldrum, Oswald and Ramakotaiah-Rao.

proofs i n t h i s area are p r e t t y long, available,

S i n c e many

quite technical,

but easily

we o n l y c i t e a n u m b e r o f r e s u l t s .

We now s t a r t w i t h t h e m o s t g e n e r a l c a s e ,

i n w h i c h S i s some

a r b i t r a r y subset of

E n d N ( r ) . A l t h o u g h many r e s u l t s c a r r y o v e r

t o t h e g e n e r a l case,

we c o n f i n e o u r a t t e n t i o n t o t h e s i t u a t i o n

where S c o n t a i n s t h e z e r o endomorphism 0 f o r t h e sake o f brevity.

H e n c e we make t h e

§ 9 MORE CLASSES OF NEAR-RINGS

362

CONVENTION A l l n e a r - r i n g s

MS(T) i n t h i s c h a p t e r have U c S,

hence a r e zero-symmetric.

( r ) i s a zero-symmetric nearS S u r p r i s i n g l y , a l l s u c h n e a r - r i n g s a r i s e as

Since i d c M S ( r ) i n any case, ring with identity. some M S ( T ) ,

M

Maxson p o i n t e d o u t :

as C.

9.191 PROPOSITION I f N i s a z e r o - s y m m e t r i c n e a r - r i n g w i t h a n d some S G E n d N ( T ) w i t h

i d e n t i t y t h e n t h e r e i s some

N

-

r

MS(T).

Proof. -__-

x+xn

i s even a monogenic MS(T)-group.

r:= (N,+)

Take

and S:=

are the right translations.

fa: T h e map h : N + { f a l a a near-ring {

nf(1)

=

if

r

=

given by a + f a ,

=

Let f(1)

f

t

very strong results, soon,

NI,

= na f o r

N1

=

Hence M S ( r ) i s as g e n e r a l conditions

t

n,meN}.

N ={fala

For a E N l e t

i

Now M

{ftM(T)lf(tim)

=

=:a.

= f(n1)

Then f ( n )

nf(m) =

as i t can be, a n d one c a n n o t e x p e c t

MS(r)

I n ( 2 ) Maxson and Q s w a l d c o n s i d e r e d

i s regular,

simnle,

p r i m i t i v e and

t o connruence r e l a t i o n s i n

o r i f S * i s a union o f groups.

A n o t h e r p a p e r s o l v e s t h e q u e s t i o n a s t o when M S ( T ) h a p p e n s t o be a n e a r - f i e l d . f o r some s

E

For t h a t ,

S and l e t

sS

l e t

y 1 R sy 2 (y1,y2tT) i f s ( y l )

be the equivalence closure o f

MS(r)

9.192 THEOREM ( M a x s o n - M e l d r u m (1)).

-

near-field (a) A l l

(b)

r

with

S

y2

=

RS.

= Endjr)

i s a

iff

elements i n

r*

are equivalent w.r.t.

has p r o p e r s u b g r o u p s Ti

y

E

r”

s

E

S either s(ri)

( i

E

=

{o}

%s’

I ) such t h a t each

i s c o n t a i n e d i n e x a c t l y o n e Ti

and k e r s = {o},

=

M S ( r ) , h e n c e f = f a a n d we g e t

and s t u d i e d t h e c o n n e c t i o n s

r

S( r )

MS(r).

o f course.

under which

Sy f o r some Y E

i s obviously

isomorphism (see 1.86/1.89).

f M ~( r ) l f o s n = s n - f }

for a l l

N-N,

where sn:

n-t an be t h e c o r r e s p o n d i n g l e f t t r a n s l a t i o n .

N-t N,

=

{snln L N},

o r S(ri)

=

r J.

and f o r each f o r some j

E

K

P“s(r),

363

9 h Bicentralizer near-rings

u

Sy (with well-ordered J ) , and y 1. r'l1 .iEJ i then the set-of a l l 6 . where 6k ( k E J ) fulfills J' r = ySSk a n d s?,, = slyn ( s , s ' ~ S ) + s 6 , = sSn

(c) If

r

=

kcJ

for a l l m s n , is just T i .

In the paper mentioned i n 9.192, the authors give the followinq 9.193 E X A M P L E Let p 2 3 be a prime a n d r be generated by two elements y , 6 o f order p such that p[y,6] = [y,[y,A]] = = [S,[y,S]1 = o (the brackets denote commutators). Then r is a nonlabelian group of order p3 and exponent p. Let T i ( i E I ) be the set o f all cyclic subgroups f ( 0 1 o f r and < y > the cyclic subgroup generated by y. Fix some i o E I. Let S: = {id}v { s t3 End(r)l ker s = , Z ( r ) + + and s ( 6 ) t3ri 3 . Then M S ( r ) fulfills the conditions 0

(Zp,+;)).

i n 9.192, hence is a near-field ( i n fact, MS(T):

turn our attention to the question as to when MS(r) We happens to be (semi)simple. For this we need an extension o f the concept of fixed-point-freeness. 9.194 DEFINITION S .rEnd(T') is fixed-point-free if 3 ( a ) v s F S : Ker s = Ker s 2 = Ker s = . . . ( b ) fl Ker s = C O I . sES ( c ) Y E r i s l , s 2 r-- S : s l ( y ) = s2(y) o => s 1

+

=

s2.

Obviously, if S is a group of automorphisms, this concept of fixed-point-freeness coincides with o u r well-known concept for automorphism groups (cf. e.g. 4.52). It can be shown that, if S is fixed-point-free a n d finite, S can be written as S = G , u... u G n u { G I , where G ,,... ,Gn are groups with identities e , ,.. . ,en a n d e . e . = 6 . (hence S i s a completely regular inverse 1

semigroup).

J

1 j

See also Maxson-Smith ( 1 1 ) .

§ 9 MORE CLASSES OF NEAR-RINGS

364 9.195

Let N E

THEOREM ( M a x s o n - S m i t h ( 8 ) ) .

n,

be a f i n i t e

n e a r - r i n g . T h e n N i s s e m i s i m p l e a l l o f i t s s i m p l e summands e i t h e r non-rings o r f i e l d s i f f

being

N i s isomorphic t o

some M S ( T ) , w h e r e S i s a f i x e d - p o i n t f r e e s e t o f e n d o morphisms o f

r

finite,

r.

More g e n e r a l l y ,

M S ( r ) , w i t h S S Endr,

i s semisimple i f f S i s a completely regular

i n v e r s e semigroup.

More i n f o r m a t i o n c a n be o b t a i n e d i f S i s s p e c i a l i z e d .

The f i r s t

c o l l e c t i o n o f r e s u l t s w h i c h we m e n t i o n c o n c e r n s a o n e - e l e m e n t set S = {s},

i n t h e second s e r i e s o f r e s u l t s , S i s a semigroup

o f " l i n e a r " maps. O f c o u r s e ,

MCsl(r)

=

M

Let

r

<S>

(r),

where i s > i s t h e

subsemigroup generated by s . 9.195

THEOREM ( M a x s o n - S m i t h ( 2 ) ) .

s =

E

End(r).

Then

be a f i n i t e g r o u p and

t h e f o l l o w i n g a s s e r t i o n s h o l d f o r N:

=

M{s,a3(T):

( a ) I f s i s n o n - n i l p o t e n t and n o t i n v e r t i b l e t h e n N i s n o t 2 - s e m i s imp1 e.

( b ) L e t s be n i l p o t e n t o f degree n > I , =

{f E NI m- 1

sn-'(y)

( y ) = 6 ) , b u t f o r n o yl

: s

w i t h maximal m. a n d N/2,(N) (c)

If

0 3 a n d A : = IS

=

r

T h e n a,(N)

r

E

E

L(y):=

ker slvy

E

r:

we h a v e s m ( y ' ) = E l

fl

L ( 6 ) = {f ~ N l f / *= U j . 6EA g M o ( A ) ; hence N/g2(N) i s s i m p l e . =

i s a v e c t o r space and s i s l i n e a r t h e n N i s

simple i f f N i s 2-semisimple.

9.197

THEOREM (Maxson-Smith ( 3 ) ) . identity, w i t h fr:

L e t R be a f i n i t e r i n g w i t h

RN a f i n i t e u n i t a r y R-module and S : = V + V,

v-frv.

( a ) MS(V) = I f E Mo(V)I ( b ) If R i s simple,

{ f r l r € R)

Then

V

r E R v m

E

M:

f(rm) = rf(m)l.

so i s M S ( V ) , and MS(V) i s a n e a r - r i n g

i f f R i s a f i e l d w i t h dimRV>l.

( c ) If R i s s e m i s i m p l e ,

w i t h n o n e o f i t s s i m p l e summands

b e i n g a f i e l d then MS(V) i s a r i n g . ( d ) If R i s n o t a f i e l d b u t i f M S ( V ) i s s t i l l s i m p l e t h e n

MS(V)

i s t h e r i n g EndR(V).

9h Bicentralizer near-rings

365

Now we t u r n o u r a t t e n t i o n t o t h e s t r u c t u r e o f M o ( r ) f o r some G G 2 A u t ( T ) . Even s t r o n g e r t h a n b e f o r e one c a n s a y t h a t MG(T) = =

M
i n Aut(T).

f G c A u t ( T ) and

denotes t h e group generated by G

H e n c e we may c o n f i n e o u r c o n s i d e r a t i o n r e c o r d i n g t o

the f o l l o w ng MA(T) w i l l always

CONVENTION F o r t h e r e s t o f t h i s c h a p t e r ,

(Aut(T). d e n o t e t h e c a s e w h e r e A = G u { o } w i t h G ~~. Due t o l i m i t a t i o n s o f s p a c e , we o n l y c i t e m o s t o f t h e r e s u l t s . We s t a r t t h e d i s c u s s i o n w i t h some e l e m e n t a r y f a c t s c o n c e r n i n g

MA(T) a n d p r o c e e d b y c h a r a c t e r i z i n g t h o s e c a s e s i n w h i c h a monogenic MA(r)-group, a g.d.

near-ring,

is

i n which MA(T) i s a r i n g , a n e a r - f i e l d ,

or a regular near-ring.

t h e r a d i c a l s and t o s t u d y

r

In o r d e r t o c o m p u t e

(semi-)simplicity

a n d p r i m i t i v i t y , we

take a look a t the l e f t ideal structure o f MA(T).

After that,

we c o n s i d e r t h e p r o b l e m , when a n e a r - r i n g c a n b e w r i t t e n a s a bicentralizer near-ring with various properties.

I n particular,

i t turned o u t t h a t b i c e n t r a l i z e r n e a r - r i n g s a r e as g e n e r a l

t h e y can be (9.191).

as

F i n a l l y , we c l o s e w i t h some c l o s u r e p r o p e r -

t i e s a n d some o t h e r i n t e r e s t i n g q u e s t i o n s a n d r e s u l t s .

Since

some r e s u l t s c o n c e r n m o r e t h a n o n e o f t h e s e t o p i c s m e n t i o n e d a b o v e , w e ' l l m e n t i o n some r e s u l t s t w i c e o n d i f f e r e n t p l a c e s and hope t h a t t h i s w i l l p r o v i d e a b e t t e r o v e r a l l l o o k . We h a v e t o a d j u s t a c o n c e p t , situation.

f i r s t m e n t i o n e d on 8 . 3 8 ,

t o our

F o r a E A a n d y c r we p r e f e r t o w r i t e a y i n s t e a d o f

a(v). 9.198

DEFINITION I n MA(T), l e t f o r Y E denote the A - s t a b i l i z e r

Obviously,

for all y

i s a subgroup o f A*.

~ s tr ( y )

r

st(y):=

l a ~ A 1 a . y = y}

o f y. i s a s u b s e m i g r o u p o f A and s t ( y ) *

The f o l l o w i n g r e s u l t t u r n e d o u t t o be b a s i c

t o t h e u n d e r s t a n d i n g o f w h a t ' s g o i n g on i n M A ( r ) .

39

366

MORE CLASSES OF NEAR-RINGS

9 . 1 9 9 PROPOSITION ( " B e t s c h ' s Lemma"). I n

MA(r), l e t

y,6

E

r.

Then

t h e r e i s some m c M A ( r ) w i t h m ( y ) = 6 i f f s t ( y ) c _ s t ( A ) . Proof. I f m(y) = 6 then c l e a r l y ay = y i m p l i e s a 6 = 6 . C o n v e r s e l y , l e t s t ( y ) s s t ( 6 ) . We d e f i n e f b y f ( y ) : = 6 , f ( a y ) : = a 6 f o r a c A and f ( @ ) = o f o r f i g A * y , t h e o r b i t o f y under A*. f i s w e l l - d e f i n e d , s i n c e a 1 y = a 2 y i m p l i e s a - l a 2 E s t ( y ) s s t ( d ) , whence = a 2 d . C l e a r l y , f E M ~ ( T )a n d f ( y ) = 6.

a,6

The f a c t t h a t f ( a y ) = a f ( d ) d e t e r i t l i n e s a l l v a l u e s o f f u n i q u e l y i n t h e o r b i t of y g i v e s us t h e f o l l o w i n g 9.20G C O R O L L A R Y Let

( y i ) i E I be a complete s e t o f r e p r e s e n t a t i v e s

r under A . I f Ai E r ( i E I ) f u l f i l l s t ( y i ) < s t ( A i ) then t h e r e i s a unique f F M A ( r ) with

o f t h e o r b i t s of

f(yi)

=

6i for all

i

E

I.

The p r o o f o f t h e n e x t r e s u l t i s s t r a i g h t f o r w a r d . 9 . 2 0 1 PROPOSITION F o r y Hence e l e m e n t s of stabilizers.

T a n d a E A* we h a v e s t ( a y ) = a ( s t ( - y ) ) a - ' . r* i n t h e same o r b i t h a v e c o n j u g a t e d

I

I f A i s a b e l i a n ( s e e e . g . Maxson-Smith ( 1 ) ) t h e n c l e a r l y e l e m e n t s i n t h e same o r b i t h a v e e q u a l s t a b i l i z e r s . Example 1 i n ! l a x s o n S m i t h ( 5 ) shows t h a t t h i s d o e s n o t n e c e s s a r i l y h o l d f o r n o n a b e l i a n A. F o l l o w i n g a s u g g e s t i o n of G . B e t s c h , we f i x a very s u g g e s t i v e

n o t a t i o n f o r elements in 9.202

r.

N O T A T I O N I n t h e M A ( T ) - s i t u a t i o n we w r i t e

( a ) For y , 6 E r we s a y t h a t y d i v i d e s 6 a n d w r i t e y 1 6 i f 6 = f y f o r some f c M A ( T ) , i . e . i f s t ( y ) s t ( 6 ) . i . e . f o r s t ( y ) = st(&), we s a y t h a t y a n d 6 a r e e q u i v a l e n t and w r i t e y - 6 .

( b ) For y16

9h Bicentralizer near-rings

(c) If YET', function

l e t e

367

be t h e ( b y 9.200

Y

uniquely determined)

i n MA(T') w h i c h f i x e s a l l e l e m e n t s i n t h e o r -

b i t o f y under A e l e m e n t - w i s e and sends a l l o t h e r e l e ments t o zero.

I

Obviously, in

r/% ,

r

i s a preorder r e l a t i o n i n

where

?i

s i t u a t i o n i n which

I

o r b i t s of "usual

r

I f y l , ...,yn then e

(b).

The

i s d i s c r e t e w i l l prove t o be e q u i v a l e n t

t o t h e r e g u l a r i t y o f M A ( r ) (see 9.207). idempotent.

a n d an o r d e r r e l a t i o n

d e n o t e s t h e e q u i v a l e n c e o f 9.202 The maps e

Y

are clearly

are representatives o f the non-zero

,...,

e a r e o f t e n r e f e r r e d t o as t h e y1 'n idempotents". I n t h i s case, they are also orthogonal.

Now we d r a w o u r a t t e n t i o n t o t h e c a s e w h e r e I' i s a n i o n o a e n i c MA(r)-group.

9.203

F r o m 9 . 1 9 9 we g e t

PROPOSITION ( B e t s c h ( l o ) ,

(11)).

The f o l l o w i n g c o n d i t i o n s

are equivalent: (a)

r

i s a monoqenic M A ( r ) - g r o u p .

n and s t ( 6 )

( b ) T h e r e e x i s t s P + A s r w i t h A*n = all

(c) 3 (d)

y r :~ s t ( y )

r*

=

{id) for

6 ~ b . =

Cidl.

I.

has a s m a l l e s t e l e m e n t w . r . t .

L,!hen

i s MA(r) a ring, a near-field,

9.204

THEOREM ( M a x s o n - P e t t e t - S m i t h

d.g.

or regular ?

(1)) Let

r

be f i n i t e .

M A ( T ) i s a r i n g i f f i t i s a d i r e c t sum o f f i e l d s . 9 . 2 0 5 THEOREM ( M a x s o n - S m i t h ( 5 ) )

Let

r

be f i n i t e .

Equivalent are:

( a ) MA(T) i s a n e a r - f i e l d ( b ) A a c t s t r a n s i t i v e l y o n I' (c) 9.206

r*

THEOREM

i s a s i n g l e o r b i t under (Maxson-Smith ( 1 2 ) ) L e t

M A ( r ) i s d.g.

A*.

r

be f i n i t e and s o l v a b l e . I f

t h e n Y has d e r i v e d l e n g t h 2 and MA(T) i s a

r i n g (go t o 9.204).

P9

368

MORE CLASSES OF NEAR-RINGS

9 . 2 0 7 T H E O R E M (Meldrum-Oswald

(I)).

MA(T)

is regular i f f

s t ( y ) s s t ( 6 ) implies s t ( y ) = s t ( 6 ) f o r a l l y , 6 ~ r . The p r o o f o f t h i s r e s u l t c a n a l s o be f o u n d i n M e l d r u m ( 2 3 ) . O b v i o u s l y , M A ( T ) i s r e g u l a r i f A* i s f i x e d - p o i n t - f r e e . O n t h e o t h e r hand we h a v e 9.208

E X A M P L E (Meldrum-Oswald ( 1 ) , Meldrum ( 1 3 ) ) . Take r = A 5 and A = I n n ( r ) . Then A - A and t h e s t a b i l i z e r of Y E r 5' i s just i t s normalizer. Since t h e i r orders a r e 3,4 and 5 , no p r o p e r c o n t a i n m e n t o f s t a b i l i z e r s i s p o s s i b l e by L a g r a n g e ' s t h e o r e m . 9 . 2 0 7 shows t h a t M A ( r ) i s r e g u l a r .

I n M e l d r u m - Z e l l e r ( I ) , Meldrum-Oswald ( 1 ) and Maxson-MeldrumOswald ( I ) , a l l t w o - s i d e d i n v a r i a n t s u b n e a r - r i n g s o f r e g u l a r M ( r ) ' s (under various conditions) a r e determined. B a s i c a l l y , A t w o - s i d e d i n v a r i a n t s u b n e a r - r i n g s of M A ( r ) c o n s i s t s of a l l

whose " r a n k " i s s m a l l e r t h a n a g i v e n c a r d i n a l . T h i s g e n e r a l i z e s 7.34, of c o u r s e . f €MA(T)

( r ) we n e e d some k n o w l e d g e a b o u t A l e f t ideals. A f i r s t result i s

For c o m p u t i n g t h e r a d i c a l s o f M ( s t i c t l y ) maximal

9 . 2 0 s T H E O R E M ( S m i t h ( I ) ) . L e t T be f i n i t e . Then t h e l a t t i c e of l e f t i d e a l s o f M A ( T ) i s d i s t r i b u t i v e . 9 . 2 1 0 T H E O R E M ( M a x s o n - S m i t h ( 9 ) , c f . 7 . 2 2 ) L e t L be a minimal l e f t i d e a l i n M A ( T ) . Then t h e r e i s some Y E T w i t h L c _ M A ( r ) e Y M o r e o v e r , N = M A ( r ) e y c=> s t ( f ( y ) ) = s t ( v ) h o l d s f o r some f E L <=> L i s n o t c o n t a i n e d i n t h e i n t e r s e c t i o n of a l l maximal l e f t i d e a l s o f M A f I ' ) . We want t o know e v e r y t h i n g : when i s a l e f t i d e a l s t r i c t l y m i n i m a l ( i . e . a l s o minimal a s a n N o - s u b g r o u p ?

.

369

9h Bicentralizer near-rings

9.211 THEOREM (Maxson-Smith minimal

G e n e r a l i z i n g 7.28

9.212

( 9 ) ) F o r y c I',

MA(r)ey i s a s t r i c t l y

i d e a l i f f t h e o r b i t o f y i s a maximal one.

left

( b ) we g e t

THEOREM ( M a x s o n - S m i t h

( 9 ) ) If L i s a maximal l e f t i d e a l i n

M A ( r ) t h e n t h e r e i s some y ~ r *s u c h t h a t e i t h e r or

L = (o:y)+K

where K

Y'

Y

being contained i n MA(r)e

ma1 w . r . t .

L

=

( 0 : ~ )

i s a l e f t i d e a l o f MA(r), maxiY

.

I n t h e f i n i t e c a s e we g e t t h e f o l l o w i n g r e s u l t w h i c h n e e d s

9 . 2 1 3 NOTATION a l l

6

E

If

A

i s an o r b i t o f

r*

then LA i s the s e t o f

feMA(T) s u c h t h a t 6 p r o p e r l y d i v i d e s f ( 6 ) f o r some A.

.

9 . 2 1 4 THEOREM ( M a x s o n - S m i t h ( I ) , ( 5 ) ) . L e t r b e f i n i t e Then ( a ) I f L q 9 . M A ( T ) i s s t r i c t l y m a x i m a l t h e n L = L A f o r some

r*

orbit A o f

( b ) L A i s s t r i c t l y maximal

i f f f o r some 6 L A we h a v e

T6:= {y

i s a normal

that

E

IIfily,

but yt6i

subgroup

I-':=

1 y s r l ~ 1 y > i, f ~ y E n l y ~ y , A y ~i s~ ay ~ union 6 o f c o s e t s o f ry f o r a l l y o € r \ r g a n d i f a y - y c r 6

of

(with a c

For fi nite

A)

implies ayo

-

yoc

and r e g u l a r M A ( r ) ' s (see e.g.

r6

f o r a l l such yo.

7.18, 7.21

and 7 . 2 8 ) ,

t h e s i t u a t i o n i s much e a s i e r . 9 . 2 1 5 THEOREM ( M a x s o n - S m i t h (a) A l l

( 9 ) ) L e t MA(r)

minimal l e f t ideals o f

be f i n i t e and r e g u l a r

MA(r) a r e o f t h e f o r m

MA(r)eY and a u t o m a t i c a l l y s t r i c t l y minimal.

( b ) A l l maximal ( 0 : ~ a) n d

l e f t ideals o f MA(r) are o f the form

automatically

s t r i c t l y maximal.

T h i s p r o v i d e s v a l u a b l e i n f o r m a t i o n f r o m w h i c h one can compute the radicals o f

MA(r).

coincide by 5.3(b).

Recall that the

'yl

and t h e

rd2

radicals

39

370 9.216

M O R E CLASSES OF N E A R - R I N G S

THEOREM ( M a x s o n - S m i t h ( 1 ) , ( 5 ) , ( 9 ) ) yl,

N:=

. . . ,yn

r be f i n i t e , l e t

Let

be r e p r e s e n t a t i v e s o f a l l o r b i t s of

r*

and l e t

MA(r).

(a) go(N) = A n

B , where A i s t h e i n t e r s e c t i o n o f a l l

), where He i s m i n i m a l and B i s t h e i n t e r yi Y i i s not s e c t i o n o f a l l ( ( ( o : y . ) + K ) : N ) , s u c h t h a t Ne J Yj yj maximal and K i s d e f i n e d as i n 9.212. (o:Ne

(b)

3,,,2(Pl) is

t h e sun1 o f a l l t h o s e K

such t h a t ( o : y k ) Yk

i s n o t maximal. (c)

22(Pl)i s

the i n t e r s e c t i o n o f a l l those

d e s c r i b e d i n 9.214 J2(N)

LA which a r e

(b).

i s n i l p o t e n t i f fe a c h L A i s s t r i c t l y m a x i m a l .

C o n c e r n i n g s e m i s i m p l i c i t y we g e t 9.217

THEOREM (Maxson-Smith

(5),(9))

( a ) N i s 0 - s e m i s i m p l e <=>

Ne

Let

r

be f i n i t e and N : = M A ( r ) .

i s a maximal l e f t i d e a l

whenever t h e o r b i t o f y i s maximal. ( b ) N i s 2 - s e m i s i m p l e <->

a l l stabilizers

i n T'*

are

maximal. If a l l st(y) with y c (e.g.

r*

a r e normal subgroups o f A*

i f A i s abelian or fixed-point-free)

2-semisimple

then N i s

i f f N i s simple.

The s i m p l i c i t y o f M A ( T ) was i n v e s t i g a t e d i n s e v e r a l p a p e r s . Examples o f n o n - s i m p l e M A ( r ) ' s a r e e x h i b i t e d i n Meldrum-Oswald

( I ) , Maxson-Smith ( 5 ) and e s p e c i a l l y i n M e l d r u m - Z e l l e r ( 1 ) . r i s f i n i t e . I f M ( r ) i s f i n i t e and

T h i s can happen even i f s i m p l e t h e n MA(T)

i s 2-pi.imitive

by 4.47

A (a).

Up t o now n o c o m p l e t e c h a r a c t e r i z a t i o n o f s i m p l e M A ( T ) ' s seems t o be p o s s i b l e .

But there are important p a r t i a l results.

p r e s e n t thein w i t h o u t p r o o f .

We

9 h Bicentralizer near-rings

371

9.218 T H E O R E M (a) (Meldrum-Oswald (I), Meldrum ( 1 3 ) , Maxson-Smith (5)) L e t MA(T) be regular such t h a t all stabilizers o f elements i n T* a r e conjugated. Then there is some group A and some f i x e d - p o i n t - f r e e automorphism group F on A with MA(T) = MF(A). If dimFA is finite (see 7.7) o r if I F / IrI then MA(r) and M;(A) are simple. ( b ) (Maxson-Smith (5)). L e t r be finite. Then M A ( T ) is s i m p l e iff all stabilizers i n r * a r e conjugated (if A is a b e l i a n , the l a s t condition can be replaced by "all orbits i n r * have the s a m e size"). f c ) (Maxson-Smith (5)). If r is f i n i t e and M A ( T ) is s i m p l e , is an elementary but A n o t f i x - p o i n t - f r e e then abelian p-group, hence a vector space over Z and A * P' a c t s irreducibly a s a group o f l i n e a r automorphisms. ( d ) (Maxson-Smith (5)). If r is f i n i t e and M A ( T ) simple and non-abelian then A * is f i x e d - p o i n t - f r e e . 9.219

COROLLARY (Meldrum-Zeller ( I ) , Maxson-Meldrum-Oswald ( I ) ) . I f T is a vector s p a c e o v e r a field F and if A = F (acting by multiplication) then MA(T) is simple.

In considering V-primitivity o f MA(T), c a r e should be taken. Suppose that M A ( r ) is regular such t h a t all stabilizers i n r * are c o n j u g a t e d , but such t h a t A * is n o t fixed-point-free. Then M A ( r )z M F ( A ) by 9.21P f o r some A and s o m e f i x e d - p o i n t - f r e e F
turn to 0-primitivity. If

r

is f i n i t e w e c a n s a y quite

O b v i o u s l y , we can reformulate 0-primitivity a s follows to Get (a) o ( b ) in the next result.

§ 9 MORE CLASSES OF NEAR-RINGS

372

9.220

r

THEOREM L e t

f { o } f u l f i l l one ( a n d hence a l l ) c o n d i t i o n s

E q u i v a l e n t a r e (Betsch ( l o ) , Ramakotaiah-Rao(3)):

of 9.233.

( a ) M A ( r ) i s 0 - p r i m i t i v e on

+

r,

A o r , t h e r e a r e m c MA(r)a n d w i t h y,-y2 E A, b u t m(y,) - m(y2) $ A .

( b ) If { o }

( c ) T h e r e i s some ~ ( 0

: r\A6)

yl,

y2

r

E

(AA a s i n 9 . 2 0 3 ( b ) ) s u c h t h a t

E

does n o t c o n t a i n a non-zero l e f t i d e a l o f

MA(T) w i t h L2 = {O}.

9.221

THEOREM ( M a x s o n - S m i t h ( 9 ) ) I f r' i s f i n i t e t h e n M A ( T ) 0-primitive

is

i f f ( a ) and ( b ) h o l d :

( a ) A l l m a x i m a l o r b i t s i n 1'* a r e c o n j u g a t e d (b) If the o r b i t o f maximal

left

Y E

r*

i s maximal t h e n M A ( r ) e

i s a

Y

ideal.

An i m p o r t a n t r e s u l t o n 2 - p r i m i t i v i t y c a n b e d e r i v e d f r o m 9 . 2 1 8 ( a ) : 9.222

THEOREM (Betsch ( 9 ) , ( 1 0 ) ) (see 9.203)

If r i s a m o n o g e n i c M A ( r ) - g r O U p

t h e n M A ( I ' ) i s 2 - p r i m i t i v e on

r

i f f F.*

is

fixed-poi nt-free.

As a n i l l u s t r a t i o n o f t h e s e r e s u l t s we g i v e i n f o r m a t i o n i n a n i n t e r e s t i n g s p e c i a l case.

2.223

THEOREM ( M a x s o n - O s w a l d ( I ) ) . and A* o v e r F.

the general

Let

F be a f i e l d ,

as an R-module.

(a) If

ylay then

abelian

l i n e a r group o f r e g u l a r nxn-matrices

Suppose t h a t A* c A u t ( r ) .

S i n c e A*

simple r i n g R o f a l l nxn-matrices over

r

r

generates t h e

F , we m i g h t v i e w

Then t h e f o l l o w i n g r e s u l t s h o l d : ayly

for a l l Y E

( b ) A l l minimal l e f t i d e a l s of

r,

a€A*.

MA(T) a r e g i v e n by M A ( r ) e

( c ) MA(T) has no n i l p o t e n t l e f t i d e a l s e x c e p t { a } . (d) Every non-zero l e f t ideal o f MA(T) idempotent.

Y

.

contains a non-zero

9h Bicentralizer near-rings

373

(e) MA(r) i s 0-primitive. (f)

MA(T) i s simple i f f i t i s 2 - p r i m i t i v e .

( 9 ) I f n = 1 then MA(T) i s s i m p l e i f f dimRr = 1 ( i n t h i s case,

MA(T) i s a f i e l d isomorphic t o F).

( h ) By ( b ) , ~ I / ~ ( M A ( ~ =) ) { o } . T h i s r e s u l t s complement a l s o obtained "sharp"

and g e n e r a l i z e

9.197.

The a u t h o r s

u p p e r and l o w e r bounds f o r J 2 ( M A ( r ) ) .

Now we c o l l e c t r e s u l t s w h i c h c o n c e r n t h e q u e s t i o n a s t o when A*

has t o be f i x e d - p o i n t - f r e e .

9.224

THEOREM ( a ) A*

i s fixed-point-free

i f f M A ( r ) i s r e g u l a r and

r

i s

a rnonogenic M A ( r ) - g r o u p ( B e t s c h ( 1 0 ) ) .

(b) If

r

i s monogenic t h e n A*

r

M A ( T ) i s 2 - p r i m i t i v e on ( c ) See a l s o 9 . 2 1 8

Also,

(a),

i s fixed-point-free

iff

(see 9.222).

( c ) and ( d ) .

we m i g h t a s k , w h i c h n e a r - r i n g s h a v e a " r e p r e s e n t a t i o n "

as a b i c e n t r s l i z e r near-ring.

We r e c a l l 9 . 1 9 1 a n d g o f u r t h e r

on. 9.225

THEOREM ( M a x s o n - S m i t h ( I ) ,

Maxson-Meldrum ( 1 1 ,

Maxson-

Pettet-Srni t h ( 1 ) ) (a) I f N i s a near-ring w i t h i d e n t i t y then N some g r o u p (b)

If

r

MS(T)

a n d some S c E n d ( r ) ( s e e 9 . 1 9 1 ) .

N i s a n e a r - f i e l d t h e n t h e r e i s some g r o u p

a n d some G S A u t (C) I f N =

(r) N

with

N,a...a Nk, where

simple near-rings ring,

for

N z

N1

,..., N k

are f i n i t e

s u c h t h a t a t l e a s t o n e Ni

but not a field,

r

0

MG(r). i s a

t h e n N c a n n o t be i s o m o r p h i c

t o some M ; ( i - 1 . (d) Let F1< F2S

... 5

Fn b e f i n i t e f i e l d s .

Then t h e r e i s

some v e c t o r s p a c e V o v e r F 1 a n d a g r o u p G o f l i n e a r transformations

on V s u c h t h a t F I G . .

. O F n z ME(V).

374

89

MORE CLASSES OF NEAR-RINGS

See t h e s e p a p e r s f o r more d e t a i l s . For i n s t a n c e , Maxson-Meldrum gave an example o f a n e a r - f i e l d N w i t h N 2 M ( r ) , where S 5 E n d ( r ) , S b u t S $ A u t ( r ) . The t h i r d p a p e r mentioned in 9 . 2 2 5 b r i n g s t h e r e a d e r a l s o back t o t h e F r o b e n i u s g r o u p s which come u p i n t h e s t u d i e s o f p l a n a r i t y . Cf. a l s o 9 . 2 1 0 ( a ) .

A t y p e of c l o s u r e b r i n g s us back t o t h e b e g i n n i n g of t h i s c h a p t e r ; Then o b v i o u s l y A * ( A u t N ( r ) =: ALAut(r) suppose t h a t N ( M A ( T ) . 0

and NcMX(T)

cMA(T).

O f t e n , t h e way from A t o

A

i s a "closure":

9.226 THEOREM ( a ) (Maxson-Smith ( 1 0 ) ) . N

MA(r) f o r

=

Let N(Mo(r) be s i m p l e . Then some A i f f N = M A u t N ( , ) ( r ) . On t h e o t h e r

h a n d , t h e r e e x i s t s some g r o u p A such t h a t N i s n o t

i s o m o r p h i c t o some M A ( A ) w i t h A * ( A u t ( T ) . ~

( b ) ( B e t s c h ( 9 ) , ( 7 0 ) ) . Let N be d e n s e l \ ! ( s e e 4 . 2 6 ) embedded i n MA(r) ( w . r . t . 3 ) which i s monogenic o n r . Then 0

A*

=

A u t N ( r ) , hence

get A*

=

AutMA(

MA(r)

I n p a r t i c u l a r , we ( s e e 7 . 1 3 ! a n d 7 . 3 7 ! ) a s well a s

r )( r )

=

MA(T).

I s n ' t t h a t a b e a u t i f u l f o r m u l a ? The proof w i l l be p u b l i s h e d i n a f o r t h c o m i n g p a p e r o f G . B e t s c h . These c o n s i d e r a t i o n s a l l o w us t o g e t b a c k t h e i n g r e d i e n t s ( r a n d A ) , M A ( T ) = : M i s made o f . Me s o we g e t back t h e monogenic I f Y E r * t h e n o b v i o u s l y My Y' M-subgroups of r . A l s o , f o r n E M , t h e map e Y ne + t n w i t h tn:Me +Me me + (me ) ( e ne ) = me ne t u r n s o u t t o be a s e m i Y Y' Y Y Y Y Y Y group isomorphism between ( e Me - 1 a n d EndM(Mey). I f we plug Y Y' t h i s t o g e t h e r a n d assume t h a t y g e n e r a t e s a l l of r we g e t a very s a t i s f a c t o r y r e s u l t . S e e a l s o t h e p r o o f of 4 . 5 6 . 9.227

r b e ( b y y) a monogenic M : = N A ( r ) in p a r t i c u l a r g r o u p . Then I ' - ~Mey a n d E n d M ( f ) - e Me Y Y' A * z group of u n i t s o f ( e Me -)

T H E O R E M (Betsch ( 9 ) ) . Let

Y

Y'

We n o w i l l u s t r a t e t h e s e i d e a s o n K l e i n ' s f o u r g r o u p .

9h Bicentralizer near-rings 9 . 2 2 8 EXAMPLES ( B e t s c h ( 9 ) , ( 1 1 ) ) . L e t

r

=

375

Z2xZ2

=

{O,a,b,c}.

Then A u t ( r ) : S 3 ( s e e A p p e n d i x , n e a r - r i n g s o f low o r d e r , E ) ) . ( a ) I f A * : = S 3 , M A ( T ) i s i s o m o r p h i c t o t h e f i e l d Z 2 . The s t a b i l i z e r s o f a , b and c a r e a l l e q u a l and c o n s i s t o f i d and t h e t h r e e t r a n s p o s i t i o n s . Hence 9 . 1 9 2 , 9 . 2 0 7 and 9.218 a p p l y . ( b ) L e t A * : = A 3 . T h e n A* i s f i x e d - p o i n t - f r e e . H e n c e M A ( r ) i s 2 - p r i m i t i v e on r. M A ( T ) h a s 4 e l e m e n t s ; i t s a d d i t i v e qroup i s isomorphic t o r . a b cb ) } . T h e n r i s a m o n o g e n i c M A ( r ) {id, (O * a g r o u p w i t h g e n e r a t o r b . A* i s n o t f i x e d - p o i n t - f r e e a n d M A ( T ) i s n o t r e g u l a r . M A ( T ) h a s 8 e l e m e n t s . The

( c ) Take A*:=

j2-radical

i s t h e a n n i h i l a t o r of a and has o r d e r 4 .

Moreover, ebMA(r)eb = {O,eb,f} with f ( 0 ) = f ( a ) = 0 , f ( b ) = c , f ( c ) = b . I t s group of u n i t s i s C e b , f } and i n f a c t i s o m o r p h i c t o A*. We g o on w i t h some t o p o l o g i c a l p r o p e r t i e s . o f 4.25.

Recall the notation

S(m,y)

+

8, 9.229 T H E O R E M (Rarnakotaiah-Rao ( 3 ) ) . For e a c h A s r * , A M A ( T ) i s c l o s e d i n M o ( r ) w . r . t . t h e t o p o l o g y 7, g e n e r a t e d b y t h e s u b b a s e { S ( m , i ) 1111 E Mo(r),6 E A}. F u r t h e r m o r e , fA i s H a u s d o r f f i f f b, = r * . Of c o u r s e ,

’&*

i s just t h e f i n i t e topology

J(see

4.26) in

Mo(r). With t h i s n o t a t i o n , o n e c a n r e f o r m u l a t e t h e d e n s i t y t h e o r e m 4 . 3 0 f o r 0 - p r i m i t i v e n o n - r i n g s N on N T ( s e e 4 . 1 4 a n d t h e f o o t n o t e on p . 1 1 6 ! ) : 9.230 T H E O R E M (Ramakotaiah-Rao ( 3 ) , Betsch ( 1 1 ) ) . I f t h e nonr i n g N = N o i s 0 - p r i m i t i v e on r a n d A = A u t N ( r ) V fo} t h e n 1\1

i s dense in

=

MA(r) w.r.t.

I n 9.230,

Eft. M o ( I ’ ) I

v

a

E

A v

y O~1 : f ( a y ) = a f ( y ) l =

i t i s s t r a i g h t f o r w a r d t o see t h e e q u a l i t y of t h e t w o

near-rings involved.

376

59

M O R E CLASSES OF N E A R - R I N G S

A n o t h e r t o p i c was p u r s u e d by M .

Holcombe:

9 . 2 3 1 DEFINITION L e t N be a n e a r - r i n g w i t h i d e n t i t y .

let K(Na)

be t h e s e t o f t h o s e n E N w i t h Nan

=

F o r EN Na a n d

w i t h 3 m N~ V x E N A : xmn = x n m = x . L e t X a be t h e s e t o f a l l Nb which a r e maximal f o r b e i n g c o n t a i n e d i n Na. Then H ( N a ) : = { f n : N + N , x + x n / f n i n d u e s a p e r m u t a t i o n on X a } i s c a l l e d t h e holonomy g r o u p o f N a . I f N i s f i n i t e , t h e s e m i g r o u p o f a l l f n ( n & N ) c a n be c o v e r e d b y a w r e a t h p r o d u c t o f holonomy g r o u p s b y a w e l l - k r o w n r e s u l t

,.

o f ( E i l e n b e r g ) . Let y 1 . . , y n be a c o m p l e t e s e t o f r e p r e s e n t a t i v e s of r * u n d e r t h e a c t i o n o f A * < A u t ( T ) . F o r Na, Nb p u t Na-Nb i f t h e r e e x i s t s x,y

3-. 2 3 2

E

N w i t h N a x s N b and N b y 6 N a .

T H E O R E M (Holcombe ( 9 ) , ( 1 0 ) ) . L e t N t h a t r * h a s k o r b i t s u n d e r A*.

=

Then we g e t

MA(r) a n d

suppose

( a ) I f a E N* then Na-N(el+ . . . + e S ) , where e l , . . . e s E 3 a n d s i s t h e number o f o r b i t s i n E { e y ,... , e 1 YS

a(r)*. ( b ) H ( N ( e l +. . . + e , ) )

=

C f & M A ( r )1 f i s a b i j e c t i v e map

on ( e l + ' . . . + e s ) r } . ( c ) The s e m i g r o u p { f n : N + N , x + x n I n E NI c a n be c o v e r e d b y a w r e a t h p r o d u c t o f k holonomy g r o u p s o f t h e t y p e H ( N ( e l +. . . + e , ) )

(with s

=

1,2

,... , k ) .

Clay-Maxson-Meldrum ( 1 ) e x t e n d e d t h e s t u d y of t h e u , ~ i t s01' ivlA(r) ( i . e . t h e g r o u p U A ( r ) o f i n v e r t i b l e f u n c t i o n s i n M A ( T ) m e n t i o n e d a t t h e e n d o f 9 7 a ) ) t o t h e c a s e t h a t A* i s n o t n e c e s s a r i l y f i x e d - p o i n t - f r e e . U A ( Y ) t u r n s o u t t o be ( a s a g r o u p ) isomorphic t o a d i r e c t product of c e r t a i n wreath p r o d u c t s . I f M,,,(r) i s r e g u l a r , t h e s e w r e a t h p r o d u c t s a r e o f t h e t y p e mentioned i n $ 7 c ) . Also a " d e t e r m i n a n t - l i k e ' ' f u n c t i o n D i s d e f i n e d o n a p a r t P of M A ( r ) . I f t h e r e a r e o n l y f i n i t e l y many o r b i t s o f r !under A * , P = M A ( r ) . I f o n e v i e w s t h e s e o r b i t s as g e n e r a l i z a t i o n s o f one-dimensional s u b s p a c e s of a v e c t o r s p a c e t h e n t h e f o l l o w i n g r e s u l t s show t h a t 0 r e a l l y b e h a v e s l i k e a determinant function:

9h Bicentralizer near-rings r e g u l a r i t y o f MA(T)

( a ) Under c e r t a i n c o n d i t i o n s (e.9. f i n i t e l y many o r b i t s ) , (b) I f f E M A ( T )

D(f)

377

with

0 iff f i s invertible.

maps t w o d i s t i n c t o r b i t s t o t h e same o r b i t

t h e n D ( f ) = 0. ( c ) I f g d i f f e r s f r o m f o n l y i n one o r b i t t h e n D ( g ) i s a

'' mu 1 t i p l e Also,

'I

of

D ( f ) ( "mu 1 t i 1 i n e a r it y " ) .

t h e concept o f

"

eigenvalues"

i s discussed i n t h i s paper.

I n ( 1 6 ) , Maxson shows an i n t e r e s t i n g c o n n e c t i o n t o g e o m e t r y .

A

s o - c a l l e d t r a n s l a t i o n Sperner space w i t h o p e r a t o r s l e a d s i n a n a t u r a l way t o a n a s s o c i a t e d n e a r - r i n g w h i c h c o n s i s t s o f m a p p i n g s on a group (which " c o o r d i n a t e s " c e r t a i n endomorphisms. type.

t h i s s p a c e ) w h i c h commute w i t h

Hence t h i s n e a r - r i n g i s o f t h e MS(T)-

Maxson c h a r a c t e r i z e s t h e c a s e i n w h i c h t h i s b i c e n t r a l i z e r

near-ring i s a near-field. B i c e n t r a l i z e r n e a r - r i n g s a r i s e i n many o t h e r s i t u a t i o n s a s w e l l . i n H i i c k e l ' s t h e o r y o f m o l e c u l a r o r b i t s i n quantum mechanics, instance,

for

one s t u d i e s t r a n s f o r m a t i o n s w h i c h commute w i t h t h e

Hamiltomian operator.

A s Maxson-Smith ( 1 ) p o i n t e d o u t ,

the study o f M A ( r )

i s also

m o t i v a t e d by i n v e s t i g a t i o n s c o n c e r n i n g automorphisms o f l i n e a r automata.

T h i s b r i n g s us s t r a i g h t i n t o o u r l a s t ( p r o p e r ) c h a p t e r .

59

378

MORE CLASSES OF NEAR-RINGS

i ) N E A R - R I N G S A N D AUTOMATA _____ -_ In

h i s papers ( 1 1 ) - ( 1 4 ) ,

M.

Holconibe has e s t a b l i s h e d an i n t i m a t e

c o n n e c t i o n between n e a r - r i n g s

and l i n e a r automata.

In fact,

l i n e a r i t y i s n o t necessary f o r v a r i o u s concepts and r e s u l t s ,

and

near-rings might develop i t s

f r o m w h a t we h a v e s e e n u p t o n o w ,

f u l l power j u s t i n t h e n o n - l i n e a r case ( b u t even f r o m l i n e a r a u t o m a t a one does n o t g e t r i n g s , Roughly spoken,

but a f f i n e near-rings.

automata c o n s i s t o f i n p u t s ,

t o g e t h e r w i t h maps w h i c h d e s c r i b e how " n e w " s t a t e and t h e o u t p u t . p l a y any r o l e .

inputs affect the

F o r many c o n s i d e r a t i o n s , o u t p u t d o n o t

H e n c e we s t a r t w i t h " o n e h a l f o f a u t o m a t a " :

which o n l y have i n p u t s and s t a t e s . (Kalman) o r L i d l - P i l z

those

F o r a much more e x t e n s i v e

tre?.tment o f t h e s e c r e a t u r e s see e.g.

We w i l l

s t a t e s , and o u t p u t s ,

(Eilenberg),

(Holcombe),

(1).

s e e s h o r t l y t h a t i t i s a d v i s a b l e now t o c h a n g e o u r

confession f o r o p p o r t u n i s t i c reasons

( s e e t h e l i n e s a f t e r 9.24A);

i n t h i s c h a p t e r we w i l l w r i t e maps f r o m t h e r i g h t ( h e n c e we write x f instead o f f ( x ) ) ;

t h i s i m p l i e s t h a t we a r e now d e a l i n g

l e f t near-rings instead o f r i g h t near-rinqs w i t h _-

(see 1 . 2 ) .

The a u t h o r i s f u l l y a w a r e o f t h e c r i m e c o m m i t t e d h e r e .

But t o

t h e s u b j e c t now! 9.233

DEFINITION A semiautomaton i s a t r i p l e S = (Q,A,F), Q and A a r e s e t s

(called the s t a t e s e t and t h e i n p u t s e t )

and F i s a f u n c t i o n f r o m QxA i n Q, transition function. additively),

where

called the state-

I f Q i s a g r o u p (we a l w a y s w r i t e i t

we c a l l S a g r o u p - s e m i a u t o m a t o n a n d a b b r e v i a t e

t h i s b y __ GSA. For q

E

Q and a

E

A we i n t e r p r e t e F ( q , a )

as t h e "new s t a t e

o b t a i n e d f r o m t h e o l d s t a t e q b y means o f t h e i n p u t a " .

If

S = (Q,A,F)

i s a semiautomaton,

mappings f a f r o m Q t o Q, qf,:

we g e t a c o l l e c t i o n o f

one f o r e a c h a & A , = F(q,a).

which are given by

9i Near-rings a n d a u t o m a t a

Hence f a " s c r i b e s

379

t h e e f f e c t o f t h e i n p u t a on t h e s t a t e s e t

Q o f S. I f t h e i n p u t al E A i s f o l l o w e d by t h e i n p u t a2,

t h e semiautomaton

and t h e n i n t o 1 We e x t e n d ( a s u s u a l ) A t o t h e f r e e m o n o i d A * o v e r A

from the state q

"moves" (qf,

F

Q f i r s t into qfa

) f 1 a2 c o n s i s t i n g o f a l l f i n i t e sequences o f elements o f A, t h e empty sequence

A)

i.e.

i s a monomorphism f r o m A*

t h e map a + f a

including

and g e t

t i o n monoid o v e r Q w i t h f A

i d

=

9'

i n t o the transforma-

I n the case o f GSA's,

we a r e

also able t o study the superposition fa +fa2 (defined pointwisely) 1 o f t w o " s i m u l t a n e o u s " i n p u t s al ,a2 E A . Hence i t i s n a t u r a l t o consider

{fala

E

A } "{f,}

composition o f maps). course, 9.234

a n d a l l o f i t s sums a n d p r o d u c t s ( = The o b v i o u s f r a m e w o r k f o r t h a t i s ,

o f

t h e s t r u c t u r e of a n e a r - r i n g .

DEFINITION L e t S = (Q,A,F)

N(S) o f

M(Q)

be a GSA.

The s u b n e a r - r i n g

generated by i d Q and a l l f a ' s

(a € A )

is

called the syntactic near-ring o f S. Thus N(S) i s always a n e a r - r i n g w i t h i d e n t i t y . then N(S) i f f i n i t e , y e t most i m p o r t a n t , 9.235

too.

If Q i s finite

We n o w b r i e f l y d i s c u s s t w o s p e c i a l ,

cases. L e t Q and A be a d d i t i v e groups

The homomorphism case.

w i t h z e r o 0 a n d F a homomorphism from t h e d i r e c t p r o d u c t

Q x A . We t h e n c a l l qfa

=

F(q,a)

(Q,A,F)

= F(q,O)

+

a homomorphic GSA. (0,a))

Because o f

F ( q , O ) + F(0,a)

= qfo

+

+ O f a

we g e t f a = f o + f a , w h e r e f o i s a h o m o m o r p h i s m ( i . e .

a

d i s t r i b u t i v e e l e m e n t i n N ( Q ) ) , w h i l e f a i s t h e map w i t h c o n s t a n t v a l u e Of,.

0 f o r a l l a E A,

i.e.

if Ofa

d.g.

near-ring,

=

I f no i n p u t can change t h e zero s t a t e ,

consisting o f

N(S) o b v i o u s l y i s a ? -sums o f powers o f f o then

(which a r e endomorphisms).We a l s o g e t a d.g.

near-ring

380

li9 MORE CLASSES OF NEAR-RINGS i f F i s a d d i t i v e i n t h e f i r s t component.

I f this i s not

t h e c a s e , we h a v e t o t a k e a c l o s e r l o o k , w h a t N ( S ) c o n s i s t s of

(go t o 9.237).

F o r homomorphic GSA's one sees by

fyI+. . . +f a n - , f o + f a n )

= fE + ( f a i n d u c t i o n t h a t fa a 1 2"-a" 1 w h e r e t h e map i n b r a c k e t s i s c o n s t a n t .

f!

9.236

Each power

i s a homomorphism.

The l i n e a r c a s e i s a s p e c i a l c a s e o f t h e h o m o m o r p h i s m case i n which Q and A a r e a b e l i a n groups R - m o d u l e s f o r some r i n g

( o r more g e n e r a l l y ,

R) a n d w h e r e F i s l i n e a r . We a r r i v e

a t t h e case o f l i n e a r semiautomata,

the type o f (semi-)

a u t o m a t a w h i c h a r e s t u d i e d m o s t e x t e n s i v e l y ( s e e e.g. (Holcombe) o r (Kalman)).

T h i s c a s e d e s e r v e s more w o r d s .

L e t Q and A b e f r e e R-modules w i t h f i n i t e b a s e s X,Y respectively.

1x1

Let

=

/ Y I = m.

n,

Then t h e a c t i o n o f

F can be d e s c r i b e d by an m x ( n + m ) - m a t r i x Z = ( z . . ) o v e r

'J R i f we r e p l a c e e a c h e l e m e n t o f Q a n d o f A b y i t s d e -

composition fa = fo+fainduces a "decomposition"

o f Z such

that

F(q,a)

Z.(q,a)

=

=

. ... .

.a

. ... . 'ml* =:

*

"mm

B.q+C.a

We t h e n c l e t qfa, a2...ak If, i n particular,

N(S) i s a ring,

= B k . q + B k - I .Cal+...+B.C.ak-,+C.ak

C = 0,

i f B = 0 then qfa

... d . ak

i f f c.(ak-q;) = fa,... 1 a k' each f a (and hence each fa f o r =

map f r o m Q t o Q.

BK.q and

=

.ak

g e n e r a t e d by B and t h e u n i t m a t r i x I .

On t h e o t h e r hand, Anyhow,

we g e t q f a

I f Q i s f r e e on X w i t h

CLE

C.ak.

=

A*)

1x1

=

We g e t

i s an a f f i n e n t h e n we

c a n use a method due t o B l a c k e t t ( 3 ) t o e x t e n d t h e i d e a o f m a t r i x r e p r e s e n t a t i o n s f r o m l i n e a r maps t o a f f i n e maps. L e t f b e a n a f f i n e map.

9i Near-rings a n d automata

38 1

Then f d e c o m p o s e s a s f = f o + c w h e r e f o i s a homomorphism a n d c i s c o n s t a n t . L e t F be t h e m a t r i x f o r f o w . r . t . X . I n v e n t a symbol e w i t h e + e = e e = e a n d e r = r e = e f o r a l l r E R . Then

e s t a b l i s h e s an isomorphism between M a f f ( Q ) r i n g of a l l ( n + l ) x ( n + l ) m a t r i c e s o v e r Rufe} 9.237 THEOREM Let S

*

N(S) = { 1

(Q,A,F)

=

f a , / " ,E A * )

a n d a subnear-

.

be a homomorphic G S A . =:

Then

N

1

Proof. N s H ( S )

i s clear.

C o n v e r s e l y i t s u f f i c e s t o show t h a t

, since obviously N contains a l l f a ( a E A ) a n d i d Q = f . I n f a c t , we show t h a t N i s a s u b n e a r - r i n g o f M f 4). ( a ) T a k e f = I: 2 f a . E N , g = I: f f EM. I t is clear i 1 j 'j t h a t f + g E PI. S o c o n s i d e r f g : N i s a near-ring

.

f g = ( C ? f a ) ( C ' f ) = c ' ( cj ? f a i ) f E j i i j 'j j ( b ) Hence we o n l y l o o k a t t h e l a s t e x p r e s s i o n i n a ) . L e t R j = a l a 2 . . . a n € A * . Then

: f Ni I f nJ.

( L 2 f c c . ) f a f a . . .f a i 1 1 2 n ( c ) We f i r s t f o c u s o u r a t t e n t i o n t o n = 1 a n d p u t a , = a f o r a moment. (1 t f a , ) f a = ( c ! f a , ) f o + f a = (1 i f , l . f o ) + 1 i 1 i 1 i + -fa = (; T f ( y , o ) + Ta = (1 2 f + - fo + fa EN. 1 1 i ( d l T h e r e f o r e we g e t yk E A* w i t h (;

1

=

We r e m a r k t h a t N ( S ) c a n a l s o be c h a r a c t e r i z e d a s t h e s u b n e a r r i n g of M ( Q ) generatd by {Tala E A } u { f o , i d } . B u t the e x p l i c i t e e x p r e s s i o n i n 9.237

i s much e a s i e r t o h a n d l e .

382

$ 9 MORE CLASSES OF NEAR-RINGS

9 . 2 3 8 C O R O L L A R Y L e t S = ( Q , A , F ) be a GSA. a ) I f S i s homomorphic t h e n b y t h e l a s t f o r m u l a i n 9 . 2 3 5 a n d by 9.237 N(S) i s a g . d . 9 . n e a r - r i n a w i t h i d e n t i t y . b ) I f S i s e v e n l i n e a r , we c a n a r r a n g e t h e f a ' s s o t h a t e a c h e l e m e n t o f N ( S ) i s t h e sum o f l i n e a r and a c o n s t a n t map. Hence N ( S ) i s an a f f i n e n e a r - r i n g w i t h i d e n t i t y i n t h i s c ? s e . So n o t e v e n i n t h e l i n e a r c a s e we c e t p r o p e r n e a r - r i n g s which a r e n o t r i n g s . We w i l l s e e i n 9 . 2 4 1 t h a t N ( S ) can be q u i t e g e n e r a l f o r a r b i t r a r y S . I n t h e homomorphic c a s e we e x h i b i t a n i n t e r e s t i n g f e a t u r e of t h e zero-symmetric

p a r t No(S)

:=

(N(S))o.

9 . 2 3 9 PROPOSITION L e t S = ( Q , A , F )

be homomorphic. Then N o ( S ) c o n s i s t s o f a l l f i n i t e sums o f e l e m e n t s o f t h e f o r m c t f - c with f

2

E

3

{ i d , f o , f o , f o ,... 1 a n d c

E

{Z

+- F a ,

lai

E

A*}.

1

All e l e m e n t s c ? f - c a s a b o v e a r e i n N o ( S ) . C o n v e r s e l y , t a k e g = Z ? f a , E N o ( S ) . Then 0 = Og

Proof.

--

=

1

By s t a n d a r d g r o u p t n e o r y , =

'i C 2 (fo

we c a n a r r a n g e

(1

=

1 i fa.

+

T,,) 1

i n t o sums a n d d i f f e r e n c e s o f e l e m e n t s

n 1l

o f t h e form c + f ,

- c , w h e r e c i s t h e sum o f some

Ta ' s . i

9.240 -

C O R O L L A R Y L e t S be l i n e a r . Then ( w i t h f i : = i d ) 1 N ~ ( s )= { z o f : + z , f o + . . . + z n f : l z i E z , n E N ~ I Hence N o ( S ) i s t h e s u b n e a r - r i n g o f M a f f ( Q ) g e n e r a t e d by

{ i d , f o l . Since

=

1

(MaffiQ))o

i s a ring, No(S)is a r i n g , too.

S i n c e t h e r e a r e p l e n t y of n e a r - r i n g s N where N o i s n o t a r i n g w i t h t w o g e n e r a t o r s , we s e e t h a t f a r n o t e v e r y n e a r - r i n g a r i s e s a s a N ( S ) w i t h l i n e a r S . What h a p p e n s i n t h e g e n e r a l c a s e ? Which n e a r - r i n g s a r i s e a s N ( S ) f o r v a r i o u s t y p e s o f S ? Compare t h e next r e s u l t w i t h 9.191.

9i Near-ringsand automata

383

9 . 2 4 1 T H E O R E M F o r e v e r y n e a r - r i n g N w i t h i d e n t i t y t h e r e i s some GSP, S w i t h N - N ( S ) . P r o o f . By 1 . 8 6 we c a n f i n d a g r o u p Q s u c h t h a t N i s i s o m o r -~ p h i c t o a s u b n e a r - r i n g fl o f M ( Q ) . L e t A be an i n d e x Then s e t f o r N, i . e . fl = { f a l a C A I . L e t F ( q , a ) : = qf,. N-N = N(5) w i t h S = (Q,A,F). S i n c e e v e r y n e a r - r i n g c a n be embedded i n a n e a r - r i n g w i t h i d e n t i t y ( 1 . 1 0 2 ) we g e t 9 . 2 4 2 C O R O L L A R Y E v e r y n e a r - r i n g can be embedded i n t h e n e a r - r i n q o f some G S A . 9 . 2 4 3 T H E O R E M For

a near-ring N t h e r e e x i s t s a l i n e a r G S A 5 with

N z N(S) i f f ( a ) (N,+) i s abelian ( b ) N h a s an i d e n t i t y 1 ( c ) T h e r e i s some d c : N d s u c h t h a t N o i s g e n e r a t e d b y C l , d } . P r o o f . One i m p l i c a t i o n i s c o v e r e d by 9 . 2 4 0 . S o l e t N b e a ___ n e a r - r i n g w i t h ( a ) - ( c ) . By 1 . 9 1 a n d 1 . 5 6 we know t h a t

PI i s i s o m o r p h i c t o a s u b n e a r - r i n g R o f M ( N , + ) . L e t d and be t h e i m a g e s o f d a n d 1 i n K. S i n c e d i s

-

distributive, -

1

=

a

i s an endomorphism of ( N , + )

and

a,

i d N . No i s g e n e r a t e d by i d a n d whence n N o = I z o i d + zla + . . . +zna I z i E Z , ~ EN^}.

Now l e t ( A , + )

and F ( q , a ) : = qa + Oa. Then ( Q , A , F ) i s a l i n e a r GSA, s i n c e ( N , + ) i s a b e l i a n . = f o we g e t No = N o ( S ) . F u r t h e r m o r e , t a k e Since frs. N c ( S ) . We g e t f = O f = O(C +_ f a , ) = C 2 ( O f a . ) :=

(Q,+)

:=

(N,+)

a

1

with Ofa,

= fa

l ,.n O - I +...+'fan = 0Ta

1

a n - l + . . . + O i a om E

1 n = Nc. T h i s shows N c ( S ) c N c . C o n v e r s e l y , e v e r y T € f l c ( w i t h c o n s t a n t v a l u e c ) i s i n Nc(S) s i n c e = Tc. Hence N ( S ) = K G N . -

1

-

c

I t i s customary in a l g e b r a i c automata theory t o consider the s e m i g r o u p - e p i m o r p h i s m A * + N(S) g i v e n b y a + f a . The i d e a of

=

89

384

MORE CLASSES OF N E A R - R I N G S

simultaneous i n p u t s ( l i n e s before 9.224)

e n a b l e s us t o t r a n s f e r

t h i s epimorphism f r o m semigroups t o n e a r - r i n g s . instance,

i n t e r p r e t a,a2

" i n p u t s e q u e n c e ala2

2a2 as b e i n g t h e complex i n p u t

together w i t h t h e simultaneous i n p u t a We e x t e n d A t o t h e f r e e n e a r - r i n g A

( i n double, s t r e n g t h ) " . o v e r A.

+

We c a n , f o r

...,a n )

I f a# = w(al,

a

i s a w o r d i n A # we d e f i n e

,...,

f ) fw(a, ,a,,) 1 an a n d F # ( q , a # ) : = ; f a # . T h u s we g e t a n e x t e n d e d s i m u l t a n e o u s # # # # : = (Q,A , F ) . L e t I be { a E A I f # i s t h e s e q u e n t i a l GSA S a z e r o m a p } . T h e n I i s a n e a r - r i n g i d e a l a n d we g e t b y t h e := w(fa

,. . .

homomorphism t h e o r e m : 9.244

#

THEOREM A / I

=

N(S

/I )

=

N(S).

I f we h a d u s e d r i g h t n e a r - r i n g s ,

we w o u l d h a v e N ( S ) a n t i i s o m o r p h i c

# # t o A l I . Hence N ( S ) c a n be v i e w e d as a homomorphic image o f A It i s ,

however,

impossible t o give a nice "canonical" # f o r a l l elements o f A

.

form

.

A p o s s i b l e r e l i e f comes f r o m t h e o b s e r v a t i o n t h a t o n e m i g h t r e p l a c e A # b y Azr

,

the free algebra i n a variety v o f near-rings

c o n t a i n i n g N(S) ( f o r i n s t a n c e ,

one m i g h t t a k e

as t h e v a r i e t y

generated by N ( S ) ) . ATTENTION

! If A a l r e a d y b e a r s some a d d i t i v e s t r u c t u r e ( a s i n

t h e cases 9.235/9.236,

f o r instance),

(and i n most cases w i l l ) i n A!

t h i s new a d d i t i o n c a n

be d i f f e r e n t f r o m t h e g i v e n a d d i t i o n

I n p a r t i c u l a r , o u r new a d d i t i o n i s o n e i n A # a n d

not

in

A*.

I n t h e l i n e a r c a s e we saw t h a t N ( S ) i s an a f f i n e n e a r - r i n g . S i n c e t h e c l a s s o f a l l a f f i n e n e a r - r i n g s i s known t o f o r m a v a r i e t y (9.821,

i t makes s e n s e t o l o o k a t f r e e a f f i n e n e a r - r i n g s ,

t h e m o r e s o s i n c e we know how t h i s m o n s t e r s l o o k l i k e .

9i Near-rings and automata

385

9 . 2 4 5 PROPOSITION L e t A be a s e t , A* t h e f r e e monoid o v e r A a n d A t h e f r e e a f f i n e n e a r - r i n g o v e r A . Then e v e r y e l e m e n t o f A i s a f i n i t e sum o f e l e m e n t s ? a i w i t h cii E ( A u { O ) ) * . P r o o f . S i n c e x ( y + z ) = x y + x z , ( x + y ) z = xz-xzO+yz-yzO+zO a n d ( - x ) y = -xy+yxG+yG a r e l a w s i n t h e v a r i e t y o f a f f i n e n e a r - r i n g s , we c a n b r i n g a l l e x p r e s s i o n s i n t o + - s u m s o f e l e m e n t s which a r e p r o d u c t s o f e l e m e n t s i n A u { O l ( o b s e r v e t h a t we u s e l e f t n e a r - r i n g s ! ) .

We n o w t u r n t o t h e c o n c e p t o f a c c e s s i b i l i t y .

9 . 2 4 6 DEFINITION Let

S = (Q,A,F)

be a G S A a n d A # t h e f r e e n e a r -

r i n g o n A . q 1 E Q i s a c c e s s i b l e f r o m q 2 E Q i f t h e r e i s some w i t h q2fn = 9 , . S i s a c c e s s i b l e i f each s t a t e q i s a c c e s s i b l e from each o t h e r s t a t e .

R E A#

N ( S ) i s n o t o n l y a n e a r - r i n g , b u t i t a l s o o p e r a t e s on Q . O b v i o u s l y Q i s a n N(S)-group v i a q f a in t h e u s u a l meaning. q , i s a c c e s s i b l e f r o m q 2 i f f q 1 E ~ ~ N ( S A) l.t e n a t i v e l y , Q c a n be v i e w e d a s a n A # group v i a qa:= q f a . 9 . 2 4 7 PROPOSITION L e t

S b e a G S A . S i s a c c e s s i b l e i f f Q i s an

N : = N ( S ) - g r o u p with O N = Q. Proof. I f S i s a c c e s s i b l e then obviously O N suppose t h a t Q = qNo+ONc

+ qNc

= =

=

Q.

Conversely,

O N = O N c . I f q E Q t h e n qN = q N o + q N o + Q = Q , a n d S i s shown t o be

accessible. I t m i g h t be m o s t u s e f u l t o e x a m i n e t h e r e l a t i o n s h i p b e t w e e n q e n e r a t o r s , p r i m i t i v i t y a n d a c c e s s i b i l i t y more c l o s e l y . Mow we l o o k a t c o n s t r u c t i o n s of s e m i a u t o m a t a a n d t h e i r c o r r e s p o n d i n g syntactic near-rings. 9 . 2 4 8 DEFINITION Let S = ( Q , A , F ) a n d S ' = ( Q ' , A , F ' ) be G S A w i t h i d e n t i c a l i n p u t s e t s . A g r o u p homomorphism h : Q + Q ' i s c a l l e d an GSA-homomorphism i f h ( q f a ) = h ( q ) f i h o l d s f o r a l l q E Q a n d a E A (with f;(q'):=F'(a,q'),of course).

89

386

MORE CLASSES OF NEAR-RINGS

We t h e n w r i t e h : S - S ' .

9.249 -

P R O P-___ O S I T I O N ( C f . Holcombe ( 1 1 ) ) . L e t h : S - S '

be a GSA-

epimorphism. Then t h e r e e x i s t s a n e a r - r i n g epimorphism h from N(S) t o N ( S ' ) w i t h h(qn) = h ( q ) K ( n ) f o r a l l q s Q and n

E

N(S).

,..., f a k ) = 1 f a k . Then h ( q f w ) =

I f n E N ( S ) , n i s a word n = w

Proof.

=

w(fa

,...,

i n fa fw(al ,ak) 1 = h ( q ) f b f o l l o w s f r o n 9.248 b y i n d u c t i o n on t h e

,. . .

l e n g t h o f w. = h(q)fl;, f W

=

f,, '

fi i s w e l l - d e f i n e d

D e f i n e % ( f w ) : = fl;.

s i n c e f w = fl; i m p l i e s h ( q ) f l ; for a l l q follows.

F.

h(qfw) = h(qfw,) =

=

Q.S i n c e h i s s u r j e c t i v e ,

Obviously,

5 i s a near-ring

epimorphism and h ( q n ) = h ( q f w ) = h(q)f; i s also true f o r a l l q

E

Q and n

E

=

h(q)E(n)

N(S).

I f we h a v e m o r e i n m i n d , we h a v e t o c o n s i d e r o u t p u t s a s w e l l . 9.250

a u t o m a t o n i a a q u i n t u p l e A = (Q,A,B,F,G), D E F I N I T I O N An where (Q,A,F) __ s e t ) and G:

o f A).

i s a semiautomaton, QxA-t B a f u n c t i o n

B a set (the output

(called the output function

I f Q i s a group, A i s c a l l e d a group-automaton

GA). ( a b b r e v i a t e d by -

S i m i l a r t o 9.235/9.236

a h o m o m o r p h i c G A i f Q,A,B . a r e g r o u p s a n d F,G morphisms. A i s c a l l e d a l i n e a r GA

we c a l l A a r e homo-

o r l i n e a r automaton

o r l i n e a r s e q u e n t i a l m a c h i n e i f Q,A,B

a r e R-modules f o r

some r i n g R a n d F,G a r e R - l i n e a r maps. S i n c e f o r e v e r y ( g r o u p - , h o m o m o r p h i c - ,1 i n e a r - ) a u t o m a t o n A = =

(Q,A,B,F,G),

attributes,

S : = (Q,A,F)

i s a s e m i a u t o m a t o n w i t h t h e same

we d e f i n e N ( A ) a s N ( S ) .

Hence t h e s y n t a c t i c n e a r -

r i n g o f a GA has n o t h i n g t o do w i t h o u t p u t s a n d o u t p u t maps! I t i s easy t o a d j u s t 9.244

I n many c a s e s , For instance, i n series.

and 9.248/9.249

t o t h e GA-case.

h o w e v e r , o u t p u t s do p l a y a n e s s e n t i a l r o l e . i f one wants t o c o n n e c t two

For doing t h a t ,

( o r more) automata

c o n s i d e r A = (Q,A,B,F,G)

and

9i Near-rings and automata

A'

=

387

The o u t p u t s o f A s h a l l b e t h e i n p u t s of A ' .

(Q',B,C,F',G').

I

L

A

________-----

'

A'

J

Series connection A s A ' More f o r m a l l y , (F(q,a),

=

A s A':=

F'(q',G(q,g))

(QxQ',A,C,F",G")

w i t h F"(q,g'),a):=

and G"((q,q'),a):=

G'(G(q,a),q').

A n y b o d y who k n o w s a b i t o f a u t o m a t a t h e o r y knows t h a t N ( A s A ' ) w i l l n o t be e a s i l y e x p r e s s a b l e i n terms o f N(A) and N ( A ' ) .

For

l i n e a r G A ' s we g e t w i t h t h e s e n o t a t i o n s 9.251

T H E _OREM (Holcombe ( 1 2 ) ) . I f A and A ' a r e l i n e a r N(A s A ' )

i s the near-ring

GA then

N(A) s N ( A ' ) a d d i t i v e l y k (k €No),

g e n e r a t e d by a l l p a i r s o f t h e f o r m ( f Q , f A )

) ( ~ E A a) n d a l l 'G(0 ,a) w i t h Po:Q+Mc(Q'), q+fG(q,o)'

constant-map-pairs

(k

~m,)

(Fa¶

N(A) s N ( A ' ) can be obt.ained,as

Holcombe remarked,

a wreath product construction f o r a.a.n.r.'s.

the

(O,kpo)

b y means o f

(See a l s o 9.285).

He a l s o g o e s o n t o r e l a t e N ( A ) s N ( A ' ) t o t h e s p l i t t i n g s h o r t e x a c t sequence 0 + Q ' + Q x Q '

-f

Q + 0 o f N(A) s Pl(A')-groups

i n the

category o f a.a.n.r.'s-groups. Now we t u r n t o t h e i n p u t - o u t p u t

b e h a v i o u r o f a GA A = (Q,A,B,F,G).

As i n t h e b e o i n n i n ? o f t h i s c h a p t e r , t h e f r e e m o n o i d s o v e r A a n d B, s =

q

:A*+B* G(q,al)

and P* denote

respectively.

For q

E

Q l e t

b e d e f i n e d b y s (A):= A , s ( a ) : = G ( q , a ) , s q ( a l , a 2 ) : = 9 9 G(F(q,al),a2) = ~ ~ ( a , ) s ~ ( ~ , ~ , a ) n( da p~r o! c e e d

inductively with

. . .q n _ , ) , a n ) . 9.252

l e t A*

sq(ala2

.... , a n )

=

s (ala 2...an-,)G(F(q,q, 9

DEFINITION s output-)

:A*+B* i s called the sequential (inputq f u n c t i o n o f A a t q . I f A i s a GA, so=: sA i s

c a l l e d t h e s e q u e n t i a l f u n c t i o n o f A.

Furthermore,

call

,.

89

388 q,q'

MORE CLASSES OF NEAR-RINGS

Q equivalent states

E

(4-q')

if s

q

= s

q1

(i.e.

if q

a n d q ' i n d u c e t h e same " i n p u t - o u t p u t - b e h a v i o u r " ) . I t m i g h t make s e n s e t o e x t e n d s

B

#

q

f r o m A # t o B',

where A# and

a r e t h e f r e e n e a r - r i n g s i n a v a r i e t y which c o n t a i n s t h e one

generated by N(A) sq(al)

=

s

9

+ s

if we d e f i n e s ( a l + a 2 ) : = G ( q , a l ) + G ( q , a 2 ) 9 ( a 2 ) and o b s e r v e t h e " a t t e n t i o n " a f t e r 9.244.

=

9 i s certainly well-defined.

If A = (Q,A,B,F,G)

i s h o m o m o r p h i c we g e t f o r q , q ' , q "

E Q :

a n d s o on, Similarly,

s

qn

i f q-q'

a

E

A and

(a) = G

= G(q'fal* q n - q ' n . We t h e r e f o r e g e t

9.253

PROPOSITION L e t A b e a h o m o m o r p h i c G A . T h e n - i s

a congruence

r e l a t i o n i n t h e N(A)-group Q. 9.254

COROLLARY -( H o l c o m b e ( 1 1 ) ) . I f A i s a h o m o m o r p h i c GA t h e n ( a ) Q o : = r q E Q I qsO3 Q N ( A ) Q ( b ) G(q,O) = 0 f o r a l l q E Q O .

We m i g h t a s k w h a t

q z q ' means i n d e t a i l . T h e l a s t f o r m u l a i n 9 . 2 3 5

s u g g e s t s t h e answer f o r homomorphis G A ' s .

9i Near-rings and automata

9.255 -_

THEOREM ( H o l c o m b e ( 1 1 ) ) .

389

L e t A be homomorphic and

k g o : Q + B , q + q g o = G ( q , O ) . Then q - q ' w , ' j k E N 0 : q ( f o q o ) = k = q'(f0go). P r oo f . L e t 9 - 9 ' . We u s e i n d u c t i o n o n k a n d s t a r t w i t h I f a E A then s (a)

k = 6 .

9

=

G(q,a)

G(q,O)

=

+

Since s (a) = s ( a ) we g e t 9 q' q g o = q ' g o . Now s u p p o s e t h a t 9 . 2 5 5 h o l d s f o r a l l o f l e n g t h k-1 = : t . Then w o r d s CY = a l a 2 . . . a k - l t A *

+ G(0,a)

=

qg,

+ G(0,a).

for a l l a EA,

s ( a a ) = s q , ( u a ) , hence G(qfu,a) = q t t-i G ( q ' f a y a ) . B y 3 . 2 3 5 , G ( q f u , a ) = G(qf:+ C fa fo ,a) = i=l i t t-i = G(qf:,O) + 7 G(fa,fo ,O) + G ( O , a ) i =I 1 t S i m i l a r l y , G ( q ' f a , a ) = G ( q ' f ok, O ) + C G ( f a f;-',O) + i=l i k k + G ( O , a ) , h e n c e G ( q f o , O ) = G(q'f,,O) a n d we g e t

k k qfoGo = q ' f o g o . 9.256

The c o n v e r s e i s s h o w n s i m i l a r l y .

D E F I N I T I O N A G A A = (Q,A,B,F,G) equality.

i s reduced i f

If A i s accessible (i.e.

i f (Q,A,F)

-

i s the

i s accessible)

and reduced t h e n A i s c a l l e d minimal. Obviously, 9.257

a homomorphic GA

{O}.

s reduced i f f Go =

PROPOSITION L e t A = ( Q A,B,F,G)

be a GA.

Then

i s accessible a (Qa i s c a l l e d the accessible p a r t o f Q )

( a ) A,:=

(Q(N(A))=:Qa,

( b ) By 9.247,

A,B,FIQ

x A y

G/Q,xA)

Q = ON(A).

( c ) A / - : = ( Q / - , A,B,F- , Q - ) a n d G.( [ q ] , a ) : = G ( q , a ) ( d ) A,/i s minimal.

w i t h F.([q],a):= i s reduced.

[F(q,a)]

The o r o o f s a r e s t r a i g h t f o r w a r d and o m i t t e d , I n l o o k i n g f o r c r i t e r i a t o d e c i d e i f a g i v e n G A A i s m i n i m a l o r n o t , we o b v i o u s l y h a v e t o v i e w Q n o t o n l y a s a n N ( A ) - g r o u p b u t a l s o h a v e t o c a r e a b o u t B.

390

$ 9 MORE CLASSES OF NEAR-RINGS

9 . 2 5 8 T H E O R E M ( c f . Holcombe ( 1 1 ) ) .

L e t A be a homomorphic G A

Then A i s r e d u c e d i f f N ( A ) Q h a s n o n o n - z e r o i d e a l s P w i t h Pqo

{Ole

=

I f N ( A ) Q has n o such i d e a l s t h e n Q o = { O ) and A i s r e d u c e d b y 9 . 2 5 3 a n d 9 . 2 5 4 . So s u p p o s e t h a t

Proof.

c o n v e r s e l y A i s reduced and t h a t P < N ( A ) Q has G ( p , O ) = p g o = 0 f o r a l l P E P . I f p c P , we s e e b y s i m i l a r

=

arguments a s in t h e l i n e s preceding 9.258 t h a t p - 0 , h e n c e p = 0 , whence P = { P I . From 9 . 2 4 7 a n d 9 . 2 5 8 we g e t 9 . 2 5 9 T H E O R E M ( c f . Holcombe ( 1 1 ) ) . Then A i s m i n i m a l i f f

N W Q

L e t A be a hornomorphic G A . i s g e n e r a t e d by 0 and does

not contain non-zero i d e a l s whichare a n n i h i l a t e d by g

0'

I n t h e l i n e a r c a s e , Holcombe ( 1 4 ) c o n t i n u e s t h e s e i d e a s by

defining a Jacobson-like radical for N ( A ) (involving the o u t p u t map G ) . A m i n i m a l l i n e a r G A h a s z e r o r a d i c a l ( b u t n o t c o n v e r s e l y ) . D e c o m p o s i t i o n s o f N ( A ) a r e s t u d i e d i n Holcombe ( 9 ) , ( 1 0 ) 9 . 2 3 2 ) and ( 1 3 ) .

(see

F i n a l l y , some comments seem t o be i n o r d e r . I f Q b e a r s a n y a l g e b r a i c s t r u c t u r e ( o r even i f Q i s a n a l g e b r a with r e l a t i o n a s c o n s i d e r e d i n ( G r a t z e r ) , f o r i n s t a n c e ) , t h e same c a r r i e s o v e r ( p o i n t - w i s e l y ) t o t h e s e t N = N(S) or N ( A ) of a l l f a ( a E A ) . From t h e c o n c a t e n a t i o n a , a 2 o f i n p u t s , c o m p o s i t i o n o f m a p p i n g s in N e n t e r s t h e a r e a , t u r n i n g N i n t o a semigroup ( i f Q i s j u s t a s e t a s in g e n e r a l automata t h e o r y ) or i n t o a s e m i n e a r - r i n g ( i f 0 i s a semigroup)or i n t o a n ordered n e a r - r i n g ( i f Q i s a n o r d e r e d g r o u p ) , a n d S O o n . We f i n d us b a c k a t t h e s i t u a t i o n d e s c r i b e d i n 1 . 1 1 8 . N a l w a y s o p e r a t e s on Q , t u r n i n g (1 i n t o

a n N-module-type s t r u c t u r e . A l t e r n a t i v e l y , one can s t u d y t h e f r e e s t r u c t u r e o n A a n d l e t i t a c t on Q ( s e e t h e l i n e s a f t e r 9 . 2 4 6 . Hence t h e t h e o r y o f " S - a c t s " ( s e e Weinert ( 1 3 ) ) i s ava i 1abl e .

9i Near-ringsand automata

391

Of c o u r s e , s i m i l a r i d e a s a p p l y , t o j l i n e a r a n d n o n - l i n e a r ) s y s t e m s theory (see e.g. Lidl-Pilz ( I ) ) , b u t the situation there i s more d e l i c a t e s i n c e t i m e - c o n s i d e r a t i o n s a l s o p l a y t h e i r r o l e .

In a very g e n e r a l s e t t i n g , automata a n d systems m i q h t b e c o n s i d e r e d a s ( i n p u t - o u t p u t - ) f u n c t i o n s ( c f . 9 . 2 5 2 ) . I f , a t i t o f t e n makes s e n s e , we c o n s i d e r a u t o m a t a o r s y s t e m s F i ( i I )~ s u c h t h a t t h e i r i n p u t - and o u t p u t - s e t s a r e a l l equal t o r , and i f r c a r r i e s t h e s t r u c t u r e o f an a d d i t i v e g r o u p ( s e e e . g . ( S a i n ) ) then one can o b s e r v e t h e f o l l o w i n g f a c t s : ( a ) The p a r a l l e l c o n n e c t i o n o f F i a n d F i s g i v e n b y F i + F j j* ( b ) The s e r i e s c o n n e c t i o n o f F . a n d F . i s g i v e n by F O F . . 1 J j 1 Hence o n e g e t s t h e f a c t ( c ) The c l a s s o f a u t o m a t a o f s y s t e m s which c a n be c o n s t r u c t e d b y means o f s e r i e s a n d / o r p a r a l l e l c o n n e c t i o n s of t h e F i l s i s p r e c i s e l y the subnear-ring N generated by I F i l i F I > i n M(r). Up t o now, t h i s s i t u a t i o n was o n l y c o n s i d e r e d f o r l i n e a r a u t o m a t a / l i n e a r s y s t e m s , i n w h i c h c a s e N t u r n s o u t t o be a r i n g . The n o n - l i n e a r s i t u a t i o n a n d t h e u s e o f n e a r - r i n g s seems t o be m o s t promissing. T h e s e i d e a s w i l l be p u r s u e d i n f o r t h c o m i n g p a p e r s .

Our w o r l d i s b e c o m i n g i n c r e a s i n g l y c o m p l i c a t e d a n d t h e a u t o m a t a a n d s y s t e m s i n v o l v e d a n d a r i s i n g a r e i n many c a s e s f z r away f r o m b e i n g l i n e a r . B u t i n many c a s e s t h e s t a t e s e t s Q c a r r y a n a t u r a l g r o u p s t r u c t u r e ( e . g . Q = R n ) . Hence o n e m i g h t hope t h a t n e a r - r i n g s can be o f use i n t h e n o n - l i n e a r c a s e , t h u s becoming a n i m p o r t a n t t o o l i n t h e u n d e r s t a n d i n g of o u r world.

392

89

MORE CLASSES OF NEAR-RINGS

j ) MISCELLANEOUS TOPICS

I n t h i s f i n a l s e c t i o n we i n t e n d t o u i v e b r i e f d e s c r i p t i o n s o f t o p i c s we d i d n ' t d i s c u s s i n o u r j o u r n e y t h r o u q h t h e " n r . - u n i v e r s e " u n t i l now. A g a i n i t s h o u l d be n o t e d t h a t b e i n q i n t h i s s e c t i o n s h o u l d n o t imply any d i s c r i m i n a t i o n o f t h i s s u b j e c t ( a s b e i n o " l e s s i m p o r t a n t " ) . ble h a v e t o r e a c h a n e n d o f t h i s monooraph t h e r e a d e r rniflht b e t i r e d .

9 . 2 6 0 SEMIPRIMARY NEAR-RINGS w e r e i n t r o d u c e d _-

and studied by

K a a r l i in a s e r i e s of p a p e r s . N = No i s c a l l e d semiprimary i f N c o n t a i n s a f i n i t e c h a i n of i d e a l s such t h a t each f a c t o r i s e i t h e r n i l p o t e n t o r isomorphic t o a r i n g of l i n e a r t r a n s f o r m a t i o n s on a f i n i t e - d i m e n s i o n a l v e c t o r s p a c e o r i s o m o r p h i c t o a c e r t a i n r i n g o f homomorphisms. S e m i p r i m a r y

n r . ' s have t h e D C C N ; i f a semiprimary n r . N i s a r i n g then i t i s semiprimary i n t h e sense of ( J a c o b s o n ) ( i . e . N / Y ( N ) h a s t h e D C C L ) . In ( 7 ) , K a a r l i shows t h a t N i s s e m i p r i m a r y iff a,/2(N) i s n i l p o t e n t , the N-group N/$'1/2(N) has t h e D C C I a n d no N-group o f t y p e 2 i s N - i s o m o r p h i c t o o n e o f i t s p r o p e r f a c t o r N - g r o u p s . The s t r u c t u r e and t h e r a d i c a l t h e o r y o f semiprimary n r . ' s N and t h e i r N-groups was developed i n K a a r l i ( 2 ) , ( 4 ) , ( 6 ) and ( 7 ) ( a n d sometimes mentioned i n t h i s book). 9 . 2 6 1 T O P O L O G Y IN N E A R - R I N G S The s t a r t i n g p o i n t was B e i d l e m a n - C o x

( 1 ) which c o n t a i n s

d e f i n i t i o n s and s t r u c t u r a l p r o p e r t i e s o f t o p o l o o i c a l rings.

near-

T o p o l o s i c a ? n r . ' s on r e l a t i v e l y f r e e F r o u p s were c o n s i d e r e d by T h a r m a r a t n a n ( 3 ) ( s e e 6 . 3 5 ( f j ) . B e t s c h ( 3 ) c o n s i d e r s t o D n l o n i c 3 1 S D ? C C S i n d u c s d b y . ~ - n r-i mitive iaeais i v = i . 2 ) . Nr.'s o f c o n t i n u o u s m a p p i n q s on t o p o l o a i c a l q r o u p s ( t o t a l l y disconnected topoloqical aroups, Banach-spaces, r e a l n u m b e r s , . . . ) were c o n s i d e r e d by B e t s c h ( 3 ) ,

9j Miscellaneous topics

Magill ( 1 ) - ( 3 ) ,

Hofer ( 1 ) - ( 5 ) ,

Yamamuro ( 5 ) ,

393

Palrner-Yama-

muro ( I ) , B l a c k e t t ( 4 ) - ( 6 ) . Su ( 1 ) , ( 2 ) , Holcornbe ( 3 ) , ( 4 ) , H . D . Brown ( 2 1 , R . H o f e r ( 3 ) , ( 5 ) , S e p p a l a ( I ) , S u ( 2 ) a n d Adler ( 1 ) . Fbr i n s t a n c e , Yamamuro o b t a i n s t h e f o l l o w i n o r e s u l t i n ( 5 ) : Let B be a r e a l B a n a c h - s p a c e o f d i m e n s i o n 2 2 , a n d l e t

N b e a n r . o f c o n t i n u o u s m a p p i n n s B -+ B , containin4 Maff(B). Then e v e r y a u t o m o r p h i s m o f B i s i n n e r . T h i s implies t h a t i f Bl,?ll and B z , M z a r e two c o u p l e s a s % and N2 a r e a l s o t o p o a b o v e a n d Pi1 = PI2 t h e n P i 1 l o g i c a l l y isomorphic (homeonorphic). See Wefelscheid ( 1 ) , ( 2 ) and ( 7 ) f o r t o p o l o g i c a l n e a r - f i e l d s See Neuberger ( 1 ) , ( 2 ) f o r a p p l i c a t i o n s of n r . ’ s i n f u n c t i o n a l a n a l y s i s . S e e M a g i l l ( 9 ) f o r an e x c e l l e n t summary.

1.262 r l E A R - R I N G S

IN A L G E B P A ! C

TOPQLQGv

I~

I n decomposina p o l y h e d r a s one meets n e a r - r i n q s as s t r u c t u r e s w h i c h a n n i h i l a t e hornoloay a r o u p s ( s e e

(1)). C u r j e l ( 2 ) c o n t a i n s ( a m o n n o t h e r s ) t h e f o l l o w i nn r e s u l t s : Let A be a f i n i t e c o m p l e x , C A t h e r e d u c e d s y s p e n s i c n II t b e n e a r - r i n o (k!ith i d e n t i t y ) O F o f A and N ( 1 A ) Curjel

7 :

homotopy c l a s s e s o f b a s e - p o i n t p r e s e r v i n o s e l f m a p s o f T f i . U s i n g t h e induced endomorphisms o f H , ( Z A ) , t h e followincl a s s e r t i o n s c a n be shown t o b e e q u i v a l e n t : (a)

m,nEN:

nn-nm

i s o f f i n i t e additive order.

( b ) The g r o u p o f i n v e r t i b l e e l e m e n t s i n t h e n o n o i d

(N,*)

( = i t s qroup k e r n e l ) i s f i n i t e .

(c)

nEN:

n n i l p o t e n t =>

I f the Betti-numbers of

n i s o f f i n i t e additive order.

a r e knOVJn, one can d e c i d e w h e t h e r o r n o t ‘I b a s t h e s e o r o p e r t i e s b y a P e c h a n i c a l a p p l i c a t i o n o f t t i l t o n ’ s formclla f o r t h e c o n o t o p y a r o u p s o f d d n i o r . of S o h e r e s . A l s o , EA

89

394

MORE CLASSES OF NEAR-RINGS

9.263 V A L U A T I O I i TH_EORY O N FIEPiR-RI>JGS T h i s i s d e v e l o p e d i n Zenimer ( 3 ) , ( 4 ) and ( f o r n e a r - f i e l d s )

i n Wefelscheid ( 6 ) , ( 7 ) .

9.264 EXTENSIONS A N D H O M O L O G Y Maxson ( I ) , C h o u d h a r i ( I ) , ( ? ) , S e t h - T e w a r i (l), Mason ( 3 ) , ( 4 ) , B a n a s c h e w s k i - N e l s o n ( I ) , Oswald ( 7 ) , Maxson-flswald ( I ) , Meldrum ( 8 ) a n d P r e h n ( 1 ) - ( 3 ) c o n s i d e r e x a c t s e q u e n c e s of N-groups, i n j e c t i v i t y , p r o j e c t i v i t y a n d t h e c o n n e c t i o n s

t,o s e m i - s i m p l i c i t y ( s e e 5 . 4 9 , 5 . 5 0 a n d 9 . 1 5 5 ) . Steinegger ( 1 ) d e s c r i b e s extensions o f near-rings by s e t s of f u n c t i o n s ( s i m i l a r t o t h e r i n g c a s e ) . For d g n r . Is, h o m o l o g i c a l i n v e s t i g a t i o n s w e r e c a r r i e d o u t b y F r o h l i c h ( 5 ) - ( 8 ) ( " n o n - a b e l i a n homological a l g e b r a " ) ; c f . Lausch ( 1 ) , ( 3 ) a n d L o c k h a r t ( 4 ) . 9 . 2 6 5 NEAR-RINGS A N D CATEGORIES

L e t c be a c a t e q o r y w i t h f i n i t e p r o d u c t s and a f i n a l XEC' be a q r o u p o b j e c t . Then Mor(X,)o = ( c f . 1 . 4 ( a ) ) i s a nr. w i t h t h e obvious o p e r a t i o n s = M(X)

o b j e c t . Let

(Holcombe ( 3 ) , ( 7 ) , ( 8 ) ) . Holcomhe s t u d i e s t h e s e n e a r - r i n q s

i n v a r i o u s c a t e a o r i e s , Hom010nv a n d c o h o m o l o n y clroups c a n be v i e w e d a s c e r t a i n N - o r o u p s f o r some nr. N. S i m i l a r c o n s i d e r a t i o n s ( i n a d d i t i v e c a t e o o r i e s ) can be f o u n d i n H u q ( 1 ) a n d Ai j a z - H u q ( 1 ) . A categorical investigation t o radical theory i s i n Holcombe ( 7 ) a n d Holcombe-Walker ( 1 ) . In ( 1 5 ) , ( 1 6 ) , ( 1 7 ) , C l a y g i v e s a d e t a i l e d account on n r . ' s ( " f i b e r e d p r o d u c t n e a r - r i n g s " ) a r i s i n g i n t h e s t u d y of c a t e g o r i e s with g r o u p o r cogroup o b j e c t s . F r o h l i c h ( 4 ) - ( 8 ) s t u d i e d d g n r . ' ~b y means o f c a t e g o r i c a l c o n s i d e r a t i o n s . Mahmood ( 1 ) - ( 4 ) c o n t i n u e d t h e s e s t u d i e s and showed (among o t h e r r e s u l t s ) t h e s u r p r i s i n g f a c t s t h a t p r o d u c t s ( c f . 6 . 9 ( d ) and t h e f a c t t h a t t h e d i r e c t p r o d u c t of d g n r . ' ~i s not d . g . in g e n e r a l ! ) , coproducts, l i m i t s a n d c o l i m i t s e x i s t i n t h e c a t e g o r y of d g n r . ' s ( N , D ) ( w i t h

9j Miscellaneous topics

395

( N , D ) - h o m o m o r p h i s m s a s i n 6 . 1 7 a s m o r p h i s m s ) . MahmoodMeldrum ( 1 ) showed t h a t s e v e r a l c a t e g o r i e s a r e l i n k e d b y f u n c t o r s a r i s i n g f r o m d g n r . ' ~ . Mahmood-Meldrum ( 2 ) a p p l i e d s e v e r a l o f t h o s e i d e a s t o s t u d y s u b d i r e c t p r o d u c t s of dgnr. 's. 9 . 2 6 6 N E A R - R I N G S O N A G I V E W SEMIGROUP

I n t h i s s i t u a t i o n one s t u d i e s a problem " d u a l " t o t h e one s t u d i e d i n $9 d ) . Given a m u l t i p l i c a t i v e s e m i g r o u p ( N , . ) , w h i c h a d d i t i o n s + c a n be d e f i n e d o n N i n o r d e r t o t u r n (N,+,.) i n t o a n e a r - r i n g ( w i t h c e r t a i n p r o p e r t i e s ) . For i n s t a n c e , Ligh ( 2 0 ) c l a s s i f i e d a l l f i n i t e groups ( G , . ) such t h a t G , and a l l subgroups of i t , a r e m u l t i p l i c a t i v e groups of n e a r - f i e l d s . I t t u r n s o u t t h a t G i s e x a c t l y one o f t h e f o u r t y p e s : ( a ) Z n , such t h a t e v e r y d i v i s o r d of n i s o f t h e f o r m d = p"-I ( p a p r i m e ) , ( b ) t h e q u a t e r n i o n g r o u p o f o r d e r 8 , ( c ) a m e t a c y c l i c group of o r d e r 24, ( d ) a b i t e t r a h e d r a l g r o u p of o r d e r 2 4 . See a l l p a p e r s in t h e b i b l i o g r a p h y which a r e l a b e l l e d by M'. 9 . 2 6 7 CONDITIONS F O R N TO B E F I N I T E L i g h ( 1 ) has s h o w n t h a t i f N c o n t a i n s n r i g h t z e r o d i v i s o r s ( N I S n 2 , hence N i s ( a t l e a s t o n e o f them E N d ) t h e n

f i n i t e . See a l s o Linh-Malone ( 1 ) . F o r r i n g s , t h e D C C a n d A C C on s u b r i n o s f o r c e t h e r i n n t o be f i n i t e . B e l l - L i o h

(1) extended t h i s result t o

d g n r . I s and o b t a i n e d s i m i l a r o t h e r f i n i t e n e s s c o n d i t i o n s ( m a i n l y f o r d g n r . ' s ) . See a l s o Bell

( I ) , F e i g e l s t o c k ( 1 ) a n d John ( 1 )

( 3 ) , ( 1 1 ) , Bell-Liqh and c f . 9 . 2 6 5 .

9 . 2 6 8 RESIDUAL FINITENESS Call a n algebra A r e s i d u a l l y f i n i t e i f f o r a l l a , b E A , a+b, there i s a f i n i t e algebra A i n t h e v a r i e t y g e n e r a t e d by a ,b A and a homomorphism h : A + A with h ( a ) t h ( b ) . Free n e a r a ,b a r e r e s i d u a l l y f i n i t e ( a n d word p r o b l e m s i n rings in

no

396

§ 9 MORE CLASSES OF NEAR-RINGS

them a r e all free sidually of F D ,V

solvable). If i s a v a r i e t y o f g r o u p s i n which groups a r e r e s i d u a l l y f i n i t e a n d i f D i s a r e f i n i t e semigroup then t h e " f r e e d . g . n e a r - r i n g " 6 . 2 1 i s r e s i d u a l l y f i n i t e , t o o . See John ( 1 ) .

9 . 2 6 9 NON-ASSOCIATIVE NEAR-RINGS

In R a m a k o t a i a h - S a n t h a k u m a r i ( 2 ) , ( 3 ) a n d S a n t h a k u m a r i ( I ) , zero-symmetric l o o p n e a r - r i n g s N a r e s t u d i e d ( w h i c h means

t h a t ( N , + ) i s a l o o p ) . Loop n r . ' s a r i s e f r o m t h e s t u d y o f m a p p i n g s o f a l o o p i n t o i t s e l f ( c f . 1 . 1 1 8 ) . Among o t h e r r e s u l t s , the a u t h o r s o b t a i n e d a d e n s i t y theorem f o r v-primit i v e l o o p n r . ' s s i m i l a r t o 4 . 3 0 . C f . a l s o 8 . 4 1 and 8 . 4 2 . Timm ( 5 ) - ( 7 ) s t u d i e d m u l t i p l i c a t i v e l y n o n - a s s o c i a t i v e n e a r r i n g s . C f . 8.48. See a l s o S t e f a n e s c u ( l ) - ( l O ) . 9 . 2 7 0 COMMUTATORS. DISTRIBUTORS A N D SOLVABILITY D i s t r i b u t o r s a r e d e f i n e d i n 9 . 7 9 . For a d e t a i l e d s t u d y o f t h e s e c o n c e p t s s e e Esch ( 1 ) a n d c o n f e r H . D . Brown Esch ( 1 ) a l s o c o n t a i n s r e s u l t s due t o F r o h l i c h ( 1 ) , ( 2 ) on d i s t r i b u t o r s a n d "weak d i s t r i b u t i v i t y " i n d p n r . ' s (cf. 6.16).

S e e a l s o Mason ( 1 ) , ( 2 )

a n d Maxson ( 1 ) .

N r . ' s g e n e r a t e d by t h e c o m m u t a t o r s o f a ( n o n - a b e l i a n ) g r o u p a r e s t u d i e d i n Gupta (1). S e e a l s o Curjel ( 1 ) . D a s i c ( 1 ) - ( 9 ) , D a s i c - P e r i c ( I ) , K u z ' m i n ( 1 ) . Meldrum ( 1 3 ) , Oswald ( 1 ) , ( 5 ) , R o b e r t s ( 1 ) a n d S c o t t ( 7 ) . 9 . 2 7 1 DISTRIBUTIVE NEAR-RINGS T h i s i s t h e p l a c e where t h e t h e o r i e s o f n e a r - r i n o s a n d s e m i r i n o s m e e t . !,!e m e n t i o n e d t h e s e n r . ' s a l r e a d y i n 1 . 1 5 , 1 . 1 0 7 and 1 . 1 0 8 . A l l o f 96 i s a p p l i c a b l e . T a u s s k y ( 1 ) a l s o showed t h a t i n a d i s t r i b u t i v e n r . rl e i t h e r e a c h e l e m e n t

i s a zero divisor o r N i s a rino. A simple d i s t r i b u t i v e

n r . i s a l s o a r i n q ( F e r r e r o ( l ) , Lirlh ( 1 3 ) ) . F o r more d e t a i l s s e e H e a t h e r l y ( 4 ) , ( 6 ) , H e a t h e r l y - L i g h ( I ) , H e a t h e r l y - O l i v i e r ( 3 1 , L i g h ( 8 ) , ( 1 5 ) , Malone ( 7 ) a n d ( a u n i f y i n g p r e s e n t a t i o n ) Weinert ( 7 ) - ( 1 0 ) .

9j Miscellaneous topics

N

i s s a i d t o be n - d i s t r i b u t i v e

\I

a b e l i a n and

=

(ncIN)

.

- 1 xyizi

1=1

n-distributive for a l l

neIN

i s a nr.

N is

if

Un(N)

with entries

N t o g e t h e r w i t h t h e u s u a l a d d i t i o n and

iff N i s n-distributive.

f o r m a l power s e r i e s ,

"Gaussian n e a r - r i n o s

N(i)".

Ligh (17))

Also,

Hn(P!)

one can s t u d y

o r o u p n e a r - r i n n s and

These s e t s a r e ( u n d e r t h e

usual operations) always near-rinos distributive

i s

.

m u l t i p l i c a t i o n then (Heatherly ( 4 ) , polynomials,

(N2,+)

N i s p s e u d o - d is t ri butive

I f one c o n s i d e r s t h e n x n - m a t r i c e s f r o m some n r .

if

, . . . ,yn,zl ,... ,zncf4:

x,yl

n

n x ( . c yizi) 1=1

397

i f fil i s p s e b d o -

( s e e H e a t h e r l y - L i a h ( 1 ) f o r t h i s a n d many

other results concerninn pseudo-distributive

near-rinns).

C o n f e r a l s o B e i d l e m a n (1) a n d S u p t a ( l ) , as w e l l a s 9.16'3. Sieno-Stefan0

( 1 ) showed t h a t a l l

2v c o i n c i d e

in a distri-

butive nr. F o r t h i s and 9.271, D'

9.272 -

see a l s o a l l o t h e r papers marked by

b

and

i n the bibliography.

CHARACTERIZING SERIES

L e t NT be a u n i t a r y N-group.

An N - s e r i e s

o f NT i s a s e r i e s

o f l e n g t h n: r = r o z r , z . . . 2 r n = Co) w i t h 4 -N r i for each i < n. I f I 9 N t h e n t h i s s e r i e s i s s a i d t o be a n-I charactsrizing series f o r I i f I= (Titl:Ti) and i=l Iris for 0S i n-2. I has a c h a r a c t e r i z i n g s e r i e s only i f

I k s i ( o : r ) f o r some k

E

IN.

A l l characterizing

s e r i e s f o r I 2 N h a v e t h e same l e n g t h n , a n d n i s j u s t t h e nilpotency class of I/(o:r).

F o r t h i s a n d many o t h e r r e -

s u l t s see Lyons ( 7 ) , Lyons-Meldrum ( 1 ) , ( 2 ) 9.273

CENTRAL N - S U B G R O U P S

and Meldrum ( 7 ) .

are studied i n Scott (22).

A sN r i s c e n t a l i f n i s c o n t a i n e d i n t h e c e n t e r o f ( r , + ) and n E N v I15 c A : n ( y t 6 ) = n y + n S . I f A i s c e n t r a l t h e n A iN r . I f r = T I b r 2 and r sN r has i n t e r s e c t i o n { o l w i t h r l a n d r 2 t h e n A i s c e n t a l . I f a r a~n d N r 2h a v e c e n t a l N-subgroups A1 ,A2

with Al

I(

gN

A2

by h then

rl

@

T2/A

with

89

398

MORE CLASSES OF NEAR-RINGS

3 : = { ( 6 1 , h ( ~ , ) ) ( 6 1 c ~ 1 )i s c a l l e d a c e n t r a l p r o d u c t o f

and

r2.

r

If

r,

zN

TI

t h e n any N-homomorphic image o f

%I

Nr i s a c e n t r a l p r o d u c t . 9.274

C-Z-TRANSITIVE

A N D C-Z-DECOMPOSABLE N E A R - R I N G S

N i s "C-Z-transitive"

\r

nccFIP

i f

nZEI.I C

3

noEr10 :

n O n c = n;.

ri0 N c

I n t h i s case,

"C-Z-decomposable"

i f

related t o a.a.n.r.'s Heatherly (2) rinqs.

9.275

N

4

(these nr.'s

are closely

!).

developes an i d e a l t h e o r y f o r t h e s e near-

N

N E A R - R I N G S w e r e a l r e a d y t o u c h e d i n 9.122. N i s H-monogenic

0'

hlh2 = 0

(i.e. H =

=

PIc

also P i l z (1),(6).

Cf.

H-MONOGENIC

If HsN

i s s t r o n o l y monoclenic. N i s

*

2

if N g

H and H i s " i n t e g r a l "

hl=O v h 2 = 0 ) . I f N i s H - m o n o g e n i c w i t h

{ O l t h e n N h a s z e r o m u l t i p l i c a t i o n . On t h e o t h e r h a n d ,

i f N i s N-monogenic t h e n N i s i n t e g r a l .

H-monogenic n e a r -

r i n g s can be c o n s t r u c t e d b y a g e n e r a l i z a t i o n o f F e r r e r o ' s method p r e s e n t e d i n 1 . 4 ( b ) . Heatherly-Olivier

F o r t h i s and o t h e r t o p i c s see

( 3 ) and O l i v i e r ( 1 ) , ( 2 ) .

9 . 2 7 6 N-SYSTEMS ___

A nr.

w i t h r i o h t c a n c e l l a t i o n law and a " h a l v a b l e

t.lEr)O

idempotent

e

+ 0"

(i.e.

3

hEN:

h+h = e )

i s called

N - s y s t em. Every ii-system i s a b e l i 8 n (see t h e proof o f 9.13(b)) inteqral

(so 3b)Z)

near-field,

and

i s a t hand), A f i n i t e ri-system i s a

b u t t h e r e do e x i s t i n f i n i t e H-systems w h i c h

are neither rinqs nor near-fields

(see Linh-f?alone ( l ) ,

L i g h - M c Q u a r r i e - S l o t t e r b f c k ( 1 ) a n d U t C u a r r i e (1),(3)).

If

N IMo(r)

a n d ti i s a n : I - s y s t e m c o n t a i n i n o

e v e r y f u n c t i o n o f N i s odd ( c f .

9.152jb)).

id,,

then

9j Miscellaneous topics

9.277

399

AUTOMORPHISM G R O U P S OF N E A R - R I N G S Scott (18) studied the group Aut(N) o f a l l automorphisms o f a n e a r - r i n g N t h e n an : x A

+

nxn-'

n invertible)

i s the

m o r p h i s m s o f N. where D(N)

=

~ n , I.f

i s i n Aut(N).

(near-ring-)

n t Nd i s i n v e r t i b l e

I n n ( N ) : = { a n I n e Nd

A

(normal) subgroup of a l l i n n e r auto-

As f o r g r o u p s we g e t D ( N ) / Z ( N ) - I n n ( N ) ,

Cn € N d l n i n v e r t i b l e ] a n d Z ( N ) = i n

E

D(N)lnn=id].

N i s c a l l e d complete

I n another analogy t o groups, a n r .

( c f . 9 . 1 0 0 ( b ) ) i f Z ( N ) = { I } , N &?Io and i f a l l autornorphisms of N a r e i n n e r . I f r i s a complete group and N = A ( r ) such t h a t ,,,T i s m o n o g e n i c t h e n N i s a c o m p l e t e n e a r - r i n g w i t h Aut(N)-Aut(r). By 7 . 1 6 ,

7.59

For instance,

I(Sn],

nf6,

i s o f t h i s type.

r.

M o ( r ) i s c o m p l e t e and A u t ( M o ( r ) )

See a l s o

and 9.226.

Magill

(7) studies a nr.

c a l l e s N,

:=

(N,+,.,)

(N,+,.),

with

n

m

c h o o s e s some a E N a n d rn : = nam

a

the near-ring

l a m i n a t e d by a. For N = M (IR) , Aut(Na) i s determined. cont I n a more g e n e r a l frame, automorphisms a r e s t u d i e d by Ncbauer ( 1 2 ) and P l o t k i n ( 3 ) . 9.278

DICKSON-NEAR- RINGS

The d e f i n i t i o n s o f c o u p l i n r l maps, Dickson n r . ' s

derived nr.'s

c a n be f o u n d i n 9.90.

and

For a detailed study

o f t h e s e c o n c e p t s s e e F l a x s o n ( 8 ) a n d Timm (6),(7).

O f course,

a Dickson near-rina

One may w r i t e a DFIR a s a r i n g and

(D,+,o)

Maxson shows e.q.

iff

(D,+,-)

(=: DNR) i s abelian.

the derived nr. i n (8) t h a t

has one and

(0

(O,+,o)

dcD":

(D,+,-)

where

(D,+,.,o),

bd

=

G

i s

6).

has an i d e n t i t y

$

6.

A finite

DNR w i t h i d e n t i t y i s a n f . . The i d e a l s t r u c t u r e o f a DiIR i s a l s o c o n s i d e r e d by P i e p e r (1) i n c o m p a r i n q t h e l e f t i d e a l s o f

(D,+,o). and

(D,+,-)

The c o n n e c t i o n b e t w e e n homomorphisms o f

(D,+,o)

and (O,+,*)

a r e s t u d i e d i n Maxson (13).

Kerby ( 5 ) s e t t l e s t h e q u e s t i o n i n which cases t h e nr. quotients o f

(D,+,o)

o f quotients o f

i s a !lickson

(D,+,*).

Of

o n e w.r.t. t h e r i n n

59 MORE CLASSES OF NEAR-RINGS

400

Aside from these considerations, "changed m u l t i p l i c a t i o n s " .

9.279

M a g i l l (2!,(7)

also studies

See a l s o 9 . 2 7 7 .

N E A R - R I N G S A N D N U M B E R THEORY

C o n n e c t i o n s between n e a r - r i n g s o f f o r m a l power s e r i e s and number t h e o r y were p o i n t e d o u t b y F r o h l i c h ( 9 ) .

Other

c o n n e c t i o n s a r e e s t a b l i s h e d i n Mazzola ( 1 ) and L i g h ( 2 0 ) . 9.280

NEAR-VECTOR S P A C E S I t seems n o t t o b e q u i t e c l e a r how t o d e f i n e a n e a r - v e c t o r

space.

Beidleman ( 1 ) d e f i n e d i t as a 2 - s e m i s i m p l e N-group

(N a n f . ) ,

and developed a k i n d i f " n e a r l y - l i n e a r "

algebra.

O t h e r a p p r o a c h e s t o t h i s c o n c e p t a r e made b y A n d r @ ( 3 ) ¶ ( 5 ) ¶

( 6 ) , Bachmann ( 2 ) .

Hule-Muller

(1) study a l q e b r a i c equations

over n r . ' s . 9.281

p-SINGULAR N E A R - R I N G S

SYLOW-TYPE T H E O R E M S ; f e r r e r o (1),(2)

N

and

\ : i \ = rn,

shows t h a t

k p /m

but

p

k+l

%in

Nd

implies the existence o f a two-sided i n v a r i a n t k subqroup o f N o f o r d e r p If I N / = p - q (p,qc_IP, p < q )

.

and

N

i s

n o t a b e l i a n t h e n ii h a s n o s u b n e a r - r i n o o f o r d e r p .

I f ti i s f i n i t e a n d

pclP,

/NI,

properly divides

:I

i s called p-singular

b u t II h a s n o s u b n e a r - r i n o w h o s e

o r d e r i s d i v i s i b l e b y p . So p - s i n a u l a r n r . ' s f o r n o t f u l f i l l i n o the Sylow-theorems".

N

i s

~ ? 7 a ~n d

p,N

if p

a r e "minimal

A p-sinnular nr.

i s s t r o n o l y monooenic.

See F e r r e r o (4)¶(5),(7)¶(18),(19) a n d S c o t t ( 8 ) . 9.282

LOCAL N E A R - R I N G S i s c a l l e d l o c a l i f L : = L(t!): = I X E N J X has no Nc7lofi??, l e f t i n v e r s e } srJ H . ( t h i s h a p p e n s i f f L i s a s u b q r o u p ) . Maxson ( 1 ) , ( 3 )

A local nr.

shows:

i s indecomposable.

w i t h DCC i s s i m p l e . A n r .

N i s l o c a l i f f N has a unique L i s q r . a n d i f PI i s n o t So L
maximal N-subgroup 2-radical

Hence a 1 - s e m i s i n p l e one

(namely L ) .

401

9j Miscellaneous topics

I f IJ i s l o c a l t h e n U/t i s a n f . , h e n c e a simple n r . i s l o c a l i f f i t i s a n f . . A l o c a l nr. has o n l y 0 and 1 as i d e m p o t e n t s . The a d d i t i v e o r o u p o f a f i n i t e l o c a l nr. i s

a p-qroup. Maxson ( 6 ) q o e s on t o d e t e r m i n e a l l l o c a l n r . ' s o f o r d e r

p a n d p 2 I n ( 9 ) a n d ( 1 2 ) he p r e s e n t s l o c a l n o n - r i n n s on n o n - c y c l i c a b e l i a r i ( p - ) n r o u p s o f o r d e r 25 a n d more results in this direction. O t h e r e x a m p l e s o f l o c a l n r . ' s a r e o i v e n by b l a f f ( V ) F I X ] , where F i s a f i e l d w i t h IF123 (Maxson ( 1 ) , ( 3 ) ) , ( C l a y - D o i ( 2 ) - 7 . 9 8 ) , a n d E ( r ) , A(r) a n d I ( T ) , w h e r e r i s a q e n e r a l i z e d q u a t e r n i o n a r o u p ( Y a l o n e (7)). See K a r z e l - V e i s s n e r ( I ) , P i e p e r (1) , ( 2 ) and

.

(1) f o r a p p l i c a t i o n s o f l o c a l nr. 's t o g e o m e t r y ( c o o r d i n a t i s a t i o n ) . Local n r . ' s w i t h DCC on p r i n c i o a l N-subgroups a r e s t u d i e d i n Ramakotaiah-Reddy ( 1 ) . Armentrout-Hardy-Maxson

9.283

ASSOCIATED RINGS

Let N be a b e l i a n a n d g e n e r a t e d by a l l h n :

A(N)

N

+

N.

m

--t

mn

t h e s u b n e a r - r i n g of M ( N , t ) ) Then A ( N ) is a ring

(the "ring associated to N " ) a n d was i n v e s t i g a t e d i n Williams ( 1 ) . N a n d A ( N ) are closely related. 9.284

S H E A F S O F NEAR-RINGS A sheaf

o f near-rings

i s a d i s j o i n t union N i ( i E I ) o f near-rings (called the s t a l k s o f Y i f I a n d s' a r e t o p o l o g i c a l s p a c e s t o g e t h e r w i t h some a s s u m p t i o n s o n t h e s e topologies. S h e a f s o f n e a r - r i n g s a r e s t u d i e d i n B e t s c h ( 5 ) ( a DaunsHofman-type r e s u l t ) and ( f o r c e r t a i n I F P - n e a r - r i n g s ) S z e t o ( 8 ) - ( 1 0 ) a n d Szeto-Wong ( 1 ) .

in

8 9 MORE CLASSES OF NEAR-RINGS

402

9.285

FULLY GENERATING S U B S E T S o f a n r . N a r e s u b s e t s S s u c h t h a t each l e f t i d e a l o f N c a n be g e n e r a t e d b y e l e m e n t s o f S.

T h i s c o n c e p t was i n t r o d u c e d a n d s t u d i e d b y V a n d e r W a l t several

p r o p e r t i e s can be

t r a n s f e r r e d from S t o N i n t h i s case.

(2),(3).

As o n e m i g h t e x p e c t ,

The e l e m e n t s o f S

must be " e v e n l y and d e n s e l y d i s t r i b u t e d t h r o u g h o u t N " S f u l l y g e n e r a t e s N.

I f S S N f u l l y generates N then the

f o l l o w i n g p r o p e r t i e s t r a n s f e r from S t o N: being a near-field, results, 9.286

simplicity,

zero-symmetry,

DCCL a n d ACCL.

For more

s e e Van d e r W a l t ( 3 ) .

NEAR-RINGS

A N D T H E I R "CHILDREN"

I n many p l a c e s , all

i f

r e l a t i o n s b e t w e e n a n r . o r an N - g r o u p and

( = s u b s t r u c t u r e s and

( o r some) o f i t s " c h i l d r e n "

f a c t o r s ) a r e s t u d i e d . We m e t t h i s t o p i c j u s t b e f o r e i n 9.285

and w i l l m e e t i t a g a i n i n t h e n e x t number 9.287.

Ferrero-Cotti

( 9 ) s t u d i e d c r i t i c a l and c o c r i t i c a l near-

r i n g s and N-groups

( t h a t a r e those ones w h i c h a r e n o t

b u t do n o t b e l o n g t o t h e v a r i e t y g e n e r a t e d b y i t s

simple,

p r o p e r f a c t o r s ( t h o s e ones w i t h p r o p e r s u b s t r u c t u r e s w h i c h do n o t b e l o n g t o t h e v a r i e t y g e n e r a t e d b y i t s p r o p e r s u b structures,

r e s p e c t i v e l y ) ) . Near-rings

proper subnr.

F

??, i n w h i c h e v e r y

i s a near-field are studied i n Pellegrini-

Manara ( 1 ) . I f s u c h a n r .

i s not integral,

but without

0 then i t i s isomorphic t o a product

n i l p o t e n t elements

I f i t i s i n t e g r a l then N i s

o f two f i e l d s o f p r i m e o r d e r . s i m p l e and 2 - p r i m i t i v e .

I n any case,

such a n r . can be

generated b y 2 elements and a l l p r o p e r N-subgroups

and

l e f t i d e a l s a r e m a x i m a l . See a l s o 7 . 1 0 2 ( f ) a n d t h e l i n e s a f t e r 1.40. 9.287

ULTRAPRODUCTS a n e a r - r i n g f o r each i

L e t I b e a s e t a n d Ni a f i l t e r on I ( i . e . F 1 c F2

n

i€ 1

Ni

A

F1

E

=>

Y

F2

+

0, 0

4J ,

EY)t h e n

consisting of a l l

F

Il Ni

7

( . . . ,niy...)

E

I. I f

~ E 7 ~ => F

~

3is F*

E Y ,

i s the subnear-ring o f w i t h { iE I l n i = 0 )

€7,

9j Miscellaneous topics

403

c a l l e d t h e J - f i l t e r p r o d u c t of t h e Nils. For i n s t a n c e , i f = { J GI I I \ J i s f i n i t e } t h e n II N i = 8 N i . A maximal J iE1 f i l t e r is called u l t r a f i l t e r , the corresponding products

'3:

a r e t h e u l t r a p r o d u c t s . J i s an u l t r a f i l t e r i f f o r a l l J G I e i t h e r 3 ~ o3r I \ J E Y . I f k E I then := {JSI/kB Jlis ll N 1. = N k . O t h e r u l t r a f i l t e r s e x i s t , an u l t r a f i l t e r with

rk

k' b u t nobody h a s e v e r s e e n t h e m . Many r e s u l t s from model

t h e o r y a r e a p p l i c a b l e . F o r e x a m p l e , e v e r y n e a r - r i n g c a n be embedded i n a n u l t r a p r o d u c t o f i t s f . g . s u b n e a r - r i n g s . S e e e . g . ( G r a t z e r ) . A l s o , o n e c a n show t h a t i f N i ( i c I ) i s 2 - p r i m i t i v e o n T i a n d i f f i s an u l t r a f i l t e r o n I then 9 N i i s 2 - p r i m i t i v e o n ll r. U l t r a p r o d u c t s a r e r e l a t e d t o 7 I 1d i r e c t ( c o - ) l i m i t s ( s e e Mahmood ( 3 ) ) a n d m i g h t p r o v e u s e f u l in future research. 9 . 2 8 8 S T I L L M O R E TOPICS ( a n d s t i l l

incomplete!)

Semigroup n e a r - r i n g s a r e i n t r o d u c e d i n Banaschewski-Nelson ( 1 ) . Near-rings with involution a r e s t u d i e d in FerreroC o t t i (4),(8), S c o t t ( 1 3 ) a n d S u p p a ( 1 ) . Wreath p r o d u c t s o f n e a r - r i n g s were a l r e a d y mentioned i n $ 9 h ) a n d $ 9 i ) . S e e C l a y ( 1 4 ) , ( 1 7 ) , Clay-Maxson-Meldrum ( l ) , Holcombe ( 9 ) - ( 1 4 ) a n d Velasco ( 1 ) . A c o n n e c t i o n between n e a r - r i n g s a n d d i f f e r e n c e e q u a t i o n s i s e s t a b l i s h e d i n Lopez ( 1 ) . F i n a l l y , we m e n t i o n a l g e b r a i c e q u a t i o n s i n n e a r - r i n g s a n d t h e i r r e l a t i o n t o e x t e n s i o n s , t h e amalgamation p r o p e r t y a n d i n j e c t i v i t y ( H u l e - M u l l e r ( 1 ), K i s s - M a r k i - P r o h l e - T h o l e n

(1))-

APPENDIX Near-rings of low order 222 remarkable examples and counterexamples List of open problems Bibliography Supplementary works List of symbols and abbreviations Index

405

NEAR-RINGS OF LOW ORDER

Now we g i v e d e s c r i p t i o n o f a l l n e a r - r i n g s o f o r d e r 1 7 a n d o f s e v e r a l c l a s s e s o f n e a r - r i n g s o f o r d e r 8 a n d 1 2 . The w h o l e d i s c u s s i o n i s d u e t o C l a y ( 2 1 , ( 4 1 , ( 7 ) , ( 8 1 , (91, A n p e r e r ( I ) , A n g e r e r - P i l z ( 1 ) a n d Yearby ( I ) . B e c a u s e o f 9 . 9 2 , t h i s ammounts t o the d e s c r i p t i o n of a l l mappings $ : T + E n d ( r ) , y GY w i t h t h e p r o p e r t y mentioned i n 9 . 9 2 , where r i s a group o f " s m a l l " o r d e r . The m u l t i p l i c a t i o n - $ i n r i s t h e n g i v e n by y 6 = $ & ( y ) . This -f

w i l l be done i n t h e f o l l o w i n g way:

( a ) I f r = { y , , . . . , ~ , } , we l i s t t h e e n d o m o r p h i s m s a l , ... a k o f r . ( b ) Every isomorphism c l a s s of n e a r - r i n g s of o r d e r n i s det e r m i n e d by t h e n - t u p l e ( k ,,..., k n ) o f e l e m e n t s o f N o , where @ = a . S o y i . + y j = @y ( y i ) = c l k . ( Y i ) ' Yi ki j J ( c ) The n u m b e r s f o l l o w i n g t h i s n - t u p l e d e n o t e t h e numbers o f t h o s e automorphisms o f ( a ) which y i e l d isomorphic n e a r - r i n g s on r. F o r t h e n e a r - r i n g s l i s t e d b e l o w , much more i n f o r m a t i o n i s g i v e n . L e t t e r s f o l l o w i n g ( a ) - ( c ) h a v e t h e s e meaning f o r t h e n e a r - r i n g N considered: A...N i s abstract affine

.

N i s commutative D. . . N i s d i s t r i b u t i v e F...N i s a near-field G . . . N i s distributively generated I...N i s integral N...N i s nilpotent O...N i s planar P. . . N i s prime Q . .. N i s q u a s i r e g u l a r R...N i s regular W...N i s without non-zero n i l p o t e n t elements C..

We o b s e r v e t h a t N i s z e r o - s y m m e t r i c i f f i t s n - t u p l e s t a r t s w i t h e n t r y 0 . N i s c o n s t a n t i f f i t s n - t u p l e i s ( 1 , l ,... , l ) . T h e n - t u p l e

406

APPENDIX

a ao(a)

a14(a)

n,(a)

al(a)

b u,(b)

aI4(b)

a2(b)

nl(b)

c ao(c)

"14(c)

aZ(c)

al(c)

n l , a 2 ,. . . ,a6 a r e i s o m o r p h i s m s y i e l d i n g i s o m o r p h i c n e a r - r i n g s .

This near-ring N i s a f f i n e , commutative, distributive(1y g e n e r a t e d ) and has c a s i d e n t i t y . B u t N i s neither a near-field, n o r i n t e g r a l , n i l p o t e n t , e t c . The r a d i c a l s a r e g i v e n by = J2(N) = t O , a ) . Hence J 1 , * ( N ) = 7 1 ( N ) = {O,al a s w e l l .

2,(N)

We c l o s e w i t h s e v e r a l n e a r - r i n g s o f o r d e r 8 a n d 1 2 a n d a l i s t o f t h e t o t a l numbers o f n e a r - r i n g s on m o s t g r o u p s o f o r d e r

AfZ1 = !Ol:

This case i s trivial.

1) (O,'?); 1 ; A D G I N Q R 2) ( 0 , l ) ; 1 ; A C D F G I P Q R ; 3) (1,l); 1;AIPQRW

I=I

c

12.

11

407

Near-rings of low order

0

f

Ia021aP

0

0 1 2

1

2 ; ; ; ; ;

1,2; 1,2; 1 ,2; 1,2; 1,2;

1) ( O , o , o , o ) ; 2) (O,l,qyQj; 3) (0,3,0,0); 4) (O,O,l,Oj; 5) (O,l,l¶g); 6) (0,1,0,1); 7) (Q,2,q,l): 8 ) (fi,l,l,l); 9) (l,!,I,l); 1Q) (0,1%3,1); 11) ( g , 1 , 3 , 2 ) ; 12) ( 9 , 3 , 0 , 3 ) ;

2

2 0 1 ACDGINQR PQ IPQRW AIPQRW ACDFGIOPQR;

1; ACDGNQ 1,2; 1 , 2 ; NQ 1; P 1,2: P

1; 1,2; 1; I P R W 1; A I P R W 1; 1,2; A C D G ; I = I 1; ACDGNQ

0 0 0 0

1

2

0 2 1

1=1

1111 1000 1111 1000 1000 1010 1010 1000 1010 1010 1010 1111

1111 1111 1111 1000 1000 IOZO 1010 1000 1010 1010 1010 1111

E ) Klein's four nroup {O,a,b,cl:

c b a O

a0

0 a b c

a1

a2

0 0 0 O a a O b c O c b

a3 a4 0 5 0 6 a7 a8 a9 ~10~11a12a13a19a15

0 b a c

0 b c a

0 c a b

0 0 0 0 0 0 c a a c O O b O a O c b a a O c c b

0 0 0 b b O b O a O b a

0 c c O

APPENDIX

O y l y l , l ) ; 1 ; IPRW; 0,14,1,1); 1,3,4; O y O y l y l ) ; 1 , 3 , 4 ; ADG 0 , 1 4 , 2 , 1 ) ; 1 , 2 , 3 , 4 , 5 , 6 ; ACDG; I = c 0,0,2,1); 1,2,3,4,5,6; 0 , 4 , 5 , 1 ) ; 1 , 2 , 5 ; ACDFGIOPR; I=c O y 7 , 1 1 , I ) ; 1 , 2 , 5 ; ACDGRW; I = c 0 , 1 4 , 1 1 , 1 ) ; 1 , 2 , 3 , 4 , 5 , 6 ; I-c 0 ,7,0 , I 1 ; 1 , 2 , 3 , 4 , 5 , 6 ; 0,0,0,1); 1,2,5; 0,7,13,7); 1,2,3,4,5,6; 0 , 7 , 0 , 7 ) ; ACDG 0 , 7 , 1 3 , 9 ) ; 1 , 2 , 3 ; ADG 0,7,0,9); 1,2,3; 0 , 1 3 , 0 , 1 3 ) ; 1 , 2 , 3 ; ACDGNQ 0 , 0 , 0 , 1 4 ) ; 1 , 2 , 3 , 4 , 5 , 6 ; NQ O,O,O,O); 1 ; ACDGNQ 7 , 7 , 1 , 1 ) ; 1 , 2 , 3 , 4 , 5 , 6 ; ARW 7,7,7,1); 1,2,3,4,5,6; 7 , 8 , 1 , 2 ) ; 1 , 2 , 3 , 4 , 5 , 6 ; APRW; I = b 7,7,1 ,7); 1,2,3,4,5,6; W 7,7,7,7); 1,2,3,4,5,6; A l y l y l y l ) ;1 ; AIPRW F ) 27, = { 0 , 1 , 2 , 3 , 4 ) :

1000 1100 1100 1100 1100 1000 1000 1100 1010 1000 1010 1010 1010 1010 1111 1111 1111 1000 1010 1000 1010 1010 1000

A d d i t i o n i s modulo 5.

I

1 ) (O,O,O,O,O);

1 ; ACDGNQ

2) ( 0 , 1 , 0 , 0 , 0 ) ; 1 , 2 , 3 , 4 ; P 3) 4) 5) 6) 7) 8) 9) 10)

~ O y l y l y O y O1 ,~2 ;, 3 , 4 ; ~ 0 , 0 , 1 , 1 , 0 ~ 1; , 2 ; p ( 0 , l ,I , I ,O); 1,2,3,4; ( 0 , 0 , 4 , 1 ,O); 1 , 2 , 3 , 4 ; (0,1,4,1 , 4 ) ; 1,2,3,4; ( O , l y l y l y l ) ; 1 ; IPRW ( 1 , 1 , 1 , 1 , 1 ) ; 1 ; AIPRW ( 0 , l ,2,3,4); 1,2,3,4;

P p OP IOPRW

ACDFGIOPRW; 1=1

1000 1100 1100 1100 1100 1000 1000 1100 1010 1111 1010 1010 1010 1010 1111 1111 1111 1000 1010 1000 1010 1010 1000

Near-rings of low order

=

{0,1,2,3,4,51:

409

A d d i t i o n i s modulo 6.

a5 - a Oa 1 a2a 3 a!+ 0 1 0 1 2 3 2 0 2 4 0 3 0 3 0 3 4 0 4 2 0 5 0 5 4 3

4 2 0 4 2

5 4 3 2 1

1=1

100000 100000 100000 100000 100000 100000 100100 100000 100000 100000 100000 100100 100100 100000 100000 100000 101010 101010 100000 100000 101010 101010 101010 101010 100100 100100 100000 100000 101010 101010 100100 100100 100100 100100 100000 100100 101010 101010

111111 111111 111111 111111 100000 100000 100100 111111 111111 100000 100000 100100 100100 111111 111111 100000 101010 101010 100000 111111 101010 101010 101010 101010 100100 100100 100000 100000 101010 101010 100100 100100 100100 100100 100000 100100 101010 101010

.--.-0-00000000000000000

~.--o.-~?oooo--ooooc--oOo

~~.-o~co~oooo?oooooo---

x

u u x x o m n n n x o x u m

x u m n x o

x ~ n u m o x

~ ? - o . - - r - o o o o ~ o o o o c ~ c o o o - - - 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c c - - - - - c c - - - ? - c - - c - - - -

? 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . - 0 0 0 0 ~ ~ 0 0 0 0 ~ 0 0 0 0 . - - . ? 0 0 0

-oooooorooooroo-000---

m o x x n v O m L l o X h

~ 0 0 0 0 - ~ 3 0 0 0 - - 0 0 0 0 ~ - . - 0 0 0

- 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - c c c - - - - ? - - - c - c - - c c - - -

0

om n v x x

m

+I

I

bj Y

“

m

m

t;

t;

E. u)

d

f

I

2 N

a m m n u m m u

X x x x

X x x x

o u u u o o onnnoo o m m m o o o u o u n n

0 0 0 0 0 0

o m on x x o m n u x z ,

o 4 8 o 2 8

0

a

o m n v x x

ooooo.-o-?---

0 0 0 0 0 - 0 - - - - 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

a aa

3

,--.-c-.--.--cccc-

aaaaa

41 1

Near-rings of low order

13) !7,7,7,1,1,7);1,2,3,4,5,6;

P

14) (7,7,1,2,1,2) ; 1 , 2 , 3 , 4 , 5 , 6 ;

PRW

(7,7,1,7,1,7);1,2,3,4.5,6; (?,7,7,7,1,2);1,2,3,4,5,6; (7,7,7,7,1,7);1,2,3,4,5,6; ( 7 , 7 , 2 , 1 , 7 , 7 ) ;I ,2,3 , & , 5 , G

P

PW P

!7,7,7,1,7,7);1,2,3,4,5,6; P 2;p 3,s;

p

2,4; p 395 3 9 5

2,4 2,4

2$4 3 , 5 , CDG 3 , 5 ; PRW

? , 5 ; PW 3.5 3,5 I PRW P G CDGNQ I PRW

I ) Z,

=

A d d i t i o n i s modulo 7.

{0,1,2,3,4,5,61:

j 5

0.0

ala20.3

a4

a StY5

0 0 0 0 0 0 0

0 I 2 3 4 5 6

0 4 1 5 2 6 3

0 5 3 1 6 4 2

0 2 4 6 1 3 5

0 3 6 2 5 1 4

0 6 5 4 3 2 1

1 ) ( 0,9, 'I, 0,Q ,'I,'3) ;1; ACDGNQ

2)(0,1,0,O,O,~,O);1,2,3,4,5,6; P 3) ( 0 , 1 , 1 ,Q,Q,q,9);1,1.,3,4,5,6; P 4 ) (0.1 , 0 , 1 , Q , O , O ) ; 1 , 2 , 3 , 4 , 5 , 6 ; P 5) ( 9 , l . l . l , O , Q , C J ) ; 1 , 2 , 3 . 4 , 5 , 6 ; p 6 ) ( o , l , l , ~ , l . ~ , ~ ) P; l , ~ 7 1 ( r) , 2 ,4,'I 1,'1. 3 ) ; 1 2 3 5.6; 1

9

9

9

9

op

100011 10001 1 100011 100011 100011 100011 100011 100000 100000 100000 100000 10001 1 100000 100011 100011 100000 100011 100011 100011 100011 100011 100000 100000 100011 100011 111111 100011

?00011 10001 1 100011 100011 100011 100011 100011 100000 111112 111111 111111 10001 1 111111 100011 100011 111111 100011 100011 100011 100011 100011 100000 111111 100011 100011 111111 100011

APPEN D I X

412

14)(0,0,1,1,1,1,Q); 1,2,3; P 15)(Q,1,1,1,1,1,0]; 1,2,3,4,5,6; P 16)(0,0,6,6,l,l,r)j; 1,2,3,4,5,6;OP 17)(9,n,6,1,6,1.?); 1,2,3,4,5,6;OP 18)(~,1,1,1,1,1,1); 1 ; IPRW 19 ) ( 1 ,1 ,I , I ,1 ,1,1) ; I ; A 1 PRW 20)(0,6,6,6,1,1,1); 1,2,3,4,5,6; IOPRW Zl)(n,6.6,1,6,1,1); 1,3; IOPRW 22)(n,2,4,4,1,2,1): 1,2,3,4,5,6;IOPRW 23)(9,1,2,3,4,5,6); 1,2,3,4,5,6;ADGIOPRW; I=! 24)(3,1,2,4,4,2,1); 1,2,3; IOPRW

J) I q

= C0,1,2,3,4,5,6,7 : f l d d i t i o n i s m o d u l o 8. 3 CL 4 a 5~ a6 a ? 000c)rJ')r)r)

a0 a 1 a 2 --

0 1

2

3 4 5

6 7

1

2 3

4

5 6 7

8 9 10 I1 12 13

14 15 16

17 18 19 20 21 22 23 24 25 26 27 28

0 0 9 Q

1 2 3 4

2 4 6 n

3 6 1 4

4 0 4 n

5 2 7 d

6 4 2 Q

-

7 6 5 4

Q 5 ? i J 1 6 7

q 6 4 2 3 6 4 2 0 7 6 5 6 3 2 1 10000000 11111111

11111111 11111111 11111111 11111111 11111111 11111111 10000000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1 11111111 11111111 11111111 11111111 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10000000 10001000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1 10000000 10000000 10000000 10000000 10000000 10000000 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 10000000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1 10000000 1 1 1 1 1 1 1 1

Near-rings of low order

29

3c 31 32 33 34 35 36 37 3E 39 4':

41 42 43 41 45 46 47 48 49 50 51 52 51 54

55 56 57 58

59 6'7 61 62

63 64 65 66 67 68 I

69 1 7'7 1 71) 72) 73) 74 1 75 1 76) 77 1 78) 79) 8')) 81) 82 1

P

P P NQ NQ NQ NQ

ADG NQ NQ

413

10000000 10000000 10000000 11111111 10000000 11111111 10000000 11111111 10000000 10000000 10000000 10000000 10000000 11111111 11111111 11111111 11111111 12211111 11111111 11111111 11111111 11111111 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10000000 11111111 10000000 11111111 10101010 10101010 10101010 1 0 1 0 l 0 1 0 11111111 11111111 11111111 11111111 10000000 11111111 11111111 1 1 1 1 1 1 1 1 10000000 11111111 11111111 11111111 10000000 10000000 10000000 10000000 10001000 11111111 10000000 l l l l l l l l 10000000 11111111 10000000 10000000 10000000 10000000 10000000 10000000 11111111 1111111 11111111 1111111 11111111 1111111 11111111 1111111 10000000 0000000 10000000 0000000 10000000 0000000 10000000 3000000 10000000 1111111 10001000 0001000 10000000 1111111 10000000 0000000 10000000 0000000 10000000 0000000 11111111 1111111 11111111 l l l l l l l 11111111 1111111 11111111 1111111 11111111 1111111 10000000 1111111 101 010'1 0 0101010

414

APPEND I X

~3)~0~1,0~1,0,7,0,7)~1,3~ ~ 4 ~ ~ ~ , 1 , ~ , 1 , 1 , 1 , 1 , 1 ~ ; 1P , 5 ; 8 5 )( n , l , 0 , 1 , 4 , 1 , 4 , 1 ) : 1 . 5 ;

~6)(~,1,~,1,4,5,Q,5~;1,3; 87)(q,1,0,1,4,7,Q,7):1,3; 88)(0,1,q,3,0,1,n,3);1,5:

8 9 ) (0,1 , Q , 3 ,n , 3 , Q , 1 ) ;1,7;

9Q)~0,1,~,3,4~1,Q,3);1,5: 3 1 ) ( r l , l , q , 3 , 4 , 3 , Q , l ) ;1,7;

107)(0,1,4,5,4,5,4,1);1,7; 1 0 8 ) ( Q , 1 , 4 , 7 , 4 , 1 , d , 7 ) ;1,5; l,’a)!~.1,6,!,6.5,6,5!:1,3;

, 5 , 6 , 1 ) ;1 , 7 ; 1~1)(9,1.7,7~r),1,1,7):1,5; 112)(0,2,4,4,0,2,r),4);1,5 NQ 113) 0,2,4,6,0,2,4,6);1,5 ACDl6NQ 114) 0,2,4,2,0,6,4,6);1,3 NQ 1 1 5 ) ( 9 , 2 , 4 , 6 ,0 , 6 , 4 , 2 ) ;1,7 NQ 1 1 6 ) ( 0 , 4 , 0 , 4 , 0 , 4 , 4 , 4 ) ; 1 , 5 NQ 1 1 7 ) ( 0 , 4 , 4 , 6 , 0 , 4 , 0 , 6 ) ; 1 , 5 NQ 1 1 8 ) ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) ;1; A I P P W 1 1 9 ) (0,0,0,0,0,0,0,0) ; l ; ACOGNQ 110 ( 3 , 1 , 6 , 5 , 4

I

129)(0,0,~,n,1,0,0,n);l;P 1 2 1 j ( o , o , i , r ) , q , o , i ,rl j ;I ; l22)(~,~,l,n,l,o,l,r));l;P 1 2 3 ) ( 0 . 0 . 4 .r) . 0 . 0 . 4 .0\:1 : NQ 124)(0,1,0,1,~,1,~.1~;1;

125~(r),i,0,i,i,i,n,i);i; P 126)(Q,1,0,1,4,1,n,l);l; 127)(r),1,1,1,0,1,1,1);1; 1 2 8 ) ( Q , 1,1,1,1,1,1* 1) ;1 ; IPRW

129)(n,i,2,1,4,1,2,1);i; 13‘3)(0,1,4,1,4,1,4,1):1;

131) ( 0 , 1 , 6 . 1 , 4 , 1 , 6 , 1 )

;1;

1 3 2 ) ( 0 , 2 , 4 , 2 ,0 , 2 , 4 , 2 ) 1 3 3 ) ( 0 , 4 ,n, 4 , r), 4 , r), 4 ) 1 3 4 ) ( 0 , 4 , 4 , 4 ,n , 4 , 4 , 4 ) 135) (r),6,4,6,r),6,4,6)

; I ; NQ :1 ; ACOGNQ ;1 ; NQ ; I ; NQ

10101010 10000000 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10000000 10000000 10101010 10101010 10101010 10101010 10000000 10000000 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10000000 11111111 11111111 11111111 11111111 11111111 11111111 10101010 11111111 10000000 10001000 10000000 11111111 101 0101 0 10000000 10101010 10001000 10000000 10101010 101 0101 0 10101010 11111111 11111111 11111111 11111111

10101010 10000000 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 10101010 11111111 11111111 10101010 10101010 10101010 10101010 11111111 11111111 10101010 10101010 10101010 10101010 10101010 10101010 10101010 11111111 11111111 11111111 11111111 11111111 11111111 11111111 10101010 11111111 10000000 10001000 10000000 11111111 101 0101 0 10000000 10101010 10001000 10000000 10101010 10101 010 10101010 11111111 11111111 11111111 11111111

Near-rings of low order

K ) The dihedral group D, "

=

415

{O,a ,2a ,3a ,b , a t b ,2a+b ,3a+b}:

For more compact typing, we list the endomorphisms row-wise from now on. 3

a

La

3a

b

a+b

0

0

C

9

a a a a 3a 3a 3a 3a

2a 2a 2a Za 2a 2a 2a 2a

0

0

0 0

0

9

0

0

0

0 0

2a b atb 2at-b 3atb 2a b atb 2atb 3atb 2a 2a 2a 2a atb atb 3atb 3atb b b 2atk Za-k

0 0

7

2a b a+b 2atb 3a+b 2a b atb 2atb 3a+b 2a 2a 2a 2a ath atb 3atb 3atb b b 2atb

0 h a+b 2a+b 3atb b atb 2atb 3atb 2a b atb 2atb 3atb 2a b atb 2atb 3a+b 0

0

0 Q 3

3 3a 3a 3a 3a

'i

?atb

7 0 3 0

0

n

0

0

0

n 0 r)

rl 0 r)

0 0 0 r) r)

0 0 0 r)

0 9 0 cl

0

3

t

o

a 0 9 a a a 2a 2a 2a 3a 3a 3a 0 b b 3atb atb atb b 2 a t b 2 a t b a+b 3atb 3atb 2atb

o r)

3 rt rl

n r)

r) r)

0 0

0 0 r)

9 r)

0 0 0

2a 2a 3a 0 a 2atb 3atb b atb

a a a a 9 rt 0

3a 3a 0 a 2a atb 2atb 3atb b

r)

0 0 0

b atb 2atb 3atb 2a 3atb 2a a+b 2a 2atb 2a 5 b. b atb 2atb 3atb 0 a 2a 3a

atb 2a+h 3atb b 3a+b b atb 2atb 2a b at b 2atb Sa+b r)

0

n 0

2a b

a+b 2atb 3atb 2a+b 3atb b atb 3atb 2a atb 2a 2atb 2a b 2a

2 a + b 3a+b

0 2ath 3atb b atb 2atb 3atb h atb 2a b a+b 2a+b 3ath 2a b atb 2a+b 3atb '1

n n

0 r)

b atb 2atb 3atb 2a 3atb 2a atb 2a 2atS 2a b

a tb 2a+b atb 2atb 2a+b 3 a t b 3atb b b atb 3a 2a r) 3a a 0 2a a

0 .lath b atb 2atb atb 2a+h

3atb b 2a b

a+b

2atb 3atb 0 r)

n rl rt

2a b a t t) 2atb 3a+b 2atb 3a+b b a+b 3atb 2a atb 2a 2atb 2a b 2a 3atb 3atb b atb 2atb a 2a 3a 0

APPENDIX

416

Not all near-rings on D8 will be listed (altogether there are 1447 isomorphism classes ! ) . But we give a complete list o f representatives o f isomorphism classes o f near-rings on D8 which are either non-zerosymmetric or have an identity or are integral. 1 0 , 1, 1, 1,19,

10, 13, lr), 10, 19, 10, 10, 10, 10, 10, 19,

10,

10, 10, 19, 10,

I , 1,lq); P 11110000 11110000 1, 1 , 1,10, l , l q , l f l ) ; p 11110000 11110000 1, 1 , 1 9 , 1 0 , 1, 1 , 1); P 11110000 11110000 1 , 1 , 1 0 , 1 9 , 1, 1 , 1 0 ) ; p 11110000 11110000 1, 1 , 1 9 , 1 q , l , l 9 , 1 ) ; p iiiinooo ~ ~ ~ ~ 1 , 1 , 1 0 , 1 r ) , 1,10,1Q); P 11110000 11110000 1, 1 , 1 n , 1 f l , 1 0 , 1 , 1 ) ; p 11110000 11110000 1 , 1,10,1q,1n, 1,101; p 11110000 11110000 1 , i , i ~ , i ~ , i ~ , ii qj ;, P 11110000 11110000 1 . 1 , 1 ~ , 1 9 , 1 0 , 1 ~ , 1 Qp ) ~ 11110000 11110000 l , l O , 1,10, 1 , 1 , l Q ) 11110000 11110000 i , i o , 1.ir-1, i , i o , i n ) 11110000 11110000 10100000 10100000 5.10, i , i o , 5) l , l 9 , 5,1q, 5 , 1 0 , 1 ) 10100000 10100000 11110000 11110000 1,10, 5,1f7,19,10,10) 1,19,10,19, 1, 1, 1) 11110000 11110000 l . l r ) . l ~ . l O . 1. 1 . 1 7 1 11110000 11110000

o o o o

wn,

11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 10100000 11110000 1010010 1 101 00101 10100000 10100000 10100101 10100101

11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 10100000 11110000 101 001 01 10100101 10100000 10100000 10100101 10100101

11110000 11110000 11110000 11110000

11110000 11110000 11110000 11110000

11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000

11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000

iiiioono ~ ~ ~ ~ o o o o

~ ~ ~ ~ ioi i ion ooo o

41 7

Near-rings of low order

49)(1n,24,10.24,13,1n,i~,Z4~ 50)(15, 51)(15, 52)(15, 53)(15, 54)(15, 55)(15, 56)(15, 57)(15, 58)(15, 59)(15, 60)(15. 51)(:5.

1, 1, 1,15, 1, 1 . 1 5 ) ; 1, 1, 1 , 1 5 , 1 , 1 5 , 1 5 ) . 1, 1 , 1 5 , 1 5 , 1, I , 1 ) : 1, 1 , 1 5 , 1 5 , 1, 1 , 1 5 ) ; 1, 1 , 1 5 , 1 5 , 1,15, 1); 1, 1 . 1 5 , 1 5 , 1,15,15); 1, 1 , 1 5 , 1 5 , 1 5 , 1, 1 ) ; 1, 1 , 1 5 , 1 5 , 1 5 , 1.15); 1. 1 , 1 5 , 1 5 , 1 5 , 1 5 , I); 1, 1 , 1 5 , 1 5 , 1 5 , 1 5 , 1 5 3 ; 1 , 1 5 . 1 , 1 5 , 1, 1 , 1 5 ) ; 1 . 3 5 , 5 . 1 5 , 5 . 1 5 . 1’ 32)(:5,1:, i , 1 5 , 1 5 , 1, : . : ? I . 63)(15,15, 1,15,15, 1,15,15)’ :2‘,~15.15.:5,15,15. 1 . 1.15: 6 5 ) (15,15,15,15,15, 1,15,15) 66)(15,15,15,15,15, 1,15,15) 6 7 ) ( 1 5 , 1,15, 5,15,35,35,35) 6 8 ) ( 1 5 , 1,15,15,15, 1, 1, 1) 6 9 ) ( 1 5 , 1,15,15,15, 1, 1 , 1 5 ) 1,15, 1) 7 0 ) ( 1 5 , 1,15,15,15, 7 1 ) ( 1 5 , 1,15,15,15, 1,15,15) 1, 1) 7 2 ) ( 1 5 , 1,15,15,15,15, 7 3 ) ( 1 5 , 1,15,15,15,15, 1,15) 7 4 ) ( 1 5 , 1,15,15,15,15,15, 1) 7 5 ) (15, 1,15,15,15,15,15,151 7 6 ) ( 1 5 , 1 , 3 5 , 5,15, 1,75, 5 )

P P PW P PW P P P P P

P

77)(15,15,15,15,15.15,15,35) 78)(15,15,15,35,15,15,15,15) 79)(15,15,15,35,15,15,15,35) sn)(i5,15,15,35,15,35,i5,i5) 81)(15,15,15,35,15,35,15,35) 82)(15,15,35,15,15, 83)(15,15,35,35,15,

1,15, 1,15,

5) 1)

84)(15,35,1~,35,15,15,15,35) 85)(15,35,35,35,15, 1,15, 5 )

1, 1, l , l Q , 1, 1, 1); 1, 1, l , l Q , l , l Q , 1); 1, 1 , 1,10,1Q, 1 , I O ) ; I , 1, 1 , 1 Q , l f l , 1 9 , 1 Q ) ; 90)(1Q, 1,1n, l , l Q , 1, 1, 2 ) 9 1 ) ( 1 0 , l , l O , 1 , 1 0 , 1,1n, 1) 9 2 ( 1 0 , 1,1fl, 1 , l f l , 1 0 , 1 , l r ) ) 931 ( 1 0 , 1, l o , 1 , I n , 10,13,1n) 94)(lr), 1 , 2 4 , l , l Q , 1 , 2 4 , 1 ) 9 5 ) ( 1 0 , 1,24, l , l ’ l , l n , l r ) , l Q ) 96)(10,24,10,24,10, 1,24, 1)

86)(10, 87)(19, 88)(10, O9)(lr),

97)(10,24,10,24,19,1O,l~,lr)) 98)( 1 0 , 2 4 , 1 9 , 2 4 , 1 0 , 2 4 , In , 2 4 1; 9 9 ) ( 1 5 , 1, 1, 1 , 1 5 , 1, 1, 1) 1 0 0 ) ( 1 5 , 1, 1, 1 , 1 5 , 1 , 1 5 , 1); 1 0 1 ) ( 1 5 , 1, 1, 1 , 1 5 , 1 5 , 1,15); 1 0 2 ) ( 1 5 , 1, 1, 1 , 1 5 , 1 5 , 1 5 , 1 5 ) ; 1 ’ 9 3 ) ( 1 5 7 1 , 1 5 , 1 % 1 5 , 1, 1, 1)

P RW

PW PW PW

PRW PW P

P

11110000 11110000 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100000 10100000 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 101 001 01 101001 01 10100101 10100101 10100101 10100101 10100101 10100101 10100102 10100101 101 001 01 101001 01 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100101 10100000 10100000 10100101 10100101 10100101 10100101 101 001 01 101001 01 10100101 10100101 10100101 10100101 10100101 10100101 101001 01 101001 01 10100101 10100101 10100101 10100101 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 10100000 10100000 11110000 11110000 11110000 11110000 10100000 10100000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 10100101 10100101 10100101 10100101 10100101 10100101 101001 01 101001 01 10100101 10100101

418

APPENDIX

104 105 196 107 1 r)8

( 1 5 , 1,15, (15, 1,15, ( 1 5 , 1.15, ( 1 5 . 1.15,

1,15, 1,15, 1 ) 1,15,15, 1,151 !,15,15,15,15) 1,15,15,35,15) ( l 5 , 1,15, 1,15,35,35,35) 109 115, 1,35, 1,15, 1 , 3 5 , 1'1

110 ( l Q ,1,24, 1,1Q,24,10,24) 11 1 ( I n , l Q , 1,1n,1q, 1 , 1, 1 ) ; P 112 ( i q , i q , I , I Q , ~ ~i ,,i n , 1); P l l ? f I Q , l Q , l , l ? , l n , l Q , 1,ln); P l!J ( I n , l n , l , l Q , l ~ , l ~ ~ , l n , l Pn ) ; 1 1 5 ( l ~ , l Q , l Q , l q , l Q , 1, 1, 1 ) 116 ( l o , l Q * l Q , l q , l 9 , 1 , 1 o , 1 ) 117 ( l " , l ~ , l n , l Q , l ~ , 1.24, I ) 118 ( 1 0 , 1 0 , 1 Q , 1 0 , 1 ~ , 1 Q , 1 , l n )

119 120 121 122 123

(lQ,l~,lQ,10,1n,2~,1~,24)

(15,15, 1,15,15, 1 , 1, 1 ) ; P W ( 1 5 , 1 5 , 1 , 1 5 , 1 5 , 1 , 1 5 , I ) ; PW ( 1 5 , 1 5 , 1 , 1 5 , 1 5 , 1 5 , 1,151; P (15,15, 1,15,15,15,15,15); P 124 ( 1 5 , 1 5 , 1 5 , 1 5 , 1 5 , I , 1, 1 ) 1 2 5 ( 1 5 , 1 5 , 1 5 , 1 5 , 1 5 , 1,15, 1 ) 126 ( 1 5 , 1 5 . 1 5 , 1 5 , 1 5 , 1 5 . 1 . 1 5 ) 127 jii5,i5,i5,i5,i5,15;15,15j 128)(15,15,!5,15,15,35,1~.~5) 123)(15,15,75,15,15, 1,15, 1) 13~)(15,35,15,35,15,15,15,15) ln1)(15,35,15,~5,is,3j115,35) 132) 15,35,15,35,15, 1 , 1 5 , 1 ) 1 3 3 ) [ 0 , 1 , 1 4 $ 5,15,16,17,18); I = a 134)( 0 ,

1,14,

Number near-rings

L ) The

I=a

7,15,16,17,18); I = a 7,15,21,17,23); I = a 3 , 1,14, 5,!5,16.35,18); I=a 0 . 1,14, 7.15.16.17.31~; I = a 0 , 1,14. 7,15,21,17,3nj; G ; I = a 0 , 1, 1, 1, 1, 1, 1, 1): IPRU 1 , 1 , 1 , 1 , I , i , 1, I ) , I P R W 8 6 , 1 1 3 , 1 4 1 and ( 1 5 , I l , I , I ,15,1 ,1 , 1 ) are t h e Boolean

1 3 5 ) ( 3 , 1,14, 1 3 6 ) ( 0 , 1,14,

137)( 138); 139j( 140)( 141)(

5,15,21,17,23);

10 00000 10100000 10 00101 10100101 10 00101 10100101 10 00101 10100101 10 OClOl 10100101 10100000 10100000 1111cooo 11110000 11110090 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11 10000 11110000 11 10000 11110000 1 1 10000 11110000 11 10000 11110000 11 10000 11110000 10 OClOl 10100101 10 00101 10100101 10 00101 10100101 10 o o 1 0 4 10100101 101 00101 101 001 01 10100101 10100101 10100101 1 0 1 ~ 0 1 0 1 10100101 lOlll@lOI 13100101 101 00101 10100101 10100101 10100101 10100101 10100101 10100101 1c)100101 101001u1 10100101 10100101 10100000 10100000 10100101 10100101 10100000 10100000 I O I O O I ~ Iiriooiui 10100101 li1100101 1 0 1 0 ~ ~ oi ooi o c o o n 10000000 100000011 10100000 10100000

087

De.

q u a t e r n i o n n r o u p r) = I O , a , Z a , 3 a , b , a + b , 2 a + b , 3 a + b ) : t -

0 a 2a 3a b a+b 2a+b 3a+h

0 a 0 a a 2a 2a 3a 3a 0 b 3a+b atb b 2a+b a+b 3a+b 2a+b

-b

2a

3a

2a 3a 0 a 2atb 3atb b a+!,

3a 0 a 2a a+b 2a+h 3a+b b

b atb 2a+b 3a+b 2a 3a 0 a

a+h

2atb

3a+b

a+b 2atb 3a+b h a 2a 3a

2a+b 3a+h b a+b 0 a 2a 3a

3a+h h a+b 2atb

r)

3a 0 a 2a

419

Near-rings of low order

-0

a

“ 0

r)

r)

R l

rl

rx 2

r)

a a a a 3a 3a 3a 3a b b b b 2atb 2atb 2atb 2atb atb atb atb atb Jatb 3atb 3atb 3atb rl 2a 2a

0 0

a3

“4

r)

mEj

a7

0 0

“9

r)

“6

a9

0

mi a

r)

“1 1

“I 2 “1 3 “I4

a1 E “1 f

0 (1

0 0 0 r)

a1 7

0

a1 E

r)

(XI

s

“2c “2 7

a2 2

r)

0 0 r)

Q2:

0

Q2L

r)

“2 t

a2t Q2:

0 0 0

2a 9 2a 22

2a 2a 2a 2a 2a Zit 2a 2a

2a 2a 2a 2a 2a 2a 2a 23

2a 2a 2a 2a

2s 2a r)

fl

n

3a

0 3a 3a 3a 3a a a a a 2ath 2atb 2atb 2a4.b b b b b 3atb 3atb 3atb 3atb atb atb atb atb

h 0 b 2a4-b at b 3atb b 2atb atb 3atb a 3a ath 3atb a 3a atb 3atb

n

a 3a b 2a+b a 3a b 2atb 2a

2a

n

2a

2a

atb

2a+b

rl

atb 3a+h 2atb b 3ath atb b 2atb 3atb atb a 3a asb 3asb 3a a b 2atb 3a a 2a+b b a 3a 2a 2a

r)

2a+b b

3atb atb 2atb b 3atb atb 3a a 3atb atb 3a a 3atb atb 3a 3

2a+b h 3a a 2atb b 2a

r)

r)

?a

I f we impose the s a m e restrictions a s f o r O,,

j u s t t w o n e a r - r i n g s ( b o t i i a r e Boolean): 1 ) ( 0 , l , I , I , I , I , I , I ] ; I PRW 2 ) ( l , l y l y l , l , l , l , l ) ; IPRCl

10000000 10000000 10100000 10100000

3atb

n 7atb atb b 2atb atb 3atb 2atb b ai-b 3atb 3a a 3atb atb a 3a 2a+b b it

3a b 2atb 3a a 25. 2a r)

here a r e

APPEN D IX

420 M ) Some n e a r - r i n g s

on Z4xZ2

. . . ,7 1

{0,1,2,

=

w i t h (0,O) = 0 (1,O) = 1 (2,O) = 2 (3,O) = 3 (0,l) = 4 (1,l) = 5 (2,l) = 6 (3,l) = 7

Let a.

=

i d and a1 be g i v e n b y

1 ) (O,O,O,O,O,O,O,O);

1,

A;

2 ) ~ 0 7 0 y 0 , 0 y l , l , l , l A~ O; R ; N ) Some n e a r~-rings o n Z2xZ,xZ2

+ __ 0

=

g2 =

=

rd2

2 3 0

1 2 3

{0,2,4,61

{0,2)

=

= I = = = = = =

a2 ~

00 0 0 0 0 0

(x,O).

= o

1 2 3 4 5 6 7 0 3 2 5 4 7 6 3 0 1 6 7 4 5 2 1 0 7 6 5 4 5 6 7 0 1 2 3 4 7 6 1 0 3 2 7 4 5 2 3 0 1 6 5 4 3 2 1 0 endomorphisms:

2I l 3 4 5 6 7

=

{0,1y2y...y7}

0 1 2 3 4 5 6 7

0 I 2 3 4 5 6 7 Some

1 2 3 4 5 6 7 1

=

Jo

al(x,y)

0 0 0 0 2 2 2 2

1) (0,2,0,3,0,2,0,3);

2 3 4 5 6 7

a3

alt_ a5

a6

a7 _ at3

0 0 0 0 4 4 4 4

0 1 2 3 4 5 6 7

0 1 0 1 4 5 4 5

0 0 2 2 4 4 6 6

l o=

0 0 0 0 6 6 6 6

103;

71 a2

0 1 2 3 1 0 3 2

gl/2=

{0,41;

= = N Remark: f o r t h i s n e a r - r i n g N we have

2)

7 , ( r b l ( N ) ) = iO1 t a l ( N ) . (0,2,3,3,4,1,5,5); g o = 71/2= { 0 1 ; = 72 =

3) (4,6,7,1,4,4,7,7);

4) (2,2,8,8,2,2,8,8);

0 ; All r a d i c a l s All r a d i c a l s

A;

=

{O,l}

=

{0,1,4,51

{0,1,2,31

42 1

Near-rings of low order

0 ) Some n e a r - r i n g s

+ 0 1 2 3 4 5 6 7 8 9 1 0 1 1

on A4

=

{O,l,

..., 111

0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 1 0 3 2 7 6 5 4 1 0 1 2 3 0 1 5 4 7 6 1 1 1 3 2 1 0 6 7 4 5 9 4 5 6 7 8 9 1 0 1 1 0 5 4 7 6 1 1 1 0 9 8 2 6 7 4 5 9 8 1 1 1 0 3 7 6 5 4 1 0 1 1 8 9 1 8 9 1 0 1 1 0 1 2 3 4 9 8 1 1 1 0 3 2 1 0 6 1 0 1 1 8 9 1 0 3 2 7 1 1 1 0 9 8 2 3 0 1 5

9 1 0 1 1 9 1 0 1 1 1 8 9 0 9 8 8 1 1 1 0 1 2 3 3 0 1 2 1 0 0 3 2 5 6 7 7 4 5 6 5 4 4 7 6

A g a i n , we l i s t e n d o m o r p h i s m s r o w - w i s e . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3

2 3 4 5 6 7 8 9 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 8 8 8 8 0 0 5 5 5 5 1 0 1 0 1 0 1 0 0 0 6 6 6 6 1 1 11 11 1 1 0 0 7 7 7 7 9 9 9 9 0 0 8 8 8 8 4 4 4 4 0 0 9 9 9 9 7 7 7 7 0 0 1 0 1 0 1 0 1 0 5 5 5 5 0 0 1 1 1 1 1 1 11 6 6 6 6 2 3 4 5 6 7 8 9 1 0 1 1 2 3 5 4 7 6 1 0 1 1 8 9 2 3 6 7 4 5 1 1 1 0 9 8 2 3 7 6 5 4 9 8 1 1 1 0 3 2 8 9 1 1 1 0 4 5 7 6 3 2 9 8 1 0 1 1 7 6 4 5 3 2 1 0 1 1 9 8 5 4 6 7 3 2 1 1 1 0 8 9 6 7 5 4 2 3 8 1 0 9 1 1 4 6 5 7 1 3 9 1 1 810 7 5 6 4 1 3 1 0 8 1 1 9 5 7 4 6 1 3 1 1 9 1 0 8 6 4 7 5 3 1 4 6 7 5 8 1 0 1 1 9 3 1 5 7 6 4 1 0 8 9 1 1 3 1 6 4 5 7 1 1 9 8 1 0 3 1 7 5 4 6 9 1 1 1 0 8 1 2 4 7 5 6 811 9 1 0 1 2 5 6 4 7 1 0 9 1 1 8 1 2 6 5 7 4 1 1 8 1 0 9 1 2 7 4 6 5 9 1 0 8 1 1 2 1 8 1 1 1 0 9 4 7 6 5 2 1 9 1 0 1 1 8 7 4 5 6 2 1 1 0 9 811 5 6 7 4 2 1 1 1 8 9 1 0 6 5 4 7

422

APPEND I X

Subgroups:

A ) {O}

B) C 0 , l l

C ) I0,23D ) CO,2,31 E ) {0,4,81

F) { 0 , 5 , 1 0 } G ) C0,6,11l H) {0,7,93

I ) {0,1,2,31 3 ) {O,l , . . . , I l l

I , a n d J a r e normal; I i s t h e commutator subgroup; A i s the center.

A,

1 ) ~0,0,0,0,0,0,0,0,~,~,1,3) Nilpotent elements: 0,1 ,2,3,4,5,6,7 Non-zero d i v i s o r s : 0 ,8,1 1 Idempotent e l e m e n t s : 0 ,I 1 Left i d e n t i t i e s : none Right i d e n t i t i e s : none 0 ,8,1 1 Reg u 1 a r e 1 emen t s : Central elements: 0,1 ,2,3,4 Quasiregular elements: 11 Left d i v i s o r s of zero: none Right d i v i s o r s of z e r o : none Distributive elements: 0,1,2,3,4 Subnear-rings: A,B,C,D,E,G,I , J N-subgroups: A,B,C,D,I , J Left ideal s: A31 , J Monogenic N - s u b g r o u p s : A,E,G Ideals: A,I,J Nil l e f t i d e a l s : A91 I Modu 1 a r 1 e f t id e a 1 s : I Prime i d e a l s : Maximal N - s u b g r o u p s : E,G,I I Maximal l e f t i d e a l s : 0 - m o d u l a r 1e f t i d e a l s : I I-modular l e f t i d e a l s : I 2-modular l e f t ideals: I 0-radical : I 1/2-radical: I I-radical : I 2-radical: I Nil i d e a l s : A,I Q u a s i - r e g u l a r l e f t i d e a l s : none This near-ring i s zero-symmetric, s u b d i r e c t l y i r r e d u c i b l e , 2 - s e m i s i m p l e , b u t n o t 2 - p r i m i t i v e and n o t d . g . . a o , a 1 , a 3 , a=9 i d a r e near-ring-endomorphisms; they a r e t h e o n l y N-endomorphisms a s w e l l . O n l y a9 i s an automorphism.

42 3

Near-rings of low order

2) ( O , O , O , O , l , I , I , I ,5,5,5,5) N i 1 p o t en t e 1 emen t s : Non-zero d i v i s o r s : Idempotent elements: Left identities: Right i d e n t i t i e s : Regular elements: C e n t r a l e l enien t s : Quasiregular elements: Left d i v i s o r s of zero: Right divisors.of zero: Distributive elements:

0,1 , 2 , 3 , 4 0,4 ,8 094

none none

0,4,8 all all all all all

elements elements except 4 elements elements elements

A,~,C,D,E,I,J Subnear-rings: A , B , C , D , E , I ,J N-subgrouos : A , I ,J Left ideal s : A,E Monogenic N - s u b g r o u n s : A , I ,J Ideals: A,I Nil l e f t i d e a l s : I Modular l e f t i d e a l s : I Prime i d e a l s : Maximal N - s u b g r o u o s : EYI Maximal l e f t i d e a l s : I 0 -modu 1 a r 1 e i t i d ea 1 s : I I-modular 1e f t i d e a l s : I 2 - nio d u 1 a r 1 e f t i d e a 1 s : I 0-radical : I 1/2-radical: 1-radical : 2-radical: Nil i d e a l s : Q u a s i - r e g u l a r l e f t i d e a l s : none T h i s n e a r - r i n g i s z e r o - s y m m e t r i c , d i s t r i b u t i v e , commutative and subdirectly irreducible. ~ ~ , ~ ~ , c in e~a r , - r ia n g ~ e n~d o,m oar p~h i s~m s ; a O , c x l , c x 5 , a 9 , a 1 3 , ~ 1 7 , , ~~ 2i 21 5 , ~ rN2- se n d o m o r p h i s m s . Automorphisms a r e ~ ~ , ( co r di e r~ 3 ,~ i n n e r a u t o m o r p h i s m d e t e r m i n e d b y 8 ) a n d u Z 5 ( o r d e r 3 , i n n e r automorphism determined by 5 ) .

3) (1,9,9,9,1 , I ,9,9,1,9,9,9)

Nil p o t e n t e l e m e n t s : Non-zero d i v i s o r s : Idempotent elements: Left identities: Right i d e n t i t i e s : Regular elementb: Central elements: Quasi r e g u l a r e l e m e n t s : Left d i v i s o r s of zero: Right d i v i s o r s . o f zero: D is t r ib u t i v e e 1 e me n t s :

0 a l l elements except 5 a l l elements except 5 none 1 ,2 , 3 , 5 , 6 , 7 , 9 , 1 0 , I 1 a l l elements none 0 0,19293 O,4,8 0

424

APPENDIX

Subnear-rings: A,B,C,D,E,I , J N-subgroups: A,E,J L e f t ideals: A , I ,J Monogenic N - s u b g r o u p s : E,I Ideals: A,I ,J Nil l e f t ideals: A M o d u l a r 1 e f t ideal s : A Y I Prime ideals: 1 Maximal N - s u b g r o u p s : E Maximal l e f t ideals: 1 0 - m o d u l a r l e f t ideals: 1 I - m o d u l a r l e f t ideals: none 2-modul a r 1 e f t ideal s: none A 0-radical : l/Z-radical : J 1-radical: J 2-radical: J Nil ideals: A Q u a s i - r e g u l a r left ideals: n o n e This near-ring is weakly commutative ( = P I ) , w i t h o u t nilpotent e l e m e n t s , subdirectly irreduc b j e ; N N is simple and cyclic. Near-ring-endomorphisms are a o y a ~ , a 5 , c x l= id; a 1 and c x l o a r e N-endomorphisms. Some more near-rings on A 4 : 0 ,O ,O ,O ,O ,O ,O ,O ,O , 9 , 2 1 , 2 5 ) 0 ,O ,O ,O ,O .O ,O ,O , I , 4 , 9 , 1 2 1 D ,O ,O ,O ,O ,O , O ,O , 9 , 9 , 1 0 , l o )

o,o,o,o,o,o ,0,9,0,0,0,17) 0 ,o ,o ,o ,o , 0 , 9 , 9 , 0 , 1 9 , 0 , 1 9 ) 0 ,o ,o ,o,o ,o , 9 , 1 0 , 0 , 9 , 0 , 1 0 )

0,0,0,0,0,0,9,10,18,0,20,0) O , O , O ,O , 0 , 9 , 9 , 9 , 0 , 1 3 , 1 3 , 1 3 ) 0 ,o ,o ,o ,o , 9 , 9 , 1 9 , 1 9 , 9 , 0 , 1 9 ) 0,0,0,0,0,9,21 ,25,0,21 ,25,9) O , O , O , O , l , I , I ,1 ,1 ,9,21 ,25) 0,0,0,0,1 ,1 , I ,4,5,6,6,6) 0 ,O,O,O $1 ,1 ,3,3,5 ,5,5.8) 0,0,0,0,1 ,1 ,4,4,1 ,4,9,12) 0 ,O ,O ,O , 1 .1 , 9 , 4 , 5 , 6 , 2 9 , 5 ) 0,O , 0 , 0 , 1 , 2 , 9 , 9 , 1 , 9 , 2 , 9 ) 0 , 0 , 0 , 0 , 1 ,9,21 ,25,1 ,21 , 2 5 , 9 ) 0,0,0,0,9,9,11 ,11,9,9,11 ,111 0 ,O ,O ,O , 9 , 9 , 1 2 , 1 2 , 3 2 , 3 1 , 3 1 , 3 2 ) 0 ,O ,O ,O , 9 , 1 0 , 2 0 , 1 8 , 9 ,18 ,1 0 , 2 0 ) 0 ,O , 9 , 1 3 , 9 , 1 3 , 9 , 1 3 , I 3 , 9 , 9 , 1 3 ) 0 , 9 , 2 1 , 2 5 ,O ,O ,O,O ,O , 9 , 2 1 , 2 5 ) 1 , 1 , 1 , 1 , 1 , 1 , I , 1 ,I , 1 , I , I 1 1 , 1 , I ,1 , 1 , 9 , 2 1 , 2 5 , 1 ,21 , 2 5 , 9 ) 1 , I ,1,9,1,9,9,9,1 ,9,9,9) 9,9 ,9,9 ,9,9, 9, 9, 9, 9, 9, 9)

425

Near-rings of low o r d e r

P ) A remarkable n e a r - r i n g o f o r d e r 3 2 :

L e t N be t h e d i r e c t p r o d u c t of t h e n e a r - r i n g 1 ) o f N ) a n d of 1 0 ) i n E l . N i s o f minimal o r d e r w . r . t . h a v i n g a l l J a c o b s o n - t y p e n e a r - r i n g r a d i c a l s d i f f e r e n t . See a l s o 5.67(v).

Order

Group

1

Number of n r ' s on r

r

Number of isomorphism c l a s s e s o f n r ' s on r

1

=I

1

2

=2

3

3

3

=3

7

5

4

17

12

Z2XZ2

z4

99

23

5

=5

29

10

6

'6

98

60

s3 =7

160

39

112

24

%

350

135

'4"2 (Z2

6982

115

a lot

many

D8

9308

1447

Q

4692

28 1

Z9 z3xZ3

1190

222

8907

264

1200

329

3454

206

7 8

9 10

P

Z1 0 Dl 0

11 12 i

=1 1

1312

139

212

5522

1749

A4 M

872% I

6571

483 I

824

426

222 REMARKABLE EXAMPLES AND COUNTEREXAMPLES

The f o l l o w n g l i s t c o n t a i n s some e x a m p l e s o f n e a r - r i n g s w i t h r e m a r k a b l e p r o p e r t i e s . They a r e o r d e r e d a c c o r d i n a t o t h e i r " d om i n an t " property, followinp the course o f t h i s book. Some r c m a r s : I t i s v e r y d i f f i c u l t t o d e f i n e t h e b o r d e r l i n e between examples a n d theorems. I t i s e q u a l l y d i f f i c u l t t o d e c i d e , which p r o p e r t y i s " d o m i n a n t " . S o I t r i e d t o d o my b e s t a n d I a p p o l o g i z e f o r a l l p o s s i b l e c a s e s o f d i s a g r e e m e n t . Covered a r e o n l y p a p e r s which d e a l w i t h n e a r - r i n g s ( a n d n o t t h e o n e s on n e a r - f i e l d s , e t s . ) . E x c l u d e d a r e e x a m p l e s which a r e a l r e a d y m e n t i o n e d i n t h i s b o o k . " E " m e a n s : English v e r s i o n and " M 1 3 " means t h a t t h e p a g e number i s 1 3 i n t h e m a n u s c r i p t o f t h i s p a p e r (when i t h a s n o t a p p e a r e d y e t ) . No c l a i m i s made c o n c e r n i n g c o m p l e t e n e s s .

1)

E e r m a n - S i l v e r m a n 1, 2 9 / 3 0 :

2) p i i r

1 0 , 100:

Various examples o f

3)

Choudhari

1, 10:

N e a r - r i n g s on

4)

Choudhari

1, 13:

h

5)

Heatherly 7 .

350:

6 ) H e a t h e r l y 7 , 351: 7)

M a l o n e 1, 3 1 :

invariant

( b u t n o t normal) subgroups e t c

Another m u l t i p l i c a t i o n on R I x l t u r n i n g i t i n t o a near-ring

i

Z2x

Hom(N,N'),

N=(ZZ,

Z2 w i t h N-subgroups which a r e n o t l e f t

where h(N) i s not d l e f t i d e a l of

ideals

N'

i s simple w i t h x3=x, r e g u l a r and Nd=iO1.

,t,*)

N = ( Z 7 ,t,*) i s s i m p l e w i t h x 4 = x ,

INI=IR,

N c C N,

the automorphism of

N can be "mated"

by t h e ones f r o m N

and Nc. 8)

Malone 4,

37, 39

N near-rings, "mated"

9)

10)

Prehn 3,

M3:

N=Z2

w h e r e t h e h o m o m o r p h i s m s o n No a n d N C c a n a n d c a n n o t b e

t o g i v e a homomorphism on N

has an i d e n t i t y ,

Ferrero-Cotti-Rinaldi

1, 7 8

:

b u t NS3 i s n o t u n i t a r y

N finite,

simple,

a l l proper r i g h t i d e a l s a r e maximal,

b u t t h e r e a r e more t h a n 2 o f them

421

222 remarkable examples and counterexamples

Graves

1 , 7:

Graves

I , 11:

Graves

I , 11:

N nr.

Graves

1, 12:

tol[x'l.

Graves

1, 3 0 .

N

N dg. w i t h o u t Ore c i r n d i t i o n , N n r . on S3 w i t h l e f t

.?Z4

on

I

cr~ndition, l e f t cancellation,

flrp

351:

(N,t)

finite,

(O.y)=O 136R a n d 4 ,

L i g h 4,

3 7 a n d L i g h - M a l o n e 1, 3 7 8

11:

G67:

Liqh-Malone Marin I, Maxson

N finite

N near-rinq

136:

w i t h 1,

on

Z4w i i h

"No

+

R 4 , where

on

ly#0: xy-0,

no

left

ideals

the non-zero-divisors

do n o t

ruhijroup

N abelian,

x-0,

w i t h 1 ptid v x # 0

n e a r - r i n g on

b u t i o n i e y#O w l t h

some xC0,

but not integral

n o t every element has a unique square r o o t

lZ5,

IJI

N a n e a r - r i n g o n Li,

1, 2 6 :

Nd#rOL, - x i 0

i'l n e d r - r i n g

(-1)x-x

N near-ring

I , 375:

nn i d e n t i t y ,

idetitity.

iul-iii a

L i g h 6,

(O:x)=N for

siiiipl~,(O'O)=N,

i s n o t one-sided

L i g h 2,

law

0

multiplicative center

1 6 ) B a s k a r a n 1,

l e f t cancellation

a < ahove

Left

:

without

n o t emheddable i n t o a n e a r - f i e l d

: p r o p e i t i e s as above;

propepties

RoIIx

hut with l e f t cancellation law

Ore c o n d i t i o n ,

iy: y x = l , b u t Nd=;Ol i s nm d i v i s o r o f z e r o ,

w i t h some x w h i c h

but not l e f t

cancellable

23

1

24 i5

McQuarrie 2 ( a l l ) :

165.

Pilz 3,

1

Plasser

1,

N

N infinite,

Beidleman 1 , 49:

213)

Beidleman I ,

N:

50:

simple,

w i t h DCCI,

N dq.

Heatherly 3 , 705:

I,

witti

6

'r

N with

with

103:

N

I)

riii

OCCl.

N.

br=a(r-ic) where - e L Nd

N

lids

=lo'

?il, dg,

or

but not ACCI. n ~ l p o t e n t deals

15

not

is nilpotent.

not the DCCI,

rlear-ring

DCCl,

T h e sum o f a l l

J E ( N ) i s q r . and (N,+)

I , 123:

t

c

lrft i d e n t i t y e.

D C C N , N = N0 .

DCII,

no

FolxN ( F a f i r l d )

Ligh-Ramakotaiah-Reddy

B ~ i d l e m a nI .

Liz

b u t riot

are

32)

h

110

+ r=O.

n o t N-~.iinple, N dg.

nilpotent.

1

x

1 and r ( - l ) = r

with

E i 7 ) r ' s w i t h D C C l :dnd / u r A C C I a n d w i t h

29) B e i d l e m a n 9 , 203:

31

o r 1 I)

A non-ring

,,r

383:

27)

30)

N but

Rlx] with a,c

111.16:

26) R e i d l e m a n 3 ,

non-abelian,

i s integral,

on t h e n i l p o t e n t g r o u p

but not regular

D8,

hut a l l

(0:x)

=N.

N a d 7 r e c t sun1 o f

simple l e f t ideals,

but nut a direct

sum o f N - s i m p l e l e f t i d e a l s 33)

34

!

S c o t t 8,

M3:

N=I@J, but 1, Mb.

Choudhari-Goyal

,,r

does n o t s p l i t

N n r . on S 3 w i t h o u t proper l e f t i d e a l s ( s o Lz=L) f o r a11 l e f t b u t r i o t Mz=M f o r a l l

ideals),

35)

Polin 2,

36 !

Oswald

71:

F o r M,M'

N let

'.N

M o

M'

N-subgroups

b e t h e N - s u b g r o u p g e n e r a t e d by MM'. o

is not

associative.

1, 6 1 :

N

d.g.

w h i c h i s " s l . r ~ ~ t ls ye m i p r i m e "

w i t h M2=[0;),

hut not

v M'=iOI)

37) B e i d l e m a n 9 , 2 0 4 :

N dg. left

38

!

Betsch 3,38:

N

5

with

I,

n 11p o t e n t .

15

10:

( v M,M'

T(N1=...=)p(N)

ideals i s

M(Zq) , NZ4

(1.e.

" s t r i c t l y prime"

i s nilpotent,

dti

ideal

u i

t y p e 0. i i o t t y p e

i s t h e o n l y N-subgroup M

(N

N:

MM'= 0

t h e sum o f

all

-'M = i O J v nilpotent

1. a l , 2 ( N ) = 0 , b u t a l ( N ) = b 2 ( N )

1s

not

428

APPENDIX

39)

Laxton-Machin

40)

R l l a b a r i 2,

1, 2 2 8 :

21:

N d g . w i t h j u s t o n e N - s u b g r o u p o f t y p e 0,

N d?.,

"weakly

prime",

not

"weakly

r i a h t simple",

but not o f type 2. without faithful

mininial

FI-oroLIos. 41

1

B l a c k e t t 1, 3 2 :

N=N

finite,

eNe n o t c l o s e d w . r . t .

n o t a l l endomorphisms

t;

are right

multiplications. 42)

Chao 2 ,

43)

F a i n 1, 5 7 :

44)

L y o n s 4, 5 8 2 :

some e

N = V 0 ( Z 3 ):

450+452:

N i s idempotent,

E

and h a s N - s u b g r o u p s w h i c h a r e n o t l e f t 3 orthogonal

Maxson i , 4 8 :

46)

Holcombe 2 ,

47)

Beidleman 7 ,

"No.

101:

e h a v e AM=!Ol

a n i d e m p o t e n t e i n M (Z,)

Example o f

22:

e , w h e r e t h e summands A a n d M i n t h e

o n D8 w i t h i d e m p o t e n t

P e i r c e decomposition induced by

45)

finite,

0-primitive,

,

finite with i. N is 3-primitive

N dg,

N < Mo(Z4)

Betsch 3,

37:

49)

R a m a k o t a i a h 3,

~

N

25:

such t h a t I - e i s n o t idempotent

but without r i g h t identity

n o r s t r i c t l y p r i me n o r 2 - p r i mi t iv e 48)

ideals.

i n M(Z3) which i n d u c e a d i r e c t d e c o m p o s i t i o n

idempotents

N d.g.n.r.

,

1-primitive,

5 Mo(ZZ,)

N i s regular

b u t 1-e i s n i l p o t e n t .

and prime,

b u t n e i t h e r simple

.

simple w i t h r i g h t i d e n t i t y ,

, N 1-primitive without identity,

but J2(N)=N.

n o t dense i n

M

(22,)

.

CO

Beidleman 7 ,

103:

i n f i n i t e w i t h 1. N i s 2 - p r i m i t i v e a n d a l l

N dg.,

and m o d u l a r l e f t

135:

Ramakotaiah 2, P o l i n 1, 270 ( E

N

N 2-primitive.

263):

ideal),

N 2 - p r i m i t i v e w i t h 1;

M,

=:

5 Mo(Z3)

dense non-simple

Betsch 3,

54 )

Holcombe 2,

38:

N

20:

2 M(Z4) , N=N

55)

M. J o h n s o n 7 , 3 3 7 .

56)

Hartney 4,

invariant

subgroups,

reducible. land l e f t

containing a

2-sided

left

i n v a r i a n t subgroups.

ideals:

0,(0:1),(0:2),N.

2-semisimple but without r i g h t i d e n t i t y .

N n o t r e g u l a r , b u t 2 - s e m i s i m p l e w i t h OCCN. $ a,(N)

a,(FI)

20:

NM i s c o m p l e t e l y

s u b n e a r - r i n g N ' which c o n t a i n s a minimal N-subgroup,

N 2-primitive:

finite,

(x#0) are prime

s i m p l e w i t h 1 and a m i n i m a l N-subgroup

b u t without 2-sided

b u t no n o n - t r i v i a l

53)

(0.x)

ideals.

= S(N)

(cf.

p . 1 6 5 ) = : I , whpre N / I has no n i l p o t e n t l e f t

ideals.

571

Hartney 4 ,

21:

N do.,

5 8 ) B e t s c h 3, 3 9 :

N=No f i n i t e ,

59) L a x t o n 4 ,

N finite,

60)

16:

B e i d l e m a n 1,

103:

J2(FI).

0 = a o ( N ) # 31,2(N),

dg.,

N c n l ,

t

t S(N)

d0(N)

a,(N)

finite,

which i s n i l p o t e n t

n o t n i l p o t e n t , a,,*(N) dg.,

small.

w i t h a max.

n o t an i d e a l

a2(N) i s n e i t h e r n i l p o t e n t nor small nor s t r i c t l y

l e f t ideal which i s n o t regular or w l t h aZ(N) n o t qr.

61)

B e i d l e m a n 2,

228:

N dg..

f i n i t e w i t h 1, a,(N)

neither nilpotent nor qr.

62)

B e i d l e m a n 2,

228:

N dg..

f i n 7 t e w i t h 1, Z,(N)

is qr.

and = r a d i c a l subgroup ( =

fi

max.

N-subgroups)

6 3 ) B e i d l e m a n 8, 94+97:

N d.g.,

f i n i t e w i t h 1; 2,")

and a s m a l l

ideal,

i s not n i l p o t e n t , but "semi-nilpotent"

but not a small

left

ideal.

429

222 remarkable examples and counterexamples

64)

N d g . w i t h 1, c o u n t a b l e ,

Beidleman 8, 100:

only finitely 65)

B e i d l e m a n 9,

N d g . w i t h 1, f i n i t e ,

204:

66)

Beidleman 9,

67)

L a x t o n 3,

69)

!I d i s t r i b .

12,3:

71)

Lteldleman 3,383:

Z2x

,

(=" ZZ,)

a 2 ( N ) # 1 0 1 , N/a,(N]

ZZ2,

N i s n o t l o c a l and J 2 ( N )

wlth

1 and J ( N ) s t r i c t l y small

i s a near-

i s n o t s t r i c t l y Small.

=

0-primitive,

Choudhari-TPwari

3, MI:

73)

Hartney 2 ,

N f i n i t e dg. w i t h I . n o t s i m p l e ,

74)

Laxton-Machin

finitf

N 7s t h e o n l y

N).

72)

N dg.,

(1.e.

2

N - s u b g r o u p M w i t h a,(N)tM

1, 2 2 7 :

i s not

i s n o t an i d e a l .

not nilpotent.

N f i n i t e d.g.

219:

ideals

q e n e r a t e d by a s i n o l e n c Nd w i t h n Z = l , N f i n i t e ,

j,(W)i s

but

i s n o t an i d e a l , a2(N)

i s a d i r e c t summand o f J 2 ( N )

N a near-rlng on

M a x s o n 1. 6 6 , a n d 3 , 2 0 0 :

Scott

a,,,(N)

the nilpogtent l e f t

where ' Y l 1 2 ( N )

N dg.,

field 701

N has

n e a r - r i n g s w i t h and w i t h o u t n l l p o t e n t Y 2 ( N ) .

F i n i t e d.g.

6 8 ) L a x t o n - M a c h i n 1, 2 2 9 :

^a,(N);

w i t h 1. a 2 ( N ) i s q r . a n d n i l p o t e n t .

N dg.

215:

41:

Jo(N)=O,

t h e sum o f

nilpotent,

a 2 ( N ) i s n o t n i l p o t e n t , JlI2(N)<

marly N - s u b g r o u p s .

1, J o ( N ) c J 2 ( N ) = G - r a d i c a l

with

but 8-primitive with "critical

r a d i c a l " = 10)

( t h e r e i s no non-zero i d e a l c o n t a i n i n g a non-zero

nilpotent left

Examples o f

finite

d.g. near-rings

ideal) 75)

Meldrum 4,

336:

76)

Meldrum 7 ,

294:

Examples o f group d . g .

N d.g.

with

Nr s i m p l e ,

near-rings non-trivial.

falthful,

b u t N has a n i l p o t e n t i d e a l

# 10). 77)

H.Neumann 2 ,

78)

Ligh-Luh

1, 21:

79) M e l d r u m 7 , 80)

51-69:

Ideals

i n endomorphism n e a r - r i n g s

Examples o f

292:

N dg.,

Tharmaratnam 4 ,

137/38:

D-near-rlngs

N2 = ( 0 ) .

A d.g.

Endrn

on r e d u c e d f r e e

groups.

which a r e n o t d.g.

N has o n l y t r i v i a l

d.g.

n e a r - r i n g N w i t h (Nd,.)

representations

a group,

b u t NN i s n o t simple

(and conversely)

81)

Mahmood 3 , 8 0 :

D.g.

82)

H e a t h e r l y 1, 4 8 :

83)

Meldrum 3,

474:

near-rings without

5 E(S3], b u t N i s n o t d.g.

N

N d.g.

without

E l a c k e t t 4 , 602:

4 simple

B l a c k e t t 4 , 606:

N

B l a c k e t t 5, E l a c k e t t 6,

i d e n t i t y o v e r i n v e r s e semigroups

falthful

representation

on

subnear-rings

5 Mo(t),

(all):

N c o u n t a b l e and dense

(all):

dense subnear-rings

Clark

1, 3 9 0 :

M(Z4)

in detail

Clark

1, 3 9 1 :

M(ZZ6)

in detail

Clark

1, 3 9 4 :

M(S3) i n d e t a i l

G o r t o n 1, 7 5 :

M(

m)

N simple

of

i n Mcont(Rn) Mcont(Rn)

Near-rings

o f c o n s t a n t and " n e a r l y c o n s t a n t " mapping5 and p o l y n o m i a l

near-rings

(A-completeness,

92)

M.

J o h n s o n 3,

389:

N = Mo(r),

where l e f t

93)

M.

Johnson 3,

390:

N = M o ( r ) , where l e f t

simplicity) i d e a l s and normal N-subgroups c o i n c l d e i d e a l s and normal

N - s u b g r o u p s do n o t c o i n c i d e .

430 94 ) 95

1

APPEND I X

Gurthrie 1, ( a l l ) : K i n g 1, ( a l l ) :

E(D8)

E(Q,)

96)

Lyons-Malone

1, 75:

97)

F o n g 1 and 2 ,

98)

Malone 1 0 , ( a l l ) :

99)

Maxson 1 4 , 2 9 6 :

E(S3)

(all):

E(S4)

r

Examples o f g r o u p s

100)

Mcquarrie 5 , l a s t page:

101)

McQudrrie I .

102)

S o 1, ( a l l ) :

where E ( r ) i s a r i n g

r with

Examples o f g r o u p s Examples o f

E(1)

a ring, but

not cyclic.

E(r)l

o r d e r 2 7 , 3 2 , 6 4 and 8 1 w h e r e A ( r ) i s a r i n g

I. o f

Subnear-rings of R l x l

8,3:

Examples o f s ~ b s t r u c t u r e s , i d e a l s , homomorphisms,

r a d i c a l s e t c . c f polynomial

near-rings

R l x l which a r e n o t l e f t i d e a l s

103)

Pilr-So

I , '55

104)

Pilz-So

1 , 65 :

105)

Brenner 1, ( a l l ) :

106)

Ligh 4 ,

107)

A n s h e 7 - C I a y 2,

172.

108)

Clay 10, 324:

N f i n i t e , planar, non-abelian

109)

Lawver 3 , 8 9 :

I n t e g r a l and n o n - i n t e g r a l

110)

Szeto 3 , 271,273.

P i l r 22,

I

R I x l which

G

1s

e v e n a max,mal s u b g r o u p , b u t n o t a r i n g i d e a l .

ZZpixl

Ideals of

N n e a r - r i n g on R 2

10:

P i 1 2 2 2 , M5: 112)

of

2-subgroups

:

X

R 2 , N-Simple,

P l a n a r ( i n t e g r a l and n o t i n t e g r a l ) n e a r - i n g s o n Z

Examples o f

n e a r - r i n g s on 2 ( 5 * )

non-planar, but strongly uniform near-rings

A planar near-ring N on

( W ,+) w i t h a l f 2 ( N )

A planar near-ring N on ( W , t )

M5:

but not a near-fleld

w i t h >,(N)

=

=

101, b u t

a2(N)P

a2(N)

=

N

(01 and N

i s p l a n a r and i n t e g r a l

113)

Ferrero-Cottl-Pellegrini 1, b l j :

N planar,

114)

Ferrero-Cotti-Pellegrini

N f i n l t e , planar, not integral, but wlth proper ideals

115)

F e r r e r o 13, 4 2 3 .

1, M I :

I

N,

4

# A (8.86!)

Bell

1, 3 6 7 :

N

New b l o c k d e s i g n s f r o m p l a n a r n e a r - r i n g s =

= x,

No,

non-zero central

N

117)

Bell

118)

Choudhari

119)

Choudhari-Goyal

1 , 367:

=

No,

xn(x)

1 , 126f127:

=

( N , t ) a b e l i a n , b u t n o t e v e r y homomorphic image h a s a

idempotent. x,

no i d e n t i t y ,

2 non-trivial

1, M6.

(N,t)

not abelian.

duo-near-rings.

o n R 4 whlch a r e n o t " s t r l c t l y duo" ( = n o t e v e r y

Duo n e a r - r l n g s

N-subgroup i s an i d e a l ) . 120)

Heatherly 7 , 203:

N n e a r - r i n g o n ZZ3

without nllpotent elements, b u t a l l x4 = x . N i s

N-simple and i n t e r g a l , b u t n o t a n e a r - f i e l d .

121)

Ligh-Utumi

122)

P l a s s e r 1, 1 1 1 . 3 :

Examples o f I F P - n e a r - r i n g s

123)

P l a s s e r 1 , 111.8:

A s t r o n g I F P - n e a r - r i n g whlch

1 , 113:

124)

Plasser 1, 111.39:

125)

R a m a k o t a i a h 2, 1 3 2 :

N e a r - r i n g s w i t h a N = aNa f o r a l l a , b u t n o t Na

No

N

=

aNa a i d c o n v e r s e l y .

i s not weakly commutative

# N n e a r - r ~ n g o n 7Z2 x R 2 , w h e r e e a c h e l e m e n t I s a p o w e r o f i t s e l f

:M o ( R 4 ) ,

a n o ? - r i n g which i s " n - r e q u l a r ' '

(1.e. vx3n3a' xn

=

xnaxn);

all

radicals coincide.

431

222 remarkable examples and counterexamples Examples o f 3 - n e a r - r i n g s

126 )

Ratllff

1, 5 :

127)

Ratliff

1, 1 3 :

Examples o f 5 - n e a r - r i n g s

128)

Ratliff

1, 2 6 :

N a Boolean near-ring,

7

P(ni

129)

182:

Adam8 2 ,

(x3

n : N, P(n)

fp

:=

1

P[l+n) d

pn = p i .

1111

nl l p o t e n t a n d n u n - n i l p o t e n t groups of order

i n f i n i t e ) and on n o n - a b e l i a n n e a r - r i n q o n Z,

Integral

groups ( f i n i t e and

( h e n c e o f c o m p o s i t e characteristic)

H e a t h e r l y 5,

131)

Heatherly-Olivier

1. 2 1 8 a n d O l l v i c r 2. 1 5 .

Integral

132)

Heatherly-Olivier

1, 2 2 0 a n d O l l v i e r 7 , 1 8 .

Some i n t e g r a l n e a r - r i n g s

133)

Heatherly-Olivier

2,

RE'

Two n o i i - i s o i n o r p h i c

134)

Heattierly-Olivier

2,

90:

N finite,

135)

L i g h 4.

"

N integral

136)

O l i v i e r 2,

137)

W h i t t i n g t o n 1.

N d.g.

22:

138)

Choudharl-Tewari

139)

Lawver I ,

140)

L a w v e r 1, 3 7 6 :

141)

Malone 8, Kusel

143)

P i l r

144)

P i l z 3,

145) 146) 147)

not nilpotent,

N a.a.ti.r..

near-t.lng

n o t every

Archimed.

i s not nilpotent

N integral with left

cantellatlon

32

~

e l e m e n t~ In

on t h e i n f i n i t e d i h e d r a l

o r d e r e d n e a r - r i n g on [ R , + )

A vector-ordered

t h e ~~ .

~ ~ d

~

without

N ordered w i t h a

P i l z 3,

167:

Examples o f

Pilz 7,

341:

N fully

253:

N1,

<

h. c

identlty

0 h u t ca > cb

fully ordered near-pings

ordered,

N - N1

N w i t h p r o p e r Nc.

N1 a n d N~ a r e n o t . z e r o - n e a r . r , n g s .

N ~ h, u t

N p o r d e r e d n e a r - r i n g s . A n example and a counterexample t h a t t h e s e o r d e r s t o o r d e r N, Q N 2 .

148)

K.B.P.

R a o 1, 2 4 2 :

A strictly

149)

K.D.P.

R a o 1.

A s t r i c t l y fully ordered NT.

243:

(partially)

ordered

Nr

150)

Rhadhakrishna-Bhandari

1,2:

A f u l l y ordered near-ring.

151)

Rhadhakrishna-Dhandari

1.4:

A lattice-ordered

= cahb

G r a v e s I, 3 7 :

group

near-ring

can be c o n t i n u e d

152)

32

n l i i l t 1 p l r L a t i o n s on groups

A l l d . g . near-rings

162:

Pi12 8,

,

w i t h t h e same

.

h u t (N,+)

All i n t e q r a l n e a r - r l n g r o f o r d e r

"Trivlal"

[all):

1. 4 6 :

or! Z l i

A n e a r - r i n g o n 22 (5')

375:

1, 2 1 :

142)

integral,

a n d '5113

of order

on Z 5 , n o t i d e m p o t e n t a n d n o t a n e a r - f i e l d

near-ring

M6:

3,

n e a r - r i n g s on 22 9 ' '15

integral near-rings

8 y 1 ow - d e c o mpo s 1 t 1 o n "

w i t h 1, ( N , + )

(all):

but

1000.

130)

32:

P(l+m),

p(m).

N a n - i n t e g r a l donialni

151:

3x = 0 )

x,

=

n o n - r i n g w i t h a r b = 0,

N a Dickson near-r-iug associated w i t h 0;xl

(0 a division ring),

Euclidean. 153)

Maxson 8,

156:

A Dickson near-ring which

154)

Maxson 8,

161:

Examples o f

155)

Maxson 13, 410:

156)

Choudhari-Goval

157)

H e a t h e r l y 8,

is not a near-field.

Dickson near-rings

Examples o f homomorphisms

1, M3: A s t r o n g l y

353:

c 1 _ 0 =,aCnb

=

= 0.

The number o f

from

"6-rings".

o f Dickson near-rlngs

regular near-ring regular near-rings

on

Z,

of order 5 7

which i s

i

~

~~ l

APPENDIX

432

1 5 8 1 M.

333:

J o h n s o n 7,

o n Z 6 w i t h N # N o , N r e g u l a r , dl,2(N)

N near-ring =

unique n o n - t r i v i a l

N-subgroups

1 5 9 ) Mason 5,

(-):

160) S z e t o 4,

68:

1 6 1 ) M a x s o n 1,

M o ( r ) and M ( r ) a r e r e g u l a r ,

203:

= &(N)

The i n t e r s e c t i o n o f

=

(0:O)-

t h e maximal

(F a f i

l a x2

=

on S 3 , where

but not "strictly

regular".

Nd = { O ] , N not. a s u b d i r e c t p r o d u c t o f n e a r - f i e l d s

N = Fo[xl ideal

N nr.

ideal.

i s = 10,31.

N integral, regular,

7 2 a n d 3,

1 6 7 ) E s c h 1, 1 6 :

left

2

+

eld)

a3x3

t

i s l o c a l w i t h JZ(Nj # N and d i s t r i b u t o r

. . .I

t h e a d d i t i v e commutator o f two l e f t

ideals

i s not a l e f t

ideal 1 6 3 ) E s c h 1, 3 1 t 6 1 :

N nr.

on S 3 , w h e r e t h e d i s t r i b u t o r s u b g r o u p o f N i s n o t n o r m a l , i t e r a t e d

distributors 1 6 4 ) E s c h 1,

32t37185:

never y i e l d 0. where t h e d i s t r i b u t o r subgroups of

N n r . on D4, ideals,

i s 0;

t h e t h i r d d i s t r i b u t o r subgroup

left

i d e z l s are l e f t

another kind of

distributors

does n o t t e r m i n a t e a t 0.

5,

165) F e r r e r o - C o t t i

266:

N finite,

N'

N d i s t r i b u t l v e o n D4;

M12:

g

",NI

:=

166)

D a r i t 3,

167)

F e r r e r o 1, 1 0 :

168)

Heatherly 6,

65:

Distributive near-rings

169)

H e a t h e r l y 6,

66:

Smallest non-trivial

170)

Heatherly-Ligh

N,

i t s defect

but N/N'

does n o t f u l f i l l

and commutator

N d i s t r i b u t i v e n r . o n S3 w i t h e x a c t l y o n e " S y l o w "

1, 450/451:

ideal

on non-abel i a n groups

distributive near-ring

Examples of

xy=rt j yx=tz

(on

s3).

p s e u d o d i s t r i b u t i v e near-rings which a r e n o t

distributive

V,

171)

Jones

1, 5 :

Distributive near-rings

on

172)

Jones 1, 6 :

Distributive near-rings

on A 4

N a non-distributive near-ring

173)

Jones

,

I74j

Jones

, 10:

Distributive near-rings

175)

Jones

,

13:

N distributive,

176)

Jones

,

16-38:

1

Jones

.

46:

Ligh 1

, 383:

177

1781

7:

o n D8 w i t h x 2

0

N2 a c y c l i c g r o u p

A l l distributive near-rings o f order

N commutative on Aq w i t h o u t Near-rings

=

o n D12

~

<

15

i n v o l u t i o n ( = antiisomorphism

which are " n - d i s t r i b u t i v e " ,

of order 2).

but not distributive;

the nxn-

m a t r i c e s o v e r N form a n e a r - r i n g , 179)

S t e f a n e s c u 1, 4 3 9 :

1801

Ligh-Mcquarrie-Slotterbeck

A proper non-associative,

181)

M c o u a r r i e 1,

13:

182)

McQuarrie I ,

26:

1, 8 9 :

but dlstrlbutive "near-ring"

An N - s y s t e m ,

not a near-field

A proper N-system k N - s y s t e m N i n Ro [ X I w i t h a homomorphic

image w h i c h I S n o t an

N - s y s tern 183)

M c Q u a r r i e 1.

30:

Two e x a m p l e s o f

N-systems N f

No w i t h m o d u l a r i d e a l s I s u c h t h a t N / I

i s an N-system 1 8 4 1 M c Q u a r r i e 1, 3 6 :

A near-ring

on Z3 x

Z3

m u l t i p l i c a t i o n by ( - e l

w i t h a halvable idempotent e,

i s not fixed-point-free.

but the

433

222 remarkable examples and counterexamples

185)

E e i d l e m a n 8,

186)

Deskins

1 8 7 ) Su 2 ,

100:

1, 8 2 7 :

148:

N d.g.

I,

with

uncountable,

Examples o f t o p o l o g i c a l

left

ideal.

near-rings

1 8 8 ) T h a r m a r a t n a m 2,

301:

1 8 9 ) T h a r m a r a t n a m 3,

121,122:

190) Tharmaratnam 3.

124:

A topological

d.g.

191) T h a r m a r a t n a m 3,

134:

Embeddings o f

topological

192) Tharmaratnam 4,

137:

N a d i s c r e t e topologlcal d i v i s i o n d . g .

193) Yagill

Examples o f

inany N - s u b g r o u p s

but with only finitely

N i n f i n i t e w i t h e x a c t l y one minimal

topoloqical

Example5 o f

8 ( a l l ) : Examples o f n e a r - r i n g s

(R.8)-groups

topological

which i s no d.g.

near-ring,

on

1, 2 1 :

N dg., without

195) Banaschewski-Nelson

1, 2 2 :

N dg

near-ring

(R,S)-groups

topological

194) Banaschewski-Nelson

near-rings

d.g.

near-ring

groups.

non-trivial

i n j e c t i v e u n i t a r y N-groups

, w i t h i n j e c t i v e N-groups,

but not every unitary

N - g r o u p h a s an i n J e c t i v e e x t e n s i o n 1 9 6 ) Mason 3,

46:

197) Maxson I .

Injective objects

Z -groups

i n the category of

Nr p r o j e c t i v e , b u t n o t " s t r l c t l y p r o j e c t i v e "

53:

198) O s w a l d 6 , 2 6 8 :

i d e a l L a n d some N r s u c h t h a t

with 1 with a left

N d.g.

g h L i i o t n ( L , ~ ) ~ Y E ~ I I , Lf : ( i ) . ~ i , b u t N r

1, 3 2 / 3 8 :

199) Maxson-Smith

Some m a x i t i l a 1 N - s u b g r o u p s o f

1s

N =

not inJective

Mr%,(r)

N has a

(ncAut(r));

nilDotent Z-rad?cal.

5,

200) Maxson-Smith

A bicentralizer near-rino YA(l),

223:

which i s 2-seimisiniple,

but not simple. 201) Maxson-Meldrum

1, M3:

2 0 2 ) Yaxson-Meldrum

I, M 15: M A ( I )

A near-field

N

Mo( ' ) , where 5 i s n o t a suboroup o f Aut(:'). S

a near-field,

2 , M2: MS(,') a n e a r - f i e l d .

2 0 3 ) Maxson-Oswald

204) tlaxson-0swald 2 , M 17: 205) Maxson-Siiiith l C ,

147:

F a field with MA((F,+)). A simple subnear-rino with

206) Maxson-Smith 10,

148:

but

N

"

i s not abelian.

S_End(-),

of

FI

each v i r *

flenerates

(7 )

F. Mo(')

such t h a t t h e r e

i s no A 2 A u t ( - )

- M A ( l ) .

A subnear-field

N of

!lo(?),

w h e r e N : = YA u t N ( I

)(r)

a near-field

15

p r o p e r l y c o r i t a i n i n o \I.

MA('),

207) Maxson-Smith

10,

151; A f i e l d F

208) Maxson-Smith

12,

M2: A d . 0 . b i c e n t r a l i z e r n e a r - r i n q M A ( I ) ,

209) Meldrum-Zeller

I,

such t h a t

r

187: M A ( r ) n o n - s i ~ n p l e w i t h s t ( a y ) ' s t ( i )

conjuqacy classes of s t a b i l i z e r s , cAst(ay)

210)

H o l c o m b e 8,

21i)

Pilz-So

?12)

L o c k h a r t 4,

213)

Heatherly-Olivler

1,

i s n o t a v e c t o r space o v e r F.

24-27:

-+

s t ( a r ) = s t ( ? ) and

b u t w i t h some

22

such t h a t

1id1.

Endomorphism n e a r - r i n g s

1n v a r i o u s c a t e g o r i e s

150: Examples o f g e n e r a l l r e d d . g .

near-rings whlch are not d . g

A non-exact sequence o f endomorphism n e a r - r i n g s

151:

3,

M2:

N a c o m m u t a t i v e n e a r - r i n g w i t h some H

g

N s u c h t h a t N2

6

H

and H i s " I n t e g r a l " 214)

Jones-Ligh

1, M 3 :

215)

O l i v i e r 2,

28:

Two

Examples o f

near-rings

H-monogenic

additive

subgroup

on m u l t i p l i c a t i v e s e m i g r o u p s

near-rings

on S3 w i t h d i f f e r e n t

H,

w h ~ c hi s n o t a n

434 216)

APPEND IX

Kaarli 9 (all):

A c o u n t a b l e a b e l i a n n e a r - r i n g N,

where

I - p r i m i t i v e homomorphic image (hence

217)

Kaarli

9,

11: N e M o ( Z 8 ) ,

).I

2 1 8 ) F r o h l i c h 9,

(-):

Examples o f

ll(N)

=

near-rings

2 1 9 ) M a x s o n 1, 6 R : A l o c a l a b s t r a c t a f f i n e

220) S c o t t 15, 62:

N,

9

o( Ld0(N)

b u t N has a I - p r i m i t i v e a r i s i n g from formal

222)

Yearby

1,

112:

Examples o f

111-127:

central

ideal.

groups.

near-ring.

A c o m p a t i b l e n e a r - r l n o on a s i m p l e

"'-qroup

and a Z a r i s k i - t y p e

on i t .

221) S c o t t 22,

N) h a s a f o u r - e l e m e n t

+ 'bl( l o ( N ) ) ) .

N-suboroups of

Many m o r e n e a r - r i n n s On Aq

an N-oroup.

topolonv

435

LIST OF OPEN PROBLEMS

1 ) G e n e r a l l y , determine t h e s t r u c t u r e of o u r s p e c i a l c l a s s e s of n e a r - r i n g s

( r a d i c a l s , complete r e d u c i b i l i t y , semisimpli-

c i t y , p r i m i t i v i t y , . . . ) . For i n s t a n c e , what can one s a y a b o u t the r a d i c a l s of planar near-rings ? 2 ) Are a l l r e s t r i c t i o n s t o z e r o - s y m m e t r i c n e a r - r i n g s i n t h i s bock r e a l l y n e c e s s a r y ? 3) Study measure a n d i n t e g r a t i o n i n n e a r - r i n g s ( t h i s i s motiv a t e d by t h e M c o n t ( T ) - t y p e n e a r - r i n g s ) . 4 ) Do 2 . 6 3 , 2 . 8 5 , 5 . 5 4 a n d 5 . 6 2 h o l d f o r a r b i t r a r y i d e a l s ? 5 ) Is H o ( r ) a n e a r - r i n g i f i t c o n t a i n s a s u b n e a r - r i n g of M ( T ) which i s d e n s e i n M o ( r ) ( c f . 4 . 5 3 ( e ) ) ? 6 ) L e t N be 2 - p r i m i t i v e w i t h A C C L o r A C C N . Does N c o n t a i n a ( r i g h t ) i d e n t i t y ? I s t h e t o p o l o g y 3’ d i s c r e t e ( c f . 4 . 2 9 ) ? 7 ) Are t h e r a d i c a l s o f d i r e c t p r o d u c t s t h e d i r e c t p r o d u c t o f 8) 9) 10)

11) 12) 13) 14) 15) 16) 17)

the r a d i c a l s ( c f . 5.20) ? Suppose t h a t I ( T ) h a s A C C N . Is r n e c e s s a r i l y f i n i t e ( c f . 7.60(m ) ) ? R e c a l l t h e 4 problems o n page 178. Determine t h e i d e a l s t r u c t u r e i n t h e polynomial n e a r - r i n g s . Which o n e s h a v e t h e D C C L , D C C N , e t c e t e r a ? Do t h e r e e x i s t p r o p e r n e a r - d o m a i n s ( i . e . t h o s e o n e s w h i c h a r e not n e a r - f i e l d s - c f . p . 247) ? F i n d some e x a m p l e s o f i n f i n i t e n e a r - f i e l d s w h i c h a r e n o t Dickson n e a r - f i e l d s . Is 9 . 2 1 ( a ) c o r r e c t w i t h o u t t h e assumption t h a t N has a n identity (Bell) ? S t u d y l a t t i c e - o r d e r e d n e a r - r i n g s , F - n e a r - r i n g s and v e c t o r near-rings ( c f . p . 343). How c a n o n e c h a r a c t e r i z e t h o s e n e a r - r i n g s which c a n be f u l l y ordered ? Which ( p a r t i a l ) o r d e r s i n a n e a r - r i n g c a n be e x t e n d e d t o a f u l l order ( c f . ( F u c h s ) ) ? Cf. 9 , 1 5 2 ( g ) , ( h ) . I s 9.133 t r u e without the assumption t h a t N i s abelian ?

436

APPENDIX

Is a 0-semisimple n r . ( w i t h o r w i t h o u t f i n i t e n e s s c o n d i t i o n s ) tame ( c f . 9 . 1 7 1 ( d ) )

If N i s t a m e ,

?

d o e s ?,(N)

= 22(N) always h o l d ?

simple i f f i t i s 2-semisimple

I s N O-semi-

?

I s e v e r y tame n r . e m b e d d a b l e i n ( i s o m o r p h i c t o ) some S ( r ) o r C ( r ) ( s e e 7.121,

(S.D.

9.166(b))

?

S c o t t ) C a l l N semitame i f t h e r e a r e tame N-groups,

i n t e r s e c t i o n o f whose a n n i h i l a t o r s i s z e r o ( c f . What c a n b e s a i d a b o u t semitame n e a r - r i n g s

the

5.14(a)).

?

MA(r) b e r e g u l a r s u c h t h a t a l l s t a b i I s MA(r) t h e n s i m p l e ? C o n v e r s e l y , i MA(r) i s s i m p l e , i s M A ( r ) n e c e s s a r i l y r e g u l a r a n d a r e a l l (Meldrum-Zeller)

Let

l i z e r s are conjugate.

s t a b i l i z e r s conjugate ? C f .

9.218,

( B e t s c h ) Which s u b n e a r - r i n g s (on

r)

i n MA(I')

f

N o f MA(T) are 0-primitive

? What a b o u t N = M A ( r ) ? W h a t h a p p e n s i f N i s d e n s e ?

24) U n d e r w h i c h c o n d i t i o n s d o e s M A ( T ) 2 M B ( A ) h o l d ( c f . 7 . 3 8 ) ? i s o m o r p h i c t o E(n r i ) ? The same 25 1 I s a n u l t r a p r o d u c t II E ( T i ) 7

$

for

A(Ti),

I(ri),

P(Ti)

and C(Ti).

Can a z e r o - s y m m e t r i c

n e a r - r i n g be embedded i n an u l t r a p r o d u c t o f d . g . n e a r - r i n g s , and i s t h i s u l t r a p r o d u c t d.g. problem 9 ) .

a g a i n ( c f . 9.285,

6.11,

and

437

BI BLI OG R APHY P L E A S E

N O T E

The c l a s i i f i c a t i o n scheme f o r t h e p a p e r , ~ 'dis F o l l o w s : s e a r - r i n g s : , I . . . ~ d d i t i v eq r o u p s o i n e a r - r i n p s , n e a r - r i n g s on g i v e n x r o u p s .A'.

.

R..

.

\ t f i n e near-rings B o o l e a n n e a r - r i n g 5 and q c n e r . i l l l a t i o n s ( p - n e a r - r i n g s , I T P - n e a r - r l n g r , .

C..

.

C o n s t r u i t ~ o n s15rims r n d p r o d u c t s , subdirect p r o d u c t s , .

I..'.

.

0..

.

Computer-aided ~ n i e s t i g a t i o n s U i s t r i b ~ t i v r l ys e n e r a t e d n e a r - r i n g s

.

D i s t r i b u t o r s , distribritive elements. commutators, s o l v a b i l i t y . Dickson n e a r - r i n g s

O'..

0".

..I

..,

U . . . Distributive near-ring5 E... E'.

Elemrntarv, examples, ,ixiornafics, c h a i n i o n d i t i o n i ,

.

Embeddings Endomorphism n e a r - r i n g ( E ( ! ' ) ,

E"..

.

F..

F'..

I(r))

F r e e n e a r - r i n g and N-groups G e o m e t r i c i n t e r p r e t a t i o n s ( c o o r d l n a t i s a t i o n , i n c i d e n c e g r o u p s , . . .) Homological and c a t e g o r i c a l a s p e c t s , e x t e n s i o n s , i n j e c t i v i t y and p r o j e c t i v i t y

..

I...

Idempotenrs, h i r e g u l a r near-r ings

I,..

Integral near-rings,

n e a r - i n t e g r a l domains and g e n e r a l i c a c i o n s

L . .. Local n e a r - r i n g s PI.. Plodularity \ I * . . M u l t i p l i c a t i \ , e semi!.rotips

.

N..

.

of

near-rings

N i l p o t e n c e and n a n - n i l p o t e n c e

0. .. O r d e r e d n e a r - r i n g s P . . . P r i m i t i v e n e a r - r i n g s , N-qroups o f type Y P'.. P r i m e ( s e m i p r i m e , c u m p l e t e l y p r i m e , ) ideals

...

.

P". Planarity Po.. P o l y n o m i a l n e a r - r i n g s , n e a r - r i n g s o f f o r m a l power s e r i e s O...

Quasi-regularity Near-rings of quotients

Q'..

R...

Radical theory Regular n e a r - r i n g s

R'..

S i m p l i c i t v and r e m i s i m p l i c i t v

S... S".

. .

Sylow-type t o p i c s Relations t o s h a r p l y t r a n s i t i v e groups

T... 7'. V..

. .

T r a n s f o r m a t i o n n e a r - r i n g s (M(T), Topological considerations Valuations

S'.

H O ( r ) . MG(r)l

n . . . Near-rings without n i l p o t e n t elements X..

.

Other t o p i c s

Structiires related t o near-rings: C r . . Composition r i n q s (TO-Algebras) Na.. Near-algebras Nd..

Mear-domains ( i n t h e s e n s e o f " n o n - a s s o c i a t i v e

Rs.. O t h e r r e l a t e d s t r u c t u r e s ( s e m i n e a r - r i n g s , Ua..

near-fields")

...I

Universal algebraic context

Combined c l a s s i f i c a t i o n s g i v e more i n f o r m a t i o n o n t h e p a p e r ; f o r i n s t a n c e : P",F.. D',R

.

...

Planar near-fields o r Radical theory f o r d i s t r i b u t i v e l y generated near-rings

The b i b l i o q r a p h v a l s o C o n t a i n s a b s t r a c t s O f t a l k s O F n a p e r s o r e s e n t e d a t t h e n e a r - r i n g c o n f e r e n c e s u n t o 19fiZ. I f v o u w a n t t o o b t a i n t h e s e a b s t r a c t s O P ? d o e r s , p l e a s e w r l t e to O r . G.Betsch o r t h e a u t o r of t h i s book fov t h e C b e r w o l f a c h - a b s t r a c t s , t o the a u t h o r f o r t h e E d i n b u r g h - a b s t r a c t s and t o P r o f . F e r r e r o f o r t h e S a n - B e n e d e t t a - P t o c e e d i n g s . Y e a r - r i n g and n e a r - f i e l d c o n f e r e n c e s u n t o 1 9 5 2 : Oberwol f a c h Oberwolfach Oberwolfach Starred

...

Near-fields

G...

H.

A[r),

l a t t i c e of i d e a l s ,

.... .... ....

1348 lo?? 1976

Edinhurgh . . . . . . . . . . . . . . . . 1 9 7 8 Oberwol f a c h . . . . . . . . . . . . . . . . 1980 San B e n e d e t t o d e l T r o n t a . . . 1981

t * ) n a n e r s d e n o t e b o o k s o r s u r v e y a r t i c l e s on n e a r - r l n g s .

438

APPEND I X

AOAMS, W i l l i a m . R.. 318 Old Marlow Rd.. Concord. Mass. 01742, USA 1. Near i n t e g r a l domains and f i x e d - p o i n t - f r e e automorphisms, D o c t o r a l Oiss.. Boston Univ.. 1975. Boston, Mass., USA. 2. N e a r - i n t e g r a l domains on n o n - a b e l i a n groups, Monatsh. Math. 8 1 (19761, 177-183. M R 54-2731. 3. N e a r - i n t e g r a l domains on f i n i t e a b e l i a n groups, submitted. AOLER, I r v i n g . RFD. N o r t h B e n n i n g t o n , Vermont 05257, USA 1. Composition r i n g s , Ouke Math. J. 29 (1962). 607-625. A I J A Z . Kulsoom, Univ. of See AIJAZ-HUQ

Islamabad,

A, I Cr,H,A,E',S,T.T',E

Pakistan

A I J A Z , Kulsaom and HUO, S.A. 1. C a t e g o r i c a l i n v e s t i g a t i o n o f r - g r a d e d A-algebras. 21-36.

AN OR^ ,

A.1 A.1

P o r t u g a l i a e Math. 28 (1969).

Johannes. Fachber. Math., Un1v.d. Saarld., 6600 Saarbrucken, Germany 1. P r o j e k t i v e Ebenen uber Fastkorpern, Math. Z. 62 (1955). 137-160. MR 17-73. 2. Uber e i n e Beziehung zwischen Zentrum und Kern e n d l i c h e r F a s t k o r p e r , Arch. Math. 14 (1963). 145-146. MR 27-1528. 3. L i n e a r e Aloebra u b e r FaStkorDern. Math. Z. 136 119741. 295-313. 4. A f f i n e G e G e t r i e n uber F a s t k o r p e r n , M i t t e i l u n g e n aus'dem Mathem. Seminar Giefien I 1 4 (1975). 1-99. MR 58-2588. 5. Eemerkunyen uber Fastvektorraume. FU B e r l i n , Lenz-Festband (1976). 28-36. 6. Same t o p i c s on l i n e a r a l g e b r a a v e r n e a r - f i e l d s , Oberwolfach 1976.

ANGERER. Josef, Chemie L i n z AG, A-4020 L i n z , A u s t r i a 1. R a d i k a l e k l e i n e r F a s t r i n q e , Diss. Univ. L i n z , SEE ALSO ANGERER-PILZ

197R.

F,G,Rs F F ,P" ,X F,G

F.G.X F, P" ,X R,A,P'.N,Q,D,n,F, P" ,A ' ,R ' .C ,I '

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R ' , I ' .W.F

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MASON, Gordon, Oept. Math. Univ. o f New Brunswick. Fredericton, New Brunswick, Canada 1. Solvable and n i l p o t e n t near-rings, Proc. h e r . Math. SOC. 40 (1973). 351-357. FR 47-8635. 2. W-groups and n e a r - r i n g m d u l e s , tanad. Math. B u l l . 18 (1975). 241-244. MR 52-10817. 3. I n j e c t i v e and p r o j e c t i v e near-ring modules, Compositio Math. 33 (1976), 43-54. MR 54-75580. 4. I n j e c t i v e and p r o j e c t i v e near-ring nodules, Obemolfach, 1976. 5. Strongly r e g u l a r near-rings. Conf. Edinbg.. 1978. 6 . Strongly r e g u l a r near-rings, Proc. Edin. Math. Sac. 23 (1980). 27-36. MR 811-16047. SEE ALSO MASON-OSWALD

MAXSON, C a r l t o n J. and MELORUM. John D.P. 1. C e n t r a l i z e r representations of near-fields,

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~

~~

.

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.

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14. Upper f a i t h f u l d.g. n e a r - r i n g s , Proc. Edin. Math. SOC., t o appear. SEE ALSO CLAY-MAXSON-MELDRUM, FONG-MELDRUM. LYONS-MELDRUM, MdHMOOD-MELORUM. MAHMOOD-MELDRUM-O'CARROLL, MAXSON-MELDRUM-OSWALD, MELDRUM-OSWALD, MELDRUM-PILZ, MELDRUM-PILZ-SO. MELORUM-ZELLER MELORUM, John D.P. and OSWALD, Alan 1. N e a r - r i n g s of mappings, Proc. Royal Sac. Edinbg. Sect. A 83 (1979). MR 819-16042. MELDRUM, John D.P. and PILZ, Gunter 1. Polynomial a l g e b r a s and polynomial maps, submitted.

213-223.

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T.R'.S

P0,Ua

,

APPENDIX

MELORUM, John O.P., PILZ, Gunter and SO, Yong-Sian 1. Embedding n e a r - r i n g s i n t o p o l y n o m i a l n e a r - r i n g s , Proc. Edinbg. Math. ( 1 9 8 2 ) , 73-79. 2. N e a r - r i n g s of p o l y n o m i a l s over groups, submitted.

Sac. 25

MELORUM John D.P. and ZELLER, Mike The s i m p l i c i t y of n e a r - r i n g s o f mappings, Proc. Royal Sac. Edinbg. 90 A (1981), 185-193.

i.

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Cr (-) (-) Cr

Po,Cr

Rs,E

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.~

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2. Groups w i t h automorphisms t h a t l e a v e o n l y t h e n e u t r a l element f i x e d , Arch. Math. 7 (1956). 1-5. MR 17-580. NEUMANN, Hanna (1914-1971) 1. N e a r - r i n g s connected w i t h f r e e groups, Proc. l n t e r n a t i o n a l Conference, h s t e r d a m . I I , (1954), 46-47. 2. On v a r i e t i e s of groups and t h e i r a s s o c i a t e d n e a r - r i n g s , Math. 2. 65 (1956). 36-69. MR 17-1183. NURAUER, M i l f r i e d , I n s t . f . Algebra, Techn. Univ. Wien, A r g e n t i n i e r s t r . 8, A-1040 Wien, Austria 1. Ober d i e O p e r a t i o n des Einsetzens i n Polynomringen, Math. Ann. 134 (1958), 248-259. FR 20-4549. 2. D i e O p e r a t i o n deS Einsetzens b e i Polynomen i n mehreren Unbestimnten, J. Reine Angew. Math. 201 119591, 207-220. MR 21-7225. 3. Zur T h e o r i e d e r V o l l i d e a l e I , Monatsh. Math. 64 (1960). 176-183. MR 22-5652. 4. 2 u r T t e o r i e d e r V o l l i d e a l e 1 1 . Monatsh. Math. 64 (1960), 335-348. FR 22-8031. 5. Ober d i e A b l e i t u n g e n der V o l l i d e a l e , Math. 2. 75 (1961), 14-21. FIR 22-11012. 6. Funktionen auf kotmnutativen Ringen, Math. Ann. 147 (1962). 166-175. rn 25-1179. 7. Die O p e r a t i o n des Einsetzens bei r a t i o n a l e n Funktionen. U s t e r r . Akad. Wiss. Math.-Naturw. K1. 5.- 6.1 170 (1962). 35-84. MR 26-141. 8. Ober d i e O a r s t e l l u n 9 von u n i v e r s e l l e n Algebren durch Funktionenalgebren, Publ. Math. k b r e c e n 10 (1963). 151-154. 9. k r i v a t i o n s s y s t e m e m i t K e t t e n r e g e l , Monatsh. Math. 67 (1963), 36-49. 10. Transformationen von T e i l a l g e b r e n und K o n g r u e n z r e l a t i o n e n i n a l l g e m e i n e n Algebren, J.Reine Angew. Math. 214/215 (1965). 412-418. MR 29-3412. 11. Compatible and c o n s e r v a t i v e f u n c t i o n s on r e s i d u e - c l a s s r i n g s o f t h e i n t e g e r s , Colloqu. Math. SOC. Janos B o l y a i , 13. C o n t r i b u t i o n s t o number t h e o r y , k b r e c e n , Hungary, 1974. MR 55-12709. 12. Uber d i e Automorphismen von Kompositionsalgebren, Acta. Math. Acad. Sci. Hungar. 26 (1975). 275-278. MR 52-10552. 13. Local polynomial f u n c t i o n s : R e s u l t s and Problems, P r e p r i n t , Techn. Univ. Wien ( A u s t r i a ) , 1978. SEE ALSO LAUSCH-NUBAUER O'CARROLL, Lion, Math. Dept., Univ. of Edinburgh, M a y f i e l d Rd., SEE MAHM00D-MELDRUM-O'CARROLL

Edinburgh EH9 3 1 2 , Scotland

A

E",D E" , O , E

Cr,Po Cr,PO

Cr Cr

Cr C r ,P0.S. X Cr,E

E*T Cr,X E.S.Po,Ua

E ,Cr.Ua Ua,PO

APPEN D IX

456

OLIVIER, Horace R. 1. Endomorphism n e a r - r i n g s on c e r t a i n groups, M.S. Thesis, U n i v e r s i t y o f Southwestern L o u i s i and, 1970. 2. N e a r - i n t e g r a l domains and H-monogenic n e a r - r i n g s , Oiss. Univ. o f Southwestern L o u i s i a n a , L a f a y e t t e , 1976. SEE ALSO HEATHERLY-OLIVIER ORE, @ s t e i n 1. L i n e a r equations i n non-comnutative f i e l d s , SEE ALSO MURDOCH-ORE

Ann.

o f Math.

32 (1931). 463-477.

OSWALO, Alan, Oept. Math. Teesside P o l y t e c h n i c . Middlesbrough, Cleveland, TS1 3BA England 1. Some t o p i c s i n t h e s t r u c t u r e t h e o r y of n e a r - r i n q s , D o c t o r a l O i s s e r t a t i o n , Univ. o f York. 1973. 2. N e a r - r i n g s i n which every N-subgroup i s p r i n c i p a l , Proc. London Math. SOC. ( 3 ) 28 (1974), 67-88. bR 49-2854. 3. Semisimple n e a r - r i n g s have t h e maximum c o n d i t i o n on N-subgroups, J. London Math. Soc.(2) 11 (1975). 408-412. MR 52-3250. 4. Completely r e d u c i b l e n e a r - r i n g s , Oberwolfach, 1976. 5. Completely r e d u c i b l e n e a r - r i n g s , Proc. Edinbg. Math. SOC. 20 (1976/77), 187-191. MR 56-425. 6. C o n d i t i o n s on n e a r - r i n g s which imply t h a t n i l N-subgroups a r e n i l p o t e n t , Proc. Edinbg. Math. SOC. 20 (1976/77), 301-305. hR 56-12074. 7. A n o t e on i n j e c t i v e modules over a d. 9. n e a r - r i n g , Canad. Math. B u l l . 20 (1977). 267-269. MR 57-3200. 8. N e a r - r i n g s w i t h c h a i n c o n d i t i o n s on r i g h t a n n i h i l a t o r s , Conf. Edinbg., 1978. 9. N e a r - r i n g s o f q u o t i e n t s , Proc. Edin. Math. SOC. 22 (1979), 77-86. MR 80k-16047. 10. N e a r - r i n g s w i t h c h a i n c o n d i t i o n s on r i g h t a n n i h i l a t o r s , Proc. Edin. Math. SOC. 23( 1980). 123-128. MR 811 -16049. 11. R i g h t i d e a l s i n n e a r - r i n g s of mappings, Oberwolfach, 1980. 12. Semiarouos and r e l a t e d n e a r - r i m s . Oberwolfach. 1981. 13. C e n t F a l i z e r s o f t h e general lines; group, San Eenedetto del Tronte, 1981. 14. A n o t e on weakly d i s t r i b u t i v e n e a r - r i n g s , Teesside P o l y t e c h n i c Mathematical R e p o r t s , TPMR81-2, 1981. SEE ALSO MASON - OSWALD. MAXSON - OSWALD, MAXSON - MELORUM - OSIIALO, MELORUM-OSWALO OURRE. Glenn J., Math. Dept. Univ. of Southwestern L o u i s i a n a , L a f a y e t t e , L o u i s i a n a 70504, USA 1. The K r u l l - S c h m i d t theorem f o r n e a r - r i n g s , M.S. Thesis, Univ. o f Southwestern L o u i s i a n a , 1970. PALMER, K.J., Oept. Math. A u s t r a l . SEE PALMER - YAMAMURO.

N a t . Univ.,

-

,

Kyungpook Nat ' 1 Univ.,

SOC. 9

9', E " , R '

E , R ' .P

E,Q' E,R',P S.E

1,c

T'

Taegu, Korea

PELLEGRINI-MANARA, S i l v i a , 1st. Mat. U n i v e r s i t a . 43100 P a n a . I t a l y 1. On t h e S - n e a r - f i e l d s , San Benedetto del Tronto, 1981, 187-192. 2. I n t h e p l a n a r i t y of nubnear-rings of p l a n a r n e a r - r i n g s , Riv. Mat. Univ. P a n a 7 (1981). 245-249. 3. Sui q u a s i - a n e l l i a q u o z i e n t i q u a s i c o r p i p r o p r i , R o l l . Un. Mat. l t a l . B 1 (1982). 187-195. SEE ALSO FERRERO-COTTI - PELLEGRINI. PENNER, Sidney, 3148 Grand Concourse, Branx, New York 10458, USA 1. Geometric axiomatic8 of s u b s t i t u t i o n , M.S.lhesis, Univ. o f Chicago, 1958. 2. B i - and t r i - o p e r a t i o n a l a l g e b r a s of f u n c t i o n s , D o c t o r a l D i s s e r t a t i o n , I l l i n o i s l n s t i t u t e of Technology, 1964. PERIE. Vaselin, Dept. Math. SEE OASIC - PERIC.

I'.A

Canberra, A u s t r a l i a

PALMER, K. J. and YAMAMURO, Sadayuki 1 . A n o t e on f i n i t e dimensional d i f f e r e n t i a b l e mappings, J. A u s t r a l . Math. (1969), 405-408. FR 39-4714. PARK, Y. S., Dept. Math. SEE K I M PARK.

E"

E, P" ,F P" F

G,Cr G.Rs

Univ. Sarajevo, Yugoslavia

PETTET, M . R . , Math. Oept. Texas AlM, College S t a t i o n , Texas 77843, USA SEE MAXSON-PETTET-SMITH PIEPER, I r e n e . 2050 Hamburg 80, Ladenbecker Furtweg I l a , Germany 1. Uber gekoppelte Abbildungen auf F a s t r i n g e n , Oberwolfach, 2. On a c l a s s of near-modules, Oberwolfach, 1972.

1968.

PILZ, Gunter, I n s t . Math. Joh. K e p l e r Univ. L i n z , A-4040 L i n z , A u s t r i a 1. Ordnungstheori e i n K o m p o s i t i o n s r i ngen, Ooctoral O i s s e r t a t i o n , U n i v e r s i t y of Vienna, 1967. 2. Ordnungstheorie i n F a s t r i n g e n . Oberwolfach, 1968. 3. Uber geordnete Kompositionsringe, Monatsh. Math. 73 (1969). 159-169. MR 40-74, 4. Q -groups w i t h composition. Publ. Math. Univ. Debrecen 17 (1970). 313-320. MR 46-1688. 5. Geordnete F a s t r i n g e , Abh. Math. Sem. Univ. Hamburg 35 (1970). 83-89. MP .... A7-l?d .6. P a r a l l e l i s m i n n e a r - r i n g s , Rocky Mountain J. Math. 1 (1970). 483-487. MR 43-4868.

."..

0" ,E ,M,N L ,Na ,0" ,F

Cr,O,E

0 Cr.0 E,O,Ua

0 G.0

457

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E R,M,P

E" ,E ,R .P A' P' .R D,C,F' M R,P.X a l l f r o m A-2 P".P,R

E,P P0,X

P0.X

E PO F.G,D,P.P"

,PO,R,S,S"

97-99.

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p,x

F'

Po .R

P0,P.S

B,I.R.W

E",E',P.R E" ,E ' ,P .R .N Ua .R,E ,E"

F,D"

F,D" F,D" D",F

P,T,Ua R,M,N,P'

,Q.S

P,R,S,D'.Ua,Q,N,M

E,R,S,A

India

SOC. 25

,I ' ,E

Po,E'.R,Ua

0 0

,Cr

APPENDIX

PREHN, Renate, Hans-Scholz-Str. 28, 50 E r f u r t , German D e m c r a t i c R e p u b l i c 1. Zur l h e o r i e i n j e k t i v e r und p r o j e k t i v e r Gruppen uber F a s t r i n g e n , Oiss. Pad. Hochsch. E r f u r t (GDR), 1978. 2. l n j e k t i v e Gruppen uber F a s t r i n g e n , Publ. Math. Debrecen 26 (1979) 75-90. MR 81~-1605z. 3. F r e i e und p r o j e k t i v e Grupppen uber F a s t r i n g e n , s u b m i t t e d .

E ,E ' ,F ' ,H ,Ua r,r'.H.ua C,F',H,Ua

QUAKENBUSH. R. W., Oept. Math. and A s t r o n . , Univ. of Manitoba, Winnipeg, Manitoba, Canada R3T 2NZ 1. Near v e c t o r spaces o v e r GF(q) and ( v , q + l , l ) - B I B O ' s , Lin. Alg. and i t s A p p l i c a t i o n s 10 (1975), 259-266. M R 51-5335.

F ,G, X ,P"

RAOHAKRISHNA, A.. Oept. Math., I n d i a n I n s t . of Techn., Kanpur, 208016 I n d i a 1. On l a t t i c e - o r d e r e d n e a r - r i n g s and n o n - a s s o c i a t l v e r i n g s , I n d i a n I n s t . o f Technol., Kanour. I n d i a . 1975. SEE A L ~ O~ H A N D A RI-P~ADHAKRISHNA RAOO',

0

F., Babe$-Bolyai Ilniv., Kogalniceanu s t r . 1, Cluj-Napoca, Romania 1. On t h e d e f i n i t i o n of skew f i e l s , Arch. Math. ( B a s e l ) 32 (1979), 441-444.

RAHBARI, Mohammad H., 105 Second F l o o r , P a l i z i Ave., Abbas Abad, Teheran 15346, I r a n 1. R e p r e s e n t a t i o n s of groups on n e a r - r i n g s , Conf. Edinb. 1978. 2. Some aspects of n e a r - r i n g t h e o r y , Diss. Univ. Nottingham 1979.

0.F' P,D,F'

RAMAKOTAIAH, D a v u l u r i , k p t . Math.NagarJuna U n i v . ,Naaarjunanagar 522510 (AP), I n d i a 1. R a d i c a l s f o r n e a r - r i n g s , Math. 2. 97 (1967), 45-56. MR 34-7592. 2. Theory of n e a r - r i n g s , Ph. D. D i s s e r t a t i o n , Andhra Univ., 1968. 3. S t r u c t u r e of 1 - p r i m i t i v e n e a r - r i m s . Math. Z. 110 (19691. 15-26. MR 42-3129. 4. A r a d i c a l f o r n e a r - r i n g s , Arch. Math. ( E a s e l ) 23 (1972); 482-483. MR 47-3463. 5. Isomorphisms of n e a r - r i n g s o f t r a n s f o r m a t i o n s , J. London Math. Sac. 9 (1974). 772-278. hR 51-3734. . .. ... 6. S t r u c t u r e theorems on 1 - c o m p l e t e l y r e d u c i b l e N-groups. manuscript. 7. One-sided i d e a l s i n n e a r - r i n g s of t r a n s f o r m a t i o n s , submitted. 8. One-sided i d e a l s i n n e a r - r i n g s o f t r a n s f o r m a t i o n s , Oberwolfach, 1976. 9. A c h a r a c t e r i s a t i o n of a c l a s s of n o n - a b e l i a n groups, submitted. SEE ALSO LIGH - RAMAKOTAIAH - REDDY, RAMAKOTAIAH - RAO, RAMAKOTAIAH - REDDY, RAMAKOTAIAH - SANTHAKUMARI ~

~

~

RS,P T,T',P

T,T',P E",T',S

RAMAKOTAIAH, D a v u l u r i and RAO, G. Koteswara 1. T o p o l o g i c a l f o r m u l a t i o n o f d e n s i t y theorem f o r D - p r i m i t i v e n e a r - r i n g s , Proc. Roy. I r i s h Acad. 78 (1978). 127-135. MR 8Oa-16050. 2. On I F P - n e a r - r i n g s , J. Aust. Math. Soc. 27 (1979) 365-370. MR 81c-16053. 3. 0 - p r i m i t i v e n e a r - r i n g s of t r a n s f o r m a t i o n s , Proc. Roy. l r i s h Acad. Sect. A 79 (1979). 131-146. M R 80k-16048. 4. N e a r - r i n g s o f t r a n s f o r m a t i o n s a c t i n g 0 - p r i m i t i v e l y on a group, m a n u s c r i p t . 5. A s p e c i a l c l a s s o f n e a r - r i n g s , submitted. RAMAKOTAIAH. D a v u l u r i and REDDY, Venkateswara. Y. 1. Zero d i v i s o r s i n n e a r - r i n g s , t o appear. RAMAKOTAIAH, D a v u l u r i and SANTHAKUMARI, C. 1. On r i g h t q u a s i r e g u l a r elements i n n e a r - r i n g s , Arch. Math. 31 (1978) 451-456. M R 804-16024. 2. On l o o p n e a r - r i n g s , B u l l . Aust. Math. SOC. 19 (19781. 917-935. FaC 809-16046a. RAO. G. Koteswara, Dept. Math. Andhra Univ., srE RAMAKOTAIAH - RAO

RAO, V.

RAO, V.

P o s t g r a d u a t e Center, Guntur-52205 (A.P.),

Ind7a

Sambasiva, Dept. Math. Nagarjuna Univ., Nagarjunanagar 522510 (A?), I n d i a 1. A c h a r a c t e r i z a t i o n of semiprime i d e a l s i n n e a r - r i n g s , J . A u s t r a l . Math. SOC. 32 ( 1 9 8 2 ) . 212-214. SEE ALSO RAO-SATYANARAYANA Sambasjva and SATYANARAYANA, Bh. 1. The prime r a d i c a l i n n e a r - r i n g s , submitted.

P'

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18

P",R,N X,E,I.S

X.E.1.S Rs.R.P Rs,P,R,M.N,S C.Rs

Rs

APPENDIX

VAN ROOTSELAAR, R . , Van Nijerodeweq 914, Pmsterdam I , 11, The N e t h e r l a n d s 1. D i e S t r u k t u r der r e k u r s i v e n W o r t a r i t h m e t i k des H e r r n V. Vukovic, Indag. Math. 2 4 (1962) 192-2fl0. 2. A l g e b r a i s g h e Kennzeichnung f r e i e r W o r t a r i t h m e t i k e n , Compositio Math. 15 (1963). 156-168. 3. Zum A L E - F a s t h a l b r i n g b e g r i f f , Nieuw A r c h i e f v o o r Wiskunde 1 5 (1967), 247-249. SEE ALSO VanHOORN - VanROOTSELAAR WXHLING, Heinz, Math. Sem. Univ. Hamburg, Bundesstr. 55, 7000 Hamburg 13. Germany 1. E i n i g e Satze bber F a s t k o r p e r , Oberwolfach, 1968. 2. l n v a r i a n t e und v e r t a u s c h b a r e T e i l f a s t k n r p e r , Abh. Math. Sem. Univ. Hamburg 33 (1969), 197-202. MR 42-1869. 3. Automorphismen Dicksonscher F a s t k d r p e r , Oberwolfach, 1972. 4. Zur T h e o r i e d e r F a s t k o r p e r , H a b i l i t a t i o n s s c h r i f t . Hamburg, 1972. 5. Automorphismen D i c k s o n s c h e r F a s t k o r p e r p a a r e mit k l e l n e r Dicksongruppe, Abh. Math. Sem. Univ. Hamburg 44 (1975). 122-13R. MR 53-8033. 6. B e r i c h t uber F a s t k o r p e r , J a h r e s b e n c h t Dt. Math. Ver. 76 (1975). 41-103. MR 58-2599. 7. Automorphismen Dicksonscher F a s t k o r p e r p a a r e m l t k l e i n e r Dlcksongruppe, Abh. Math. Sem. Univ. Hamburq 44 ( 1 9 7 5 ) . 172-138. MR 53-8033. 8. Normale F a s t k n r p e r m i t k o m m u t a t i v e r bzw. z w e i s e i t i q e r lnztdenzgruppe, Math. 2 . 147 (1976). 65-18. MR 53-3879. 9. E i n Z a s s e n h a u s k r i t e r l u m f u r u n e n d l i c h e F a s t k o r p e r , Arch. Math. ( B a s e l ) 28 (1977). MR 55.12712. 10. Normale T e i l q u a s i k o r p e r e i n e a F a s t r l n a e s . n e r Satz von Cartan-Brauer-Hua. Math. Z. 158 ( 1 9 7 8 ) , 55-60. MR 57-383. WALKER, Roland, Dept. o f Pure Math.. SEE HOLCflMBE-WALKER

Rs Rs RS

F,D" F F.0" F,D",G F.0" F,D",G

F,D" F ,G

F ,D" F.Rs.X

Queens Univ. of B e l f a s t . R T - 7 I N N , N o r t h e r n I r e l a n d

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T'.F,D",Rs F ,T' ,D"

F,D,V Nd,S

F ,Nd,D" V,F,D" F,T',V,O" G ,S" ,Nd F, P " ,S" P" ,F ,D"

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N.S.

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Univ. of Southwestern

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R

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E , A , I ,D ,E "

1,E F.D" .Rs

FE",F,T'.F:,A E" ,T' E",T'.E'

,E

E " ,D ,F ,A

1.S

F ,P"

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46 5

LIST OF SYMBOLS AND ABBREVIATIONS

26

NS

30 41 X I k 44 c o r k 45 DCCI,DCCL,DCCN. ST, Sn 61 'LV 'Wr

Wny

62

?(I)

Qri)

ci L,

68 86,143

88 89

(N,D)

171

171

dn.

5

rn

End

171

172

@

w ) ,I ( r )

172,206

F D ,Y 180 183 I ( N ) 184 188,189 ei 189 Li' L 191) d i m H ( r ) 192 e 6 , L6, L 197 2, 200 X(r),X 200 rk(rn) 202 C(Y) 206 E n d ( r ) , A u t ( r ) 207 Inn(r) 237

ni(r)

O(N)

101 106 G o , G 0 ,G 0 106

cN(r)

0

r N ( r ) 106 30, 0 1 107 L,% 109 {(M) 110

1,z

165 165 167

23

TOA

(n)

Ck(N)

Rs(rl)

111

m

(i)

A(X,

n

Y

209

E(j)(r)

V)

215

216

?[XI , p(p) "[x!,

V[x!

D(r)

,

V[x!

216 216 217

\ P I 218 ~ , , [ x ~ , R c [x]

[ J ) , < J > 222 - ixr 2-_I 226 1, T 227 1

rxl

0.

231

221

466

APPENDIX

*t 276 C,(N) 277

e Y LA

367 369

BIBD

T*

375

279 282

IFP (Po) nid.

y

. . . ,( P 4 ) 310

Ma(r) a.a.n.r.

313

%

316

cb C(i,k)

325

Dm 3 29 323

Am,

Dzn

n'

334

f.0.

334

7

5

309

k0

<:

335

337

la]

;(r)

338 350

;oc(r) 358 J A ( r ) 365 ;t(y)

~ 1 6 ,y - 6

5

378

fa

378

N(S)

379 382

A#,N(S#) 384 GA 386 A ,A' 307 A*,so,sA 387

- 4'

CIo,Aa,Qa 1 A 393

365 366

389

l u t ( N ) , I n n ( 1()

l(N),D(N)

va

337

?,[[xJ]

376 378

7

334

3.0.Y

376

uA(r)

No(S)

322,405

q . 3

H(Na) GSA

313

D(n;n' ,n") 37 1

nll

?!?(

399

-(N)

400

J

402

Nj

399

399

467

abelian near-ring abstract affine near-ring accessible accessible part a d d i t i o n o f the f i r s t type --- second type a f f i n e complete a f f i n e map affine plane almost small a-near-ring annihilator antichain antilexicographic order antitone element archimedian order ascending chain condition a s s o c i a t e d isomorphism associated ring automaton

11 313 385 389 172 172 2 39 9 261 292 29 7 20 327 338 343 337 3 194 401 386

Baer r a d i c a l 68,139,233 balanced block design 279 base 31,262 base of equality 34 276 bas i c b 1ock B-near-ring 290 B e t s c h ' s lemma 366 bicentralizer near-ring 361 biregular 94 block 276,279 Boolean n e a r - r i n g 300 Boolean s e m i r i n g 290 cancellable element c a n c e l l a t i o n law Cebyshev polynomial center c e n t r a l idernpoten t c e n t r a l i zer c e n t r a l N-subgroup central product chain chain r u l e characteristic characterizing series class X c l e a v a b l e n e a r - r i ng Ci-modular 1e f t i d e a l C1,C2 - n e a r - r i n g

10 11,48 246 253 91 106,361 39 7 398 246 245 251 39 7 165 342 166 29 7

cocri tical 402 collineation 262 commutative near-ring 11 compatible 235,240 compatible t r i p l e s 119 complete 279,324,399 complete block design 279 complete groups 324 completely non-abelian 355 completely reducible 54 complete near-ring 336,399 composition ideal 222 composition ring 41 composition sequence 53 conjugate chain 246 10 constant near-ring constant part 10 221 c o n s t a n t polynomi a 1 convex 333 c o u p l i n g map 243,399 critical 402 C - Z-decompos a b 1 e 398 C-Z-trans i t i v e 398 decomposable 53 degenerated 261 derivation 245,255,399 Desargues i an 262,263 descending chain condition 3 D-group 132 Dickson group 255 Dickson n e a r - f i e l d 12,255 Dickson n e a r - r i n g 399 Dickson numbers 256 d i h e d r a l group 415 dimens ion 262 d i r e c t complement 46 d i r e c t product 24 24,45 d i r e c t sum d i r e c t summand 46 10 d i s t r i b u t i ve e l emen t distributively generated 11,171 near-ring d i s t r i b u t i v e n e a r - r i ncj 11 d i s t r i b u t i v e sum 49 distributor 316,396 divides 366 187 division dgnr. D - n e a r - r i ng 29 5 D-ring 312 d u o ( n e a r - ) r i ng 290

468

APPENDIX

e f f i c i e n c y of a BIBP 279 embeddable 15 enclosing ideal 223,232 171 e n d o m o r p h i sm n e a r - r i n g e q u i Val en t e 1 emen t s 366 268 equivalent mu1 t i p 1 i e r equivalent s t a t e s 388 even element 342 e x t e r n a l d i r e c t sum 24 factor near-ring f a c t o r N-group f a c t o r of a sequence f a i t h f u l N-group field f i 1t e r finite interpolation property f i ni t e t o p o l o g y f i rs t category Fisher's inequality f i xed-poi n t - f r e e F-near-ri ng formally real free abelian near-ring f r e e near-ring f r e e N-group Frobenius group Frobenius kernel f u l l ideal f u l l y generating f u l l y ordered

133 111 277 280 9,363 343 343 30 29 29 267 26 7 222 401 332

generalized d g n r . general ized polynomial g e n e r a 1 ized n e a r - r i ng generated genera t o r 9- r a d i c a l G-regular group-automaton g r o u p - s emi au toma t o n GSA-homomorphism

187 247 3 20 2,23 75 164 165 386 378 385

H-base H-dimension hereditary class H - g roup H-monogeni c holonomy g r o u p homomorphic G A homomorphic GSA h omomorph i sm homomorph i sm t h eo rem

16 16 52 21 2 402

192 19 2 143 192 39 8 376 386 379 15 17

1 I

1

idea 1 15,16 idempotent element 10 i den t i t y 10 i n c i d e n c e group 263 incidence space 261 incidence structure 261 i ncomparabl e 334 incomplete design 2 79 indecomposable 53,218 independence 31,110 inductive Moore-syste, 2 i n n e r automorphism 399 input s e t 378 insertion-of-factorsp r o p e r t y 288 305 integral nr. i n t e r n a l d i r e c t sum 45 interpolation 133 i n v a r i a n t sequence 52 invariant subnear-ring 18 3 i n v e r s e we1 1 - o r d e r i nve r t i b 1 e e l emen t s 10 i somorphi c d e c o m p o s i t i o n s 59 isomorphic sequences 52 17,45 i s o m o r p h i sm t h e o r e m s Jordan-Holder

theorem

30 rda n- H o 1de r t h e o r y

52 53

kernel-free kernel of a near-field K l e i n ' s f o u r group K r u l l -Schmi d t - T h e o r e m k-tame Kurosh-Amitsur r a d i c a l

35 253 407 353 352 143

laminated near-ring lattice-ordered near-ring A-compl e t e 1e f t i d e a l 1e f t near-ri ng l e n g t h of a sequence length of a s e r i e s lexicographic order l i n e a r automaton 1 i n e a r polynomial 1 i n e a r s emiau toma t o n l i n e a r s e q u e n t i a l machine 1 i nes L-near-ring l o c a l l y a f f i n e complete l o c a l l y polynomially complete local near-ring

399 343 165 15 7 52 39 7 338 386 219 380 386 26 1 289 239 239 400

Index l o c a l polynomial f u n c t i o n loop near-ring lower n i l r a d i c a l maximal i d e a l ( , . . ) maximum c o n d i t i o n metacycl i c minimal automaton minimal i d e a l ( . . . ) m i n i m a l non-nil potent minimal prime i d e a l minimum condition modular m o d u l a r law monogenic monotone e l e m e n t Moore-sys tem m - s y s tern

238 396 166

20 3 244 389 20 95 64,65 3 84,166 48 75 343 2 65

(N,D)-group 182 n - d i s t r i b u t i ve 39 7 (N , D ) - ( N ' ,D ' )-homomo r p h i sm 1 8 0 near-a1 gebra 41 near-domain 247 near-f i el d 11 n e a r - i n t e g r a l domain 298 nearly constant 337 near-module 13 n e a r - r i ng 7 near-ring of quotients 26,28 N- group 13 N-homomorphism 15 ni 1 69 n i l group 323 nilpotent element J,69 nilpotent subset 69 nil radical 160,166 v-modu 1 a r 86 Noetherian quotient 20 non-f i e l d 11 non- g e n e r a t o r 119 n o n - r i ng 11 n o n - t r i v i a l i n t e g r a l nr. 306 normal endomorphism 214 normal n e a r - f i e l d 264 normal s e q u e n c e 52 normed p o l y n o m i a l 219 72 v-prime v-primitive ideal 103 103 v - p r o m i t i v e n e a r - r i ng v-radical 136 145 v - s e m i s i mpl e 39 7 N-series N-simple 19 398 N - s y s tern

R-composi t i o n g r o u p odd e l e m e n t orbit o r d e r e d n e a r - r i ng o r d e r - p r e s e r v i ng map Ore c o n d i t i o n orthogonal idempotents output function output s e t

469

42 342 9 333 335 26 91 386 386

P a i r o f Dickson numbers 256 parameters of a B I B D 2 79 partial order 333 P-chai n 24 5 P e i r c e - d e c o m p o s i t i on 11 permutation polynomial 245 perfect aroup 169 planar near-field 265 n l a n a r n e a r - r i ng 269 plane 262 p - n e a r - r i ng 298 points 26 1 polynomi a1 215 polynomial f u n c t i o n 216,217 polynomially complete 219,239 p o s i t i v e cone 334 p o s i t i v e def i ni t e 343 ~ i r- e g u l a r 431 prime i d e a l 62 prime near-ring 63 prime r a d i c a l 68,161 primitive 103,105 principal sequence 53 principal orbit 275 Pi-ring 298 productive near-field 265 projective plane 261 projective space 262 proper k-tupel 258 proper refinement 53 pseudo-distri butive 397 pseudo i n t e g r a l domain 288 p - s i n g u l a r n e a r - r i ng 400

Q- r a d i c a 1 quasi regul a r quaternion group quotient near-ring quotients r a d i c a l (map) r a d i c a l subgroup r a n k map r a n k o f a map

165 89 418 28 26 140,143 165 110 202

470

reduced automaton r e f i n e m e n t o f s. s e a u e n c e regular near-ring r e q u l a r p e r m u t a t i on residually finite r i g h t d i s t r i b u t i v e law r i gh t i deal r i g h t i d e n t i t y modulo L right invariant subnear-ring r i g h t n e a r - r i ng r- i ndependen t ring-free ?f-radi c a l ill-semi p r i m e

APPENDIX

389 52 345 258 395 7 15 84 192 7 110 353 140 140

tactical configuration tame 187 TO-ideal topological dgnr. transitive t r a n s 1 a t i on tri-operational algebra t r i v i a l ideal t r i v i a l integral near-ring t r i v i a l mu1 t i p l i c a t i o n trivial order t r i v i a l subdirect product type v type 3 t y p e I...IV

2 79 350 222 187 258 265 41 16 306 8 334 25 77 80 290

225 s a t u r a t e d po lynomi a1 s e m i au t o m a t o n 378 194 ul t r a f i 1t e r s e m i - 1 i n e a r map 403 41 u l traproduct s e m i n e a r - r i ng 403 semiprimary nr. 392 u n i t a r y 14 semiprime i d e a l 66 u n i t s 184 semiprime near-ring 67 upper n i l r a d i c a l 161,166 s e m i r i ng 290 semi tame 436 343 sequential function 387 v e c t o r n e a r - r i n g s e r i e s connection 387 weak c o m m u t a t i v i t y 289 3 sharply transitive 2 5 8 we1 1 - o r d e r e d s e t sheaf 40 1 s i m i l a r mu1 t i p l i c a t i o n s 322 simple 19 Zassenhaus-cri t e r i o n 256 small 337 Z - d i s t r i b u t i v i t y 310 10 socl e 80,187,358 zero divisor 200 s-prim1 t i v e 105,165 zero s e t s p - s y s tem 67 zero-symmetric n e a r - r i n g 10 301 zero-symmetri c p a r t s p e c i a l Boo1 e a n n e a r - r i ng 10 stabilizer 253,365 zero-symmetric polynomial 221 stalks 40 1 state set 378 state-transi tion function 378 s t r i c t l y maximal 86,168 s t r i c t l y mimimal 368 s t r i c t l y ordered 341 s t r i c t l y prime 68 strictly primitive 186 276 strong IFP 75 s t r o n g l y monogeni c strongly regular 348 s t r o n g l y uniform 284 subdirectly irreducible 25 subdirect product 24 subspace 26 1 sum of ideals 44 sum rule 245 s y n t a c t i c n e a r - r i ng 379,386