Фундаментальная и прикладная математика (1995, №1)

 Rn = f(k p m) : k p m 2 N n = k + p + mg Pn = f(k p ;m) (k ;m p) (;m k p) : k p m 2N n = k + p ; m k 6= m p 6= mg Qn = f(k ;p ;m) (;p k ;m) (;p ;m k) : k p m 2 N n = k ; p ; mg: 0  1 X p v = vn(s) exp(in y) + v
  , ! !    (2.3) \ ! "   -  .    ! 3 2vs = a2 vyyy + vyy + 1 ; M(v2 )]v ; v3 + 13 M(v3 ) (2:5) v(s y + 2)  v(s y) M(v) = 0 (2:6)   M |    !   y. 6  , D   (2.4){(2.6), !: u0 = v(s y1 ) ; v(s y2 ) y1 = t ; x y2 = t + x

(2:7)

  v = v(s y) | .  3   ! (2.5), (2.6). 0   3], 4]  ! (2.5), (2.6),   $     $    7{  F,   . $ $   $  ! (1.1), (1.2). G        , !   (2.3) | \!

" .  .    (1.1), (1.2). 0 !  

 . $  ,    

 !    !  $   . 2$ . ,   ! (2.5), (2.6) ! ,      ! $   . 0 , !  ! (2.5), (2.6)   !  3    v = v0(!0 s + n0 y) n0 2 N v0 (y + 2)  v0 (y):

(2:8)

H!        (2.1), ! 

 ( !  .  "5=2  )  !  3 

p

u = "u0(  x) + "3=2u1 (  x) = (1 + "0)t 0 = !0=n0 

(2:9)

210

. .   , . .  

$  ! (1.1), (1.2). I  . ( . (2.7)) u0 (  x) = v0 (n0 ( ; x)) ; v0 (n0( + x))

(2:10)

 2- !     u1(  x)  !         u1 (t s x). H ,   / 

 3  ! !   !  3   ! (1.1), (1.2),        .

 1.      (2.5), (2.6)          (2.8),        .         "                (1.1), (1.2)   (2.9), (2.10)    !     .

3 ! 

4, !  !  n     $ (1.3)    p   $     . "n2 ,   n ! 1    !  (;=2a2)  i( 1="2 ; 2=4a2=a). J \3"       .    .  ,   4], 5]  !   . - ,      /       !$  ! (1.1), (1.2) !     $. 0/,       ,     !      . 4  $ / |     $   $!  $  ! (1 + 2"0 )h ; hxx ; "a2 hxx ; "hxx ; "h + (3:1) + "(u20 (  x)h) = 0 hjx=0 = hjx=1 = 0 !$   (1.1), (1.2)     

  !  3  (2.9)     !      "2  3. K    

   6]  /   . $  3 $  $   .  / ! , !     (3.1)     h ! h exp(;" ),    

  > 0  . , !      $ .  ! $    

  ! $  V + I,     $  . I  . V | ,  

$   V h  a2 h000 + h00 ; 20h0 + 1 ; M(v02 (n0 y)) ; v02 (n0 y)]h ; (3:2) ; 2Mv0(n0y)h]v0 (n0y) + Mv02 (n0y)h] h(y + 2)  h(y) M(h) = 0: 4, !    $      

  ! $   $   (  ! ,   

,     

 ! ,  !  

   v00 (n0y)).    E    P      span fsinr x r = 1 : : : r0g,    r0     .      

            

211

  ,    . $3. L C(E)  !   

    !

  ;1 < < +1  $ u(  x)   !   E,       :   p 1 X 2 u ( ) sin m x  = sup ju ( )j u(  x) = m m 2 2 m  m=0 m       1 !1=2 X 2 kukC (E ) = m : m=0

L C 1(E)  !     C(E),      $ u(  x), ! @u=@ 2 C(E)M      $ kukC 1 (E ) = kukC (E ) + k@u=@ kC (E ): 0   C 2 (E)     ! . 0     (3.1), ! h1 = Ph, h2 = h ; h1, $       (1 + 2"0)h1 + P BP h1 + "a2 PBPh1 + "P BPh1 ; " ; (3:3) ; "(1 + )h1 + "Pu20(  x)(h1 + h2 )] = 0 (1 + 2"0)h2 ; h2xx ; "a2 h2xx ; "h2xx ; (3:4) ; "(1 + )h2 + "(I ; P)u20(  x)(h1 + h2 )] = 0   B = ;d2 =dx2.     . $  . 0 6  6 1  h2  (3.3)   h1  (3.4). 0!3         C 2(E)  . $  N. 0, !       !   "    kN;1 kC (E )!C 2(E ) 6 N="M (3:5)   .   . $3 N N1  . . |     ",   r0  

, !   !     

. 6 (3.3)  (3.4) , ! N = diag fN1 N2 g   = 0. 7 , $    "0 = "0 (r0 ), !  0 < " 6 "0 kN;1 (3:6) 1 P kC (E )!C 2(E ) 6 N1 =": 2 .   !$   (3.3)  h2 = 0       h1 = C0( ) + "C1 ( )] h1 = C00 ( ) + "C10 ( )]  = (1  <1 : : : r0  
  7] . . !  .  !    "2          _ = "D (3:7)

212

. .   , . .  

  D |      V + I     $ exp(ir y), r = 1 : : : r0,  W22. 4   ., !    (3.7)  !  (3.6)      ,  

$  4]. 4 .    F.    sin r x, r = 1 2 : : :, !  

 3     $    (1.3)  !  r0  !  .3,   , !

kN;1 2 (I ; P )kC (E )!C 2(E ) 6 N2 ="r0 :

(3:8) %  .   (3.3), (3.4)           C(E). 6 .  N1  N2 , $        .    $. 0       ,   , !  !       (3.6), (3.8)   !  .3 r0       $. 4     $  |

  (3.5). 6 

   .  6] , !  3 $ ! (3.1)    /   .  ,  $    $!         (3.7). ;$    $!         !  .3 r0          (3.2) (      . 4]). I3$ / | .      !  3   $  ! (1.1), (1.2) |     

$  5] .       (1.1)   = (1+"0 +"2 )t,    = (") |       !

   ",   p u = "u1(  x) + "3=2u2 (  x) + "3=2 h     u1  u2   (2.9). .       N(" )h = F(  x "  h h )

(3:9)

  N |  ! $ (3.1)  $ $  . $ ,        

 2- !     F(  x "  u v) , ! kF(  x "  0 0)kC (E ) 6 N3 " (3:10) 3 = 2 kF(  x "  u1  v1 ) ; F (  x "  u2 v2 )kC (E ) 6 N4 " fju1 ; u2 j + jv1 ; v2 jg (3:11)   N4 = N4 (R), R = maxfjuj j jvj j j = 1 2g. 6  

   $  , !    Nh = 0  

  !  3  @(u0 (  x) + "u1(  x))=@ .  /       "2   

  6] ,    . ! , !     . K  

, !      ! .   $ !     (3.9),  $  $  /    . %   N         C 2(E),    2- !    $. 7   ,     !    3     Nh = G(  x)  

    $      .      $ !  G  !  3  g

            

213



$ $  !. %3  hG  /   

,   @hG =@  .       g. 7 /,     

khG kC 2(E ) 6 N5 ";1 kGkC (E )

(3:12)

   3 $    Nh = 0,    /    

    (3.5). %   ! .    (3.9)     . G        (!  G   @ 2 (u0 + "u1 )=@ 2    g), $    .    ,   N. #  (3.10){(3.12)   .  /        $   . 2- !     h(  x " ). 6 (3.12)  , !  

$        O3 $   . $   $. 0  .     $ !     (3.9), !          = p(") + "P(" )     p(")  ! ,  P       O3. J  .

4 #   

6     !  $  $  ! (2.5), (2.6)     .         = 1=a2. / !     s=a2 ! s !  $    ! 3 2vs = vyyy + fvyy + 1 ; M(v2 )]v ; v3 + 13 M(v3 )g (4:1) v(s y + 2)  v(s y) M(v) = 0      $ $   .     .  . ;  

$  . 2 ,       (4.1) v = v0 (  ') + v1 (  ') +     = s ' = ('1  '2 : : :)  1 2 2  X v0 = n( ) exp(i'n ) + 
       n , ! 1 X n4jnj2 < 1: n=1

(4:2) (4:3)

(4:4)

214

. .   , . .  

0   /   $   ,     v1 !    1 1 3 X X 3 3 @v1 ; n @' ; 3 nkm @' @@'v1@' = f(  ') (4:5) n nkm=1 n k m n=1

     . @ 2 v0 + 1 ; M(v2 )]v ; v03 + 1 M(v3 ) 0 f(  ') = ;2 @v +  (4:6) 0 0 @ @y2 3 3 0 2 - !         '     ' .   .  . 4, ! -      'n = 'k  'm  'p   

.  n, k, m, p         3     (4.5)       $     ! , !     (4.6)       const  exp(i'n ), n = 1 2 : : :. 0   ,      n     1 n = (1 ; n2) ;  (3j j2 + 4 X j j2) n = 1 2 : : : 2 d n n n k d k=1 k6=n

   . $ $  ! (4.1). . $3,  ,    .  $  n = jn j2: 1 dn = (1 ; n2) ;  (3 + 4 X k2 ) n = 1 2 : : : : (4:7) n n n d k=1 k6=n

4, !  !  (2.3)   (4.7)  !     !    ,    $ !        w22,      .  $,        (4.4). 6 .  4]  , !    /   .   $!  !         !  !       n1  : : : nm , n1 < : : : < nm , ! m- $    $  $  ! (4.1) $   $! . H! 

      (4.7),   , !   .3    ! (4.1)     . 0  > 1  $!        ,    < 1      $!$ ,            (4.7)  $  $   $ 1 = (1 ; )=3. 2       !  !  1=k2, k = 2 : : :,   3    $!  vk (!k s + ky )   (2.8),  !         (4.7)  $  $   $ k = (1 ; k2)=3. 0 . $3  .3     vk   . 

            

215

  k ; 1  $!    (     !  !  1=(4k2 ; 3m2 ), m = 1 : : : k ; 1),    < 1=(4k2 ; 1)      $!. J ,   ! 0 !   $!   $  ! (4.1)  !

  . 0       Q      $,

     ! .

5 %  .

4      (1.1) ! "a2 uxxtt, . .           ,    $  ! utt ; uxx = "uxxt + "ut ; u2ut 

(5:1)

ujx=0 = ujx=1 = 0 (5:2) $  $  . !   $   ! (1.1), (1.2). !  Q         . W22  W21 ,   W22 , W21 |       $,    !     (5.2). 2 .  . $   ! $ 3   $ 3

$  !    (5.1)   w1 = Lu, w2 = ut,   L | ! $   $  .   ;d2 =dx2.        E = W21  W21 !   w_ = A0 Lw ; "A1 L2w + "A2 w + f(w) w = colon(w1  w2)        (5:3) 0 1 0 0 0 0 0 A0 = ;1 0  A1 = 0   A2 = 0 1  f(w) = ;w2 (L;1 w1)2 :



4 .   F.,   , !  A0 L ; "(A1 L2 ; A2 )    E .    . 0  Q .  (5.3)   .              $, ! $ . I, !   = 0   ! (5.1), (5.2) ! ,    > 0   $!  Q  3  (    $

2 + "( 2 n2 ; 1) + n2 2 = 0, n = 1 2 : : :)        | !  (Re n ! ;1  n ! 1)  !  ( n ! ;1="  n ! 1). 4  ,  !   ,     !    . 3 $  ! (5.1), (5.2)  t. S  .   x    ,   . $,    t = 0. 2       !  3 $  ! (5.1), (5.2)   .     .  . 4, !   .   .        . 2,   .   . a2 = 0. !   ,   (2.5), (2.6) !  .  

216

. .   , . .  

3 2vs = vyy + 1 ; M(v2 )]v ; v3 + 13 M(v3 ) (5:4) v(s y + 2)  v(s y) M(v) = 0: (5:5) 0 , !  ! (5.4), (5.5)        v(y) 6 0. J       $  fvg = fv(y +c) c 2 Rg. 0  v(y)      (2.1), ! (   . 2) 

  !  3  $  ! (5.1), (5.2).

 2.  1= 2 (n + 1)2 6  6 1= 2n2     n.       (5.4), (5.5)   n        fvk g, k = 1 : : : n,   fv1 g    ,   -

 . "!          "         (5.1), (5.2)   (2.1), (2.7)    !     . 4, ! .      3   ! (5.4), (5.5)  (5.1), (5.2)     !    

$  . 3 . 0/        3. ! . , 

  $   ! (5.4), (5.5).    (5.4)  M(v3 ) = 0, 1 ; M(v2 ) = c > 0, v = pcz(y)    !     M  . ! $   p (z 0 )2 = 6c (!1 ; z 2 )(!2 ; z 2 ) !1 2 = 3  9 ; 6 2 (5:6) p   0 <  < 3=2. 4 !,p , ! z(y  c) 3     (5.6),     z jy=0 = !2, z 0 jy=0 = 0. 4 ,    .,   !     !    T = T ( c). T  , !     3   ! (5.4), (5.5)       n = 1 2 : : :   T( c) = 1=n c(1 + M(z 2)) = 1 (5:7)         , c. p p p  

 4! , ! T(0 c) =  = c, T( 3=2 c) = 1, @T=@ > 0. 0/   2 n2 < 1     (5.7)   !  3   .   = n (c), !  @T ;1 0 n (c) = 2nc @ > 0  2 n2 < c 6 1: (5:8) 0    = n(c)      (5.7), ! p Z1 s2 ds : (5:9) '(c) = 1 '(c) = c + 2n!2 6c p (1 ; s2 )(!1 ; !2 s2 ) 0 7 , '( 2 n2 ) =  2 n2 < 1, '(1) > 1,   (5.8)  , ! '0 (c) > 0. 0/     (5.9)   !     c = cn.

            

217

!     $  fvk g . $ ,  

   vk (y) = pck z(y k (ck ) ck ), k = 1 : : : n,        $   $!    .   ! h00 + qk (y)h ; 2M(vk (y)h)vk (y) + M(vk2 (y)h) = h (5:10) h(y + 2)  h(y) M(h) = 0 (5:11) 2   qk (y) = ck ; vk (y), !     $  ! (5.4), (5.5). 4   , !, -,  

  !   ! (5.10), (5.11)  

,    ! .    (5.10)       L02  !   $     !   

$  V M -, = 0     

  ! ,  !  

   vk0 (y).

. #   = 0     (5.10), (5.11) |  .   . I, !    L(h) = 0,   L =  d2=dy2 + qk (y),    

 ( !  .   )  !  3  vk0 (y),    ( $     vk0 (y))  3  @z(y (ck ) ck )=@  

  @T=@ > 0  !    . I , !      !    M(vk  vk0 ) = M(vk0 ) = 0  3 .   ! L(h1 ) = vk (y) h1(y + 2)  h1 (y)M (5:12) L(h2 ) = ;1 h2(y + 2)  h2 (y): (5:13) ;   

, !, !    !  3     (5.10)  = 0      h = 1 h1(y) + 2 h2 (y) + 3 vk0 (y) (5:14)  (5:15) 1 = 2M(vk h) 2 = M(vk2 h)   

 3 . . 0    (5.14)    (5.15), 

   1 , 2 !  $    (2M(vk h1 ) ; 1)1 + 2M(vk h2)2 = 0 (5:16) M(vk2 h1 )1 + (M(vk2 h2) ; 1)2 = 0: J ,   . .    Q  . 2    h1 (y)    .    ! (;1=2k 6 y 6 1=2k) L(h1 ) = vk (y) h1(;1=2k) = h1(1=2k) = 0: (5:17) 4, !     vk (y) > 0  ;1=2k < y < 1=2k  L(vk ) = ; 23 vk3  vk (;1=2k) = vk (1=2k) = 0

218

. .   , . .  

      .  L  !    (5.17)  .  .    S . 6,   ! (5.17)   !  3,  Q 3  h1 (y) , ! h1(y) = h1(;y), h1(y) < 0  jyj < 1=2k. 0     1=2k 3=2k]

!     .  ! y = 1=2k,       .   !     2=k, !, ! , 3  $  ! (5.12). 0   h2 (y)      $  ! L(h2) = ;1 h02 (0) = h02(1=2k) = 0

(5:18)

    !  3,     !,   3  !$ (5.18)   $ $  !  !    ;1=2k 0]    !     1=k     .,  !   !  3     L(h) = 0,  $     vk0 . 0, ! h2 (y) < 0  0 6 y 6 1=2k. 2$ . ,    !  (5.18)  vk0      0 1=2k], !   h2 (0)vk00 (0) = vk (0),   ! vk (0) > 0, vk00 (0) < 0  h2 (0) < 0. 0  ., ! $    0 < y0 6 1=2k, ! h2 (y) < 0  0 6 y < y0 , h2 (y0 ) = 0. H    (5.18)  vk     0 6 y 6 y0 ,    !: Z y0 Z y0 2 3 0 0 6 vk (y0 )h2 (y0 ) = 3 vk h2 dy ; vk dy < 0: 0 0 0  3  h2 (y) 

  !           ., ! 3  $  ! (5.13). 2 3  .    !  ( . (5.16)) ., ! M(vk h2) = 0, M(vk2 h2 ) < 0, M(vk h1 ) < 0,   sign h1 = ;sign vk ,    h2 ! , .    !    1=k. 6   

$   , ! $   $!  $  fvk g         2 (0 1= 2k2 ). #  , ! / $     .   .  $  v  0   .3  . 0/ .   ! (5.10), (5.11)    2(k ; 1) .   

  ! $. J  . 0  = 0   3  $  ! (5.4), (5.5) (,   

, v  0)  s ! 1    ., . . !!         . ;   $    $  ! (5.1), (5.2)   . !.    ,        

$   .

 .   = 0          (5.1), (5.2),  

   C - ,          %  . F! $   /$   :     $   . $ (   )   $       .

            

219

6 ( 

%    ! P (@=@t @=@x)u = 0 u(t x + 2 ) = u(t x)

(6:1)

  P(v w) | $  ! . 0 , !    P(i! in) = 0    n = 0 1 : : :   $  

$  . ! = !n. J     (6.1)     3  u = expfi(!n t + nx)g n = 0 1 : : : :

(6:2)

;  2],   ., !   (6.2)     $,      n  m     

jcn ; cm j > 0

(6:3)

  cn = !n=n, c0 = 0. 4, !   (6.3)     .       !    . 6 !     !       utt ; uxx = 0,             cn. ;

       .      . 7 ,  1] ! ., !  !    !       $! ,   , !    .  !   . 0!  /  ,   $   ,  , ! !         $   (6.2) !    , - !     $! , 

 . 2

      3   $  ! (1.1), (1.2) $   $. 2$ . ,   !  .3 a2  $  ! (1.1), (1.2)   . .   .3 !   $! , !   .3   a2,    , !     .3 ,   $    (a2 = 0)         $!$ . 7 ,       ! $  $     !  3 $  t x,      | !      3 $. !  , ! !

  . .    $        ! utt ; uxx + "a2 uxxxx = "uxxt + "ut ; u2ut ujx=0 = uxxjx=0 = ujx=1 = uxx jx=1 = 0   !      ! $   

    $   .   EI = "a2 , . .  . $ $   .    ! (2.5), (2.6).

220

. .   , . .  

) 1] . .   .          , ! "     // $. "% . | 1992. | ). 51. | +,-. 2. | .. 59{65. 2] 1  ,  ,. 23  3. .. 4    . . | $.: $, 1977. 3] . ..   . $ 3  "% ,! % "3    8!  9%! -  !  %  %  3"  // . %. 9. | 1987. | ). 39. | N 1. | .. 28{34. 4] . ..   . <  ,!  -  !  %  %  3"  // $. . | 1993. | ). 184. | N 3. | .. 121{136. 5] . .   .      "%

,! -  !  % - -

  -    // $. "%  | 1991. | ). 49. | +,-. 5. | .. 62{69. 6] . ..   . =  

 8   ,! 3   ,!   

-   -  - - 3 % 

> % // ). $. %.

-  | 1978. | ). 36. | .. 3{27.

7] . ..   ,

+. +. $ . 1 , % 3  3   

 8 

 ,! 3   ,!      "%  -,% - - 3 % > % // ?  .    | 1974. | ). 10. | .. 1778{1787.

(      ):  1995.

    . . 

  70-           (14.02.1924{26.05.1989)

          

 512.55

   K |   , A K |     .

  ! " #  $  $  A- . %    &!   $    .   , $#   ' ( & !  ' $ $ . ) $!,  p- '   *{, ( & ! #   $   $      . #   $  A- '   & $ $## q-- ,  q | $ , !#  .  ,  ' $ ' - . % !,   #' q-' /! ( & $(

 $ ,  q ## # ' / ' $ '  $    $! ' $ ' q 2 ;  . # # &    # q--   ' $ ' -  #  ' &!  #. 1, $  &    # $ ' -    $  A- #    ,  A ## #       $   : $ '   ( & $   ,. $ #  # $ /! ($ ! $ # -  !,   #' $' /!, !'  $ , ( & $(  $ /!    #).

Abstract Z. S. Lipkina, Locally convex modules, Fundamentalnaya i prikladnaya matematika

1(1995), 221{228.

Let K be a non-archimedean valued 7eld, A K be its integer ring. This paper is devoted to the study of the locally convex topological unital A-modules. These modules are very close to the vector spaces over non-archimedean valued 7elds. In particular, the topology of these modules can be determined by some system ; of semipseudonorms. Monna demonstrated that p-adic analogue of Hahn{Banach theorem can be proved for the locally convex vector spaces over non-archimedean valued 7elds. One can give the de7nitions of q-injectivity, where q is the seminorm which is determined on this module, and of the strong topological injectivity. It means that any q-bounded homomorphism can be extended with the same seminorm, where q is a some 7xed seminorm in the 7rst case, and an arbitrary seminorm q 2 ; in the second one.            1995, 1, N 1, 221{228. c 1995      !, "#    \%     "

222

. . 

The necessary and su8cient conditions of q-injectivity and strong topological injectivity for torsion free modules are given. At last, the necessary and su8cient conditions for topological injectivity of a locally convex A-module in the case when A is the integer ring of the main local compact nonarchimedean valued 7eld are the following ones: a topological module is complete and Baire condition holds for any continuoushomomorphism (here topological injectivity means that any continuous homomorphism of a submodule can be extended to a continuous homomorphism of the whole module).

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.  .     | .: , 1971. A. F. Monna. Analyse non-archimedienne. | Berlyn, 1970. | 320 p. O. Goldman, C. Sah. Locally compact rings of special type // J. Algebra. | 1969. A. W. Ingleton. Hahn{Banach theorem for nonarchimedean valued #elds | Proc. Cambridge Phil. Soc. | 1952. | V. 48. | P. 41{45. '    (:   1994.

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Abstract A. A. Nechaev, Finite quasi-Frobenius modules, applications to codes and linear recurrences, Fundamentalnaya i prikladnaya matematika 1(1995), 229{254.

A simple exposition of the main properties of the quasi-Frobenius modules over -nite commutative rings with identity elements. The presented results show the special role of such modules in the theory of linear recurrences and in the theory of linear codes over rings and modules. In particular it is proved that the general weight functions of the linear code over a ring and the dual code over the corresponding QF -module are connected by the Mac-Williams identity.



  

     

                   

          

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 ,  ord i = di i 2 1 t. ,    !i

   M ,  !i(j ) = ij d1i ,  ij |   < . ; ord !i = di M  =< !1 > +_ : : : +_ < !t > = M. G  2 M n 0,   

 = 1 + : : : + t  i 2< i >,      i 2 1 t      i 6= 0, . . i = ci i 0 < ci < di . B  !i() = !i(i ) = ci =di 6= 0: 2

234

. .  

,      !  r 2 R   ! 2 M     r! : M ! Q=Z    (r!)() = !(r)  2 M: B   ,  r! 2 M  ,       M   R-. 0 ,    1.7               R M R M  . ,       1.8. (+          R-  = R M  . RM  2 0    ' : M ! M  ,       2 M  '() : M  ! Q=Z   '()(!) = !() ! 2 M  : B   ,  ' |     (M +)    (M   +). G

 6= 0,     1.7    ! 2 M  ,  !() 6= 0,

  '() 2 M  n 0, . . ' |     ,       1.7 jM j = jM j ,  ' |  . , ,  ' |   R-, . . '(r) = r'()   r 2 R. 2  ! 2 M  ,        r'() r! ,  (r'())(!) = '()(r!) = (r!)() = !(r) = '(r)(!): 2   1.9. &  M = M1 M2 |    ,  +          R-  M  = M1 M2 : 2 3   !  ~! = (!1 !2) 2 M1 M2        !~ : M ! Q=Z,      !   ~ = (1 2) 2 M1 M2 !  ! ~ ( ~ ) = !1(1 ) + !2(2). ,  ,  ! ! ~     (M +),    !   M1 M2     . ;  ,       M1 M2    M  ,      1.7 !     ,      

   . ,  ,  ! |   R-,     r 2 R         !   M  : (r !~ )(  ~ ) = ~! (r  ~ ),   (r ! ~ )(  ~ ) = !1(r1)+!2 (r2) = (r!1 r!2)(  ~ ): 2 : R M  5 ,       R L

    K    ' : R K ! R M        : R L ! R M . M-    .   1.10 ( %& ) !  R M 5   %      % ,  %    %   I / R   % %    ' : R I ! R M +     2 M  ,  '(a) = a   a 2 I (. . +   %     : R R ! R M ,  "+ '). 2 2 .,   ,  #4]. 2

       

235

9    5  ,    "    Z-.

  1.11. 6 Q=Z5 . 2 G I = Zm |   Z ' : I ! Q=Z|  ,        : Z! Q=Z,  (1) = '(m)=m. 2

  1.12. &  RM |    ,  R M  | 5 

 . .   ,   R Q = R R 5 . 2 :   -    . 3 1.10,          ' : R I ! R R  I / R     : R R ! R R .     R ,    a 2 I !  '(a)     '(a) : R ! Q=Z. 3 !      R-  R ,    a 2 I r 2 R      '(a)(r) = '(a)(re) = (r'(a))(e) = '(ra)(e). 0    ! : I ! Q=Z   !(a) = '(a)(e) a 2 I: ;    1.11     !b : R ! Q=Z,   !. 3         : R ! R    (a)(r) = !b (ar) a 2 R r 2 R: B   ,   |   R-,   a 2 I      (a)(r) = !b (ar) = !(ar) = '(ar)(e) = '(a)(r), . .   '. 2

2

   . "  # . %        

=  -       I / R   R M  K < R M   R,       

M ? I = f 2 M : I = 0g R ? K = fa 2 R : aK = 0g: ,  , M ? I |   R M R ? K |   R. B  

    ,     I J / R  K L < R M     -  R ? (M ? I)  I M ? (R ? K)  K R ? (K + L) = (R ? K) \ (R ? L) M ? (I + J) = (M ? I) \ (M ? J) R ? (K \ L)  (R ? K) + (R ? L) M ? (I \ J)  (M ? I) + (M ? J):

(2:1) (2:2) (2:3)

236

. .  

: R M     ,  QF - ,    I /R

K < R M  -  (2.1)   . 0 ,         K < R M         Rb = R=R ? K ,        !  ba = a+(R ? K) 2 Rb   2 K   ba = a. 9     M ? I         R=I .

  2.1.   R M   QF - , I J / R K L < RM . 7 % 

     +  " . (a) !  K   QF -      Rb = R=R ? K . (b) !  M ? I   QF -      R=I . (c) .  (2.3)     .

2 (a) 3 K 0 < RbK Bb = Rb ? K 0 . ; Bb = B=R ? K ,  B = R ? K 0 . b = 0g = ,,     ,  K ? Bb = f 2 K : B 0 = f 2 K : B = 0g = K ? B = K \ (M ? B) = K \ K = K 0 , . . b = Bb K ? (Rb ? K 0 ) = K 0 . 9       Rb ? (K ? B) b b    B / R. N  (b)       . (c) ;  R M  QF -,  K = M ? I L = M ? J ,  I = R ? K J = R ? L, ,   (2.2), K \ L = M ? (I + J). ,  R ? (K \ L) = R ? (M ? (I + J)) = I + J = (R ? K) + (R ? L). 9       M ? (I \ J) = (M ? I) + (M ? J): 2 @   QF -    R    .   2.2. !  RR    RR   QF - . 2 3 I / R K = R ? I . ; K      ! : R ! Q=Z   I  Ker ! . <  ,   ,         Rb = R=I ,  K |     b 3,  R ? K = I . G r 2 R ? K ,  ! 

   R. rb = r + I   Rb       K . B     1.7 br = b0 , . . r 2 I R ? K  I . ,      (2.1). B,  K < R R I = R ? K . ,  R     R  R ,           1.8. ; I |       R ,   

K ,  , 

-, ,  R ? I = K: 2 2     QF -     R            

   ,       .   2.3.     (1.1), (1.2) R M   QF -   %      % ,  %  R1 M1  : : : Rt Mt  QF - .

       

237

_ t ,  Ks = es K | 2 P  K < R M    K = K1 +_ : : : +K   Rs Ms . < ,      _ t ? Kt R ? K = R1 ? K1 +_ : : : +R



_ t ? (Rt ? Kt ): M ? (R ? K) = M1 ? (R1 ? K1 )+_ : : : +M 3!   M ? (R ? K) = K         Ms ? (Rs ? Ks ) = Ks  s 2 1 t. 9   ,   I / R    _ t ,  Is = es I |   Rs ,   R ? (M ? I) = I I = I1 +_ : : : +I         Rs ? (Ms ? Is ) = Is  s 2 1 t: 2

3

QF -

     . "  .  '  

3   -      QF -    . B  ,   R M   ,  R ? M = 0. ,  , QF -   . 2    R M     R   QF -       S(M) = M ? N(R),      1.6          R M     R M . 0 ,    NS(M) = 0,  S(M)       R = R=N .  3.1. $   %   R M       R   

 +  " : (a) R M  = R R . (b) R M | QF - . (c) S(R M) |       R M . (d) dimR S(R M) = 1. (e) S(R M) |      ( )      R M (         R M ). (f) !  R M "        . 2     (a) ) (b)    2.2. (b) ) (c) 3  2 S(M) n 0. ; N  (R ? R) 6= R. @ , R ? R = N   (b) R = M ? (R ? R) = M ? N = S(M).   

(c) ) (d) ) (e) ) (f)        1.5, 1.6. (f) ) (a) 3 K |        R M . ; R K |     ,     1.6      R-  ,      ' : R K ! R R . ;    1.12 R R |

"   ,    '         : R M ! R R . 3 ! Ker  = 0,      (f) K  Ker . @ ,  |  . G (M) 6= R ,  R ? (M) 6= 0,   R R  QF -    2.2. B    |  ,  R ? M = R ? (M),    R ? M = 6 0       R M . @ ,  |

 . 2

238

. .  

 3.2.   QF -            R  -

    R R . 2 3 2.3 1.9,    ,  R |   ,   !     !      (a) (b)  3.1. 2    3.3.   QF -  R Q 5 . 2 @. 3.2 1.12. 2 B         .    3.4. &  RQ   QF - ,    % N 2 N      L < R RN  K < R QN        RN ? (QN ? L) = L QN ? (RN ? K) = K (3:1) jLj jQN ? Lj = jRjN  jKj jRN ? Kj = jRjN : (3:2) 2 3 3.2    ,  Q = R . =  ,  1.9,    QN = (RN ) . 3 ! K = QN ? L        RN ,   

L ,  RN ? K |   !  RN ,       K ,    ,      2.2, ,  RN ? K = L , . .       (3.1). ;  QN ? L           RN =L ,   1.7 jQN ? Lj = jRN =Lj , . .       (3.2). =    (3.1) (3.2)     ,    RN = (QN )   

  1.8. 2 3 End(R M) |  !    R M (     

    

!  ). 2  !  r 2 R  rb    !    R M ,     2 M    rb() = r. b        B rb %  ,  Rb = R(M)  End(R M). ,  ,  R M |    (   ! ), b = R. B  RM E - , 

     R(M) b End(R M) = R(M). ,  ,      ,     R R,  E -.   3.5. (a) 8     R Q   E - . (b) $  %    K < R Q   % %    ' : R K ! R Q

+    r 2 R  ,  ' = rbjK | %  %   rb     K . (c) &     (b) ' |    ,   r  "  , . . '  "        R Q. 2 (a) 3  2 End(R Q). J   K < R Q2   K = f( ()) :  2 Qg . 3 L = R2 ? K |   K  R2 ,

I = fb 2 R : 9a 2 R (a b) 2 Lg:

       

239

,  , I |   R. G I 6= R,  K = Q ? I |     R Q,     (0 )  2 K ,    Q2 ? L . , ,   R Q  QF -,     3.4 Q2 ? L = K . @ , (0 ) 2 K    2 K ,        K . ;  , I = R. B     L       (;r e),  r 2 R,   K = Q2 ? L ,     2 Q      ;r + e() = 0, . . () = r  = rb. (b) @  3.3   '        : R Q ! R Q,   (a)  = rb     r 2 R. (c) 2  ,  R    (1.1)      . ; R Q K    : _ t Qs = es Q s 2 1 tH Q = Q1 +_ : : : +Q _ t  Ks = es K s 2 1 t: K = K1 +_ : : : +K 3 !     Qs Ks           Rs = es R,   '    '(Ks )  Qs  s 2 1 t,   '(Ks ) = '(es K) = es '(K)  es Q. ;   ,    (c)     ,  R |   . G K = 0,  (c)   ,      r = e. 2  ,  K 6= 0. 0 ,    ' |   ,  R ? '(K) = R ? K ,

  R Q | QF -,  '(K) = Q ? (R ? '(K)) = Q ? (R ? K) = K . @ , !  r 2 R,    -  ' = rbjK ,    rK = K . 3     ,  r 2 R ,    r 2 N(R) (  R |   ) rK  NK ,    rK = K     1.4. 2

4 )  *# QF -

=    1.2,        

QF -    ,   R  . = !  

            

  

. ,     ! ,          S(R)  R,        R.  4.1.   R Q   QF -        R, S(R Q) = = R!  dimR S(R) = t. 7 % 

(R Q) = t: (4:1) &  a1 : : : at |      R S(R),  +      1 : : : t 2 Q    aij = ij ! i j 2 1 t (4:2) %  ij |  8 ,            Q = R1 + : : : + Rt: (4:3) RQ

240

. .  

4    ,      (4.3)  ,  +    a1 : : : at     R S(R)    (4.2). 2 @  1.5 (R Q) = dimR Q=NQ. 3!     (4.1)      jQ=NQj = jS(R)j:

;  R Q |   ,  S(R) = R ? N = R ? NQ. 3 3.4, : jQ=NQj = jR ? NQ=R ? Qj = jR ? NQj = jS(R)j . J  (4.1)  . 3 a1  : : : at |     R S(R), . . _ t: S(R) = Ra1+_ : : : +Ra ;   R _ s;1+Ra _ s+1 +_ : : : +Ra _ t s 2 1 t Is = Ra1+_ : : : +Ra    Q ? Is 6 Q ? Ras ,    Q ? Is  Q ? Ras ,  Is = R ? (Q ? Is )  R(Q ? Ras) = Ras ,          

  !  a1 : : : at   R.    ,      !  1  : : : t 2 Q   s 2 (Q ? Is ) n (Q ? Ras ) s 2 1 t. ;     -  ass 6= 0 ak s = 0  k 2 1 t k 6= s: 3 ! as s 2 Q ? N = R!,   as 2 R ? N . @ , as = us !,  us 2 R . ;   ,    !  s = u;s 1 s    (4.2). 2 (4.3). 3 R1 + : : : + Rt = K . ;  K |  R Q,    ,  R ? K = 0,  K = R Q,  R Q | QF -. 3 R ? K 6= 0. ;  R ? K / R         Ra. 3      S(R)  R,  a = uI1 a1 + : : : + uItat ,  uI1  : : : uIt 2 R. ;  a 6= 0,  uIs 6= I0   s 2 1 t. B    (4.2) as = uIs ass = uIs ! 6= 0, ,   , a 2= R ? K . 3  . @ ,   (4.3). 3, ,   1 : : : t 2 Q   (4.3). 3      S(R)   (4.2). 3  Ks = R1 + : : : + Rs;1 + Rs+1 + : : : + Rt s 2 1 t: 3,  S(R) ? Ks 6= 0. ;  S(R) = R ? NQ,  S(R) ? Ks = S(R) \ (R ? Ks) = (R ? NQ) \ (R ? Ks),   2.1(c) S(R) ? Ks = R ? (Ks + NQ). G S(R) ? Ks = 0, . . R ? (Ks + NQ) = 0,  Ks + NQ = Q (  R Q | QF -),   B Q = Ks ,   ,   (R Ks ) 6 t ; 1 < (R Q). = bs 2 (S(R) ? Ks ) n 0 s 2 1 t. ; bs j = 0  j 2 1 t j 6= s. < , bs s 6= 0,      (4.3) bs Q = 0 bs = 0. 3 ! bss 2 S(R)Q = R!, . . bs s = rs! 

       

241

   rs 2 R . 3 as = rs;1 bs s 2 1 t,    !  a1 : : : at 2 S(R),     (4.2). R        R S(R),    rI1a1 + : : : + rItat = 0  

-  rI1 : : : rIt 2 R,  (Ir1a1 + : : : + rItat )s = rIsas s = 0,   (4.2) rIs! = 0, . . rIs = 0  s 2 1 t. 2

   4.2. $ QF -  R Q              R       +  " : (a) R Q |     . (b) R Q  = R R. (c) R | QF -  .

2   

(a) ) (b) ) (c)   . (c) ) (a) =  2.3,    ,  R |   . = !  

(c)  ,   3.1,  S(R) |     , . . dimR S(R) = 1,   4.1 (R Q) = 1: 2

5 ,   

 

QF -

     -

=         QF -     (   2.2)    

  ,        . B  R M  ,      : M ! Q=Z,             R M . ;

   +  R M . 3       .

 5.1. :  : M ! Q=Z     +  %    

 % ,  % 

;

8  2 M  6= ) ) 9r 2 R ((r) = 6 (r))

(5:1)

2 3  |       2 M  6=  . ; (R( ; )) 6= 0, . .   !  r 2 R ,  (r( ; )) 6= 0. R  ,  (r) 6= (r). B,    (5.1)  K < R M K 6= 0. =  2 K n 0. 3      !  r 2 R ,  (r) 6= (r0_ ) = 0. @ , (K) 6= 0: 2 ,           .

 5.2. 7    R M    %      % ,  %    

QF - .

2 3 R M | QF -. ;   3.2    ,  M = R ,   1.8         R- ' : R R ! R M  = R R ,       !  r 2 R  '(r) : R ! Q=Z,    !  ! 2 R    '(r)(!) = !(r).

242

. .  

M '(e)      R ,    !  2 R ! 6= ,  !(r) 6= (r)   r 2 R. R  ,  '(e)(r!) 6= '(e)(r),

  5.1 '(e) |      R M . ;      ,      u 2 R  '(u)      R M . 3  ,       R M    . 2  ,   R       (1.1)         R M    (1.2).

 5.3. !  RM    %      % ,  %    "     Rs Ms . 2 3  |      R M . ;      Ks  Rs Ms   R M ,  (Ks ) 6= 0,      jMs     Ms       Rs Ms . B,     Rs Ms    s |     . ;    M ,       1 2 M1  : : : t 2 Mt   (1 + : : : + t) = 1 (1) + : : : + t(t ),       R M . 2  ,  K < R M K 6= 0,  es K 6= 0   s 2 1 t,   !  s 2 es K < Rs Ms ,  s (s) 6= 0. : s 2 K (s ) = s (s) 6= 0: 2 ;  ,    ,  R |   . 3, -  , N |     R R = R=N |   q = pr ! , S = S(M) = M ? N |  R M . 2  ,  R M       . 3,  R M | QF -. =  3.1,  !   ,  dimR S = 1. 3  = jS G = Ker . ;  (S +) | !   p- , (S) 6= 0,  (S) |    p  Q=Z,  , G |    p  S . ,    L(G)        R S , 

  G, . .  -   ,    G. 3 u1  : : : ur |       R     Zp .  5.4. $    % G < (S +)       L(G) =

\r

s=1

u;s 1G:

(5:2)

.   ,   dimR S > 1,  G |  %    p S ,  L(G) 6= 0.

2 ,  ,    2 L(G)       R  G. 3! 

 2 L(G),    Tsr 2 1 t      us  2 G, . .  2 u;s 1G. @ , L(G)  s=1 u;s 1 G. B,   2 u;s 1 G s 2 1 r,  u1  : : : ur  2 G,   G |    (S +),  (c1 u1 + : : : + cr ur ) 2 G   c1  : : : cr 2 Zp , . . R  G  2 L(G). R    (5.2). =   ,  G |    p  S ,  L(G) |   ,     pr = q,  dimR S > 1,  jSj > q2 L(G) 6= 0: 2

       

243

;  ,  dimR S > 1,     G = Ker         L(G),   L(G) |   R M ,   |      R M ,       . @ , dimR S = 1,   3.1 R M | QF -. 2

6

QF -

  



=       ,         

        ,              QF -   R,       ,   ,      2.2. B                    . B        . 3        F = fi1 : : : iN g  Nk0

(6:1)

    . ? M F          : F ! M . <           %   #F ] = = ((i1 ) : : : (iN )) 2 M N , ,   ,  R M F   R M N . 0 ,    k-    2 M             ! F : #PF ] = ((i1) : : : (iN )).        %  A(x) = i2Nk0 ai xi 2 Pk    F (A) = fi 2 Nk0 : ai 6= 0g: 2     ! F   (6.1)    %    %  A(x)  F   AF = (ai1  : : : aiN ) 2 RN 

     I / Pk     F  IF = fAF : A(x) 2 I F (A)  Fg < R RF : 2  R-  M  M     M#F ]   R M F   M#F ] = f#F ] :  2 Mg: @           k-PJ3            M F .

 6.1.   M   Pk -     M ,  F  Nk0 |   -

   . 7 %   %   Pk   A(x) =

X i aix

i2F

244

. .  

   M         ,   8 2 M AF #F ] = 0: (6:2) (   I = An(M) / Pk     F         IF = RF ? M#F ]: (6:3) 2 N  (6.2)       8 2 M A(x) =   (0) = 0: (6:4) 3,        8 2 M A(x) = 0: (6:5) 3  2 M A(x) =  6= 0. ; (j) 6= 0   j 2 Nk0 . ;  M  Pk -,    1 = xj     M   -  1 2 M A(x)1 = 1 1 = xj  1 (0) = (j) 6= 0      (6.4). @ ,  (6.2)  (6.5). 3        . 2   6.2.   R Q   QF -   M < Pk Q . k7 %     An(M) = 0     ,   " %    F  N0       M#F ] = QF : (6:6) 2  I = An(M)     ,    F  Nk0      IF = 0. 3 ,   (6.3), !      RF ? M#F ] = 0. ;  R Q  QF -,    3.4 M#F ] = QF ? (RF ? M#F ]) = QF ? 0 = QF :2 . 2    ,  M |     Pk - M . =  0.1 !  ,   I = An(M)    , . .     !         (6:7) Fs (xs) = xms s ; fm(ss);1xsms ;1 ; : : : ; f1(s) xs ; f0(s) 2 Pk  s 2 1 k: 3,    

 I        I   ! S = S#m1  : : : mk ] = S#m] = 0 m1 ; 1  : : :  0 mk ; 1   R M  R- M#S]. 2     H(x) 2 Pk     Res(H(x)=;;! F (x)) = Res(H(x)=F1(x1 ) : : : Fk(xk ))

       

245

P

   A(x) 2 Pk   A(x) = i2 aixi ,    H(x)  

 (F1 (x1) : : : Fk(xk )) / Pk . R              . 3 Res(xis =Fs(xs)) |       xis  Fs(xs ),   i 2 Nk0 H i(x) = ;  H(x) =

Yk

s=1

Res(xiss =Fs (xs)) =

X ij hj x :

j2

P h xi      i X i i Res(H(x)=F) = hiH (x ): i

k 2N0

(6:8)

(6:9)

,  ,   H(x) 2 I       Res(H(x)=F) 2 I . 3!  I     F1  : : : Fk    

A(x) 2 I : F (A)  S: (6:10) ;  ,   I / Pk ,        (6.7),        I . 0   ,         M    LM (F) = LM (F1(x1 ) : : : Fk (xk )).   6.3.    k-   2 LM (F)      

%  #S]    (z) =

X z hj (j)

j2

(6:11)

%  hzj |    (6.8). $   % #S] 2 M  +       k-   2 LM (F)    #S] = #S]. (      ! #S]      R-  = R M#S]: RM  2 3  = xz . ; (z) = (0). ;  xz  = H z (x),  H z (x)    (6.8). @ ,    (0)        (6.11), . .   (6.11)  . 3 #S] 2 M  . 3  k-    2 M   (z) =

X z hj (j):

(6:12)

j2 ; #S] = #S],   H i(x) = xi   i 2 S. =   2 LM (F)      (6.12) (6.8). 3      6.3     . 2

246

. .  

J     LM (I)  k-PJ3,      

 I / Pk . > -    ,   I        (6.7),       LM (I) = LM (I)#S] = f#S] :  2 LM (I)g: @       6.1.  6.4.    2 LM (F)  "    LM (I)  %      % ,  %     %  A(x)    (6.10)       A #S] = 0: (6:13) (      LM (I) = M  ? I : (6:14) 2 ;  I     (6.7)     (6.10),    2 LM (F)    2 LM (I)     ,          A(x)  (6.10). 3 A(x) =  . ; (0) = A #S], 

 2 LM (I),  (0) = 0, . .   (6.13). B,     2 LM (F)    (6.13)   A(x)   (6.10). 2  ,   = A(x)  6= 0. ; (j) 6= 0   j 2 Nk0 , k-   1 = xj  = xjA(x)    1(0) 6= 0. J    A1 (x) = Res(xj A(x)=F). 3   A1 (x)    (6.10),

  Fs(xs ) = 0  s 2 1 k,  A1 (x) = 1 1(0) = A1 #S] 6= 0,       (6.13). @ ,  = 0  2 LM (I). J  (6.14)       . 2 @             0.2   . 2   

     !   :  R Q  QF -,        I / Pk  M = LQ (I)  QF -   S = Pk =I (! |    

(a) ) (c)  0.2). 3-     ,   I        (6.7). 3 M1 |   S M , An(M1) = J . ; S ? M1 = J=I M ? (S ? M1 ) = LQ (J). ;  ,     M ? (S ? M1) = M1   ,   M2 = LQ (J)    M1 . ;   M1 M2       (6.7),      6.3   ,  M2#S] = M1#S]. @   6.4 M2#S] = Q ? J ,    6.1 J = R ? M1 #S]. ;  R Q  QF -,     3.4      M2#S] = M1#S]. 2,       I1 / S      S ? (M ? I1 ) = I1 .  I1    I1 = J1 =I ,  J1 / Pk . ; M ? I1 = LQ (J1 ) S ? (M ? I1 ) = J2=I ,  J2 = An(LQ (J1)). J  J2 = J1 !      J2 = J1 . 3     -  J2 = R ? (Q ? J1 ),     6.1 6.4. ; 0.2           QF -            S c  . =  S       R      (   , ,

       

247

   ). ; S = R#1  : : : k ] |    -   R,

   ,  S = Pk =I ,  I = fA(x) 2 Pk : A(~ ) = 0g . 3 ! !  s     s = xs + I , s 2 1 k. ;  S |    ,  I |    . @ Q = LR (I)      S -,

   0.2 S Q  QF -,   R  QF - (. . R R | QF -). 2     QF -   S = R#x]=I,  R = Z4 , I = (x2 2x). : S = R#] |       8 !    a0 + a1 ,  a0 2 0 3, a1 2 0 1. @  S QF -  Q = LR (I) = R1 + R2 ,  1 = (1 0 0 : : :) 2 = (0 2 0 0 : ::)

.  S Q  S(Q) = R!,  ! = (2 0 0 : ::). @   S Q    

    S (. .   S )    SxQ

(0) ?

% ? ? (1 ) (1 +x 2) (2) - ?? %

?y & () ( +? 2) (2) & ? y .

"

"

(!)

.

(0 2)

(0) S , ,        QF - S Q         S    S = R#x1 : : : xk ]=I

(6:15)

 R | QF -. G     (6.15),    R    ,       0.2   2.2 QF - S Q      k-PJ3     R R : Q = LR (I):

(6:16)

@   ,     QF - S Q   S    k-PJ3            ,                2.2,   ,   S          (6.15). < ,    QF -    k-PJ3   \ - " ,        S S , . .  S S  . 3   !   , 

    Q = LR (I),  R = Zd |  . = !  

d      S = Pk =I ,   ! : (S +) ! (Q=Z +)      S    < 1=d >   Q=Z. R     (Zd  +). ;  ,    S     

248

. .  

     ! : (S +) ! (R +). J   2 LR (I)         . 3   !  A 2 S    A = A(x)+ I . M b : S ! R,     ,    8A 2 S b(A) = (0)   = A(x): @   ! b     S L(I) ! S S  . =   ,  QF - S Q     (6.16),    (i)    2 Q   R,  b 2 S  ,     ,    : 8A 2 S b(A) = (0)(e)   = A(x) (6:17) . . b(A)      (0) 2 R   e. 3 ,    b : S ! Q=Z          (S +),        .   6.5. (      ! b ,     (6.17),      S -  S Q ! S S  . 2 ;  jQj = jS j = jS  j ,    ,     1  2 2 Q   ,  b1 6= b2 . 3   1 (j) 6= 2 (j)   j 2 Nk0 . 3 s = xj s  s 2 1 2. ; 1(0) 6= 2(0),   , 1(0)(r) 6= 2(0)(r)     r 2 R, . .     r1(0) r2(0)   e   . R  ,   A(x) = rxj      b1(A) 6= b2 (A), . . b1 6= b2 : 2

7 /    QF -

3       QF -       .  7.1. $   R M     +  " : (a) R M   QF - . (b)   % N 2 N     L < R RN  K < R M N        M N ? (RN ? K) = K (7:1) N N R ? (M ? L) = L: (7:2) (c) R M |    ,    % N 2 N   %  %   K < R M N    (7.1). 2     (a) ) (b)     3.4.     (b) ) (c)   . (c) ) (a). 3   2.3    ,  R |   . = !  

 R M |  QF -, ,    3.1(b,c),   M ? N = S(R M)       K . ; R ? K = N ,  M ? (R ? K) = S(M) 6= K ,       (c). 2

       

249

<        ,    HmN   R        K < R M N ,  K     -   M N         Hx# = 0. 3 H |      R,      H , H < R RN . ;,   , H |       K      , 

K = M N ? H:

(7:3)

=         ,       ,    

  7.2.   RM |    . 7 %      

RM

      R         ,   R M   QF - .

2 G R M  QF - K < R M N ,    (7.1)   H  R,     H = RN ? K ,        K . G   K < R M N       R, . .    (7.3)    H < R RN ,    (7.1),   7.1(a,c) R M  QF -. 2 2     L < R RN   R          !  ,  ,       QF -. 0    -     R M     ,  a = a   a 2 R  2 M . @,    KmN   R M      L < R RN ,  L     -   RN         Ky# = 0# . ,  , K |      L      ,      K < R M N ,    R  K ,    RN ? K = L (7:4)

     L  R M        ,    (7.4)   K = M N ? L . = !     K      

 R K .  7.1     7.3.     L < RRN       QF -  R Q. 2 0 ,     !    . B  ,  R M = = R Q + R Q   QF -,         R       M . 0          -            R. 3 R R(N ) |     N   R. B   KmN   R M        HN n   R,     b# 2 R(N )         Hx# = b# (7:5)

250

. .  

-   R      ,  Kb# = 0# . M-  ,     H        K  !  ,    !    d    -       (7.5)   -  d = n ; rang H = rang K + n ; N . R

       .   7.4.   HN n     R      KmN  QF -  R Q.      H |    R,   "     H ,  K |    Q,   " 

   K ,    D(H) )        (7.5)   D(H) = jHj;1jRjn = jKj jRjn;N : (7:6) 2 @  7.3,     KmN  Q ,    b# 2 R(N ) b# 2 H , Kb# = 0# : :  K | ,     b# 2 H         (7.5). 3    (7.6)   . ;  K    Q,    H ,      3.4      jKj jHj = jRjN . ,      (7.6). 2 =         -             #5],      . 3 jRj = r R = f 1  : : : r g . ; +     L < R RN     Z X s1(u) sr(u) WL (x1 : : : xr ) = x1 : : :xr  (7:7)

u2L

 st (u) |      u ,   t . 9   ,  jM j = m

M = f1  : : : m g ,           K < R M N : X 1 ( ~ ) m ( ~ ) (7:8) WK (y1  : : : ym ) = y1 : : :ym  ~ 2K

 t(  ~ ) |      ~ ,   t . @,    

 L < R RN K = M N ? L   "    !-. , 

    : (M +) ! (C  )(    (M +)         (C  )  C     ) ,  (7:9) W (y  : : : y ) = 1 W ( M (y) : : : M (y)) K



1

m

M  (y) =

jLj

m X =1

L

1

r

(   )y   2 1 r:

(7:10)

=  M = R = GF (q) !  -   #5]. =  M = R = Z4      #15]. 2          #2].

       

251

3       ,      : M ! C (     )     x1 

! : M ! Q=Z (    ):      

 !(x)    (x) = exp(2i!(x))     ,         ! !                M 

             (M +). 3!,          R M  +  

,  (K) 6= 1      K < R M ,     5.2   

  7.5. 7    R M   +      %      % ,  %      . 2

3  ,       QF -  

   ,     .

 7.6. &  R M | QF -    | %  +    -

,    %  %   L < RN     "   !. (7.9). &  R M |  ,       ,  +      L < R RN ,     (7.9)     

  %   % (M +).

2 2          ,         #5]. @        .

 7.7. &   |  +    P RM ,    %  %    K < R M       2K () = 0. 2 3        (K) = G |        C . 3! X j X g = 0: 2 () = jjK G j g 2G 2K =     K = M N ? L        WK (y1  : : : ym ) =

X

~ 2K

m ( ~ ) : f(  ~ )  f(  ~ ) = y11 ( ~ ) : : :ym

2    u 2 RN   fb(u) =

X

~ 2M N

(u  ~ )f(  ~ ):

(7:11)

(7:12)

252

. .  

 7.8. &   |  +    RM ,     Xb X f(u) = jLj f(  ~ ): (7:13) u2L ~ 2K

 

2 3 (7.12), 

Xb

u2L

f(u) =

X X u2L

~ 2M N

!

(u  ~ ) f(  ~ ):

(7:14)

G  ~ 2 K ,  (u  ~ ) = 1   u 2 L ,       

  (7.14)   jLj . G   ~ 2 M N n K ,    K = fu  ~ : u 2 Lg      R M ,   7.7

X

u2L

jLj (u  ~)= jK j

X

2K

() = 0:

;  (7.13)   (7.14). 2 2    7.6 -   .  (7.11),   7.8,    Xb W (y  : : : y ) = 1 f(u): (7:15) K

1

m

jLj u2L

 (7.12),        

t(  ~ ),    -  fb(u) =

X

N  Y

1 ::: N 2M l=1



(ul l )y11 (l ) : : :ymm (l) 

 t(l ) =P1, 

Q l = t , t(l ) = 0,  l 6= t . ,,    , 

b u) = f(

N X Y

l=1 l 2M

(ul l )y11 (l ) : : :ymm (l )

! Y N X m =

l=1 =1

(ul  )y :

3     

s (u), 

    - 

b u) = f( . .     (7.10)

m Yr X

=1 =1

( l  )y

!s(u)



b u) = M1 (y)s1(u) : : : Mr (y)sr (u): f(

!

       

253

;  ,  fb(u)   (7.15)    xs11(u) : : :xsrr (u)   (7.7)     : x = M  (y),  2 1 r. ;  (7.9)   (7.15) (7.7). 2     7.6      . 3  N = 1, L = R. ; WL (x1  : : : xr ) = x1 + : : : + xr . >  ,  M = f1 : : : m g ,  1 = 0. ; K = M ? L |    R M ,  WK (y1  : : : ym ) = y1 . 3    ,  R M |  ,      . 3  |     R M . ;    7.5      K < R M ,  (K) = 1. @ ,   M = f 2 M : (R) = 1g    d > 2 ! . 3 M = f1 : : : d g . ;        (7.9)    1 W ( M (y) : : : M (y)) = 1 ( M (y) + : : : + M (y)) = r r jLj L 1 jRj 1

0

1

r X m m X X X = jR1 j (   )y = jR1 j @ (  )A y : =1 =1 =1 2R

3 !   2 M ,  (R ) = 1,

X

2R

(  ) = jRjH

  2= M ,  (R ) = G |        C ,

X jX (  ) = jjR Gj g2G g = 0: 2R @ , M WL ( M 1 (y) : : : r (y)) = y1 + : : : + yd 6= y1 = WK (y)

. .  (7.9)  . 2

/ 

#1] 9 :., :  . =      . | M.: : , 1972. | 160 . #2] Ericson Th., Zinoviev W. Spherical codes. Manuscript. To appear. #3] < 9. @. B 9. 9. P       <      L    2 // N  . . | 1994. | ;. 49. | N 5. | @. 165{166. #4] P . <  . | :.: : , 1971. | 279 .

254

. .  

#5] :-=  _. 2., @! B. 2. 9. ;  ,  

-  . | :.: @, 1979. | 744 . #6] B 9. 9. P             // 2 . . | 1991. | ;. 3. | N 4. | @. 107{121. #7] B 9. 9. P             // N  . . | 1993. | ;. 48. | N 3. | @. 197{198. #8] _ <. 9: ,  , 

. ;. 2. | :.: : , 1979. | 464 . #9] Azumaya G. A duality theory for injective modules (Theory of quasi-Frobenius modules) // Amer. J. Math. | 1959. | V. 81. | N 1. | P. 249{278. #10] Kuzmin A. S., Kurakin V. L., Mikhalev A. V., Nechaev A. A. Linear recurrences over rings and modules // J. of Math. Science / Contemporary Math. and it's Appl. Thematic survays. | 1994. | Vol. 10. #11] Kuzmin A. S., Nechaev A. A. Error correcting codes on the base of linear recurring sequences over Galois rings // IV-th Int. Workshop. Algebraic and Combinatorial Coding Theory. Proceedings. Novgorod, 1994. | 132{135. #12] Morita K. Duality for modules and its applications to the theory of rings with minimum condition // Sci. Repts Tokyo Kyoiku Daigaku. | 1958. | A6, N 15 May. | P. 83{142. #13] Nechaev A. A. Linear codes over „nite rings and QF -modules // IV-th Int. Workshop. Algebraic and Combinatorial Coding Theory. Proceedings. Novgorod, 1994. | 154{157. #14] Wisbauer R. Grundlagen der Modul- und Ringtheorie. | Munchen: Verl. Reinhard Fischer, 1988. | VI, 596 s. #15] Hammous A. R., Kumar P. V., Calderbrank A. R., Sloane N. J. A., Sole P. The Z4 -linearity of Kerdock, Preparata, Goethals and related codes. Manuscript. 1993. &"     ':   1995.

T-           

. . 

        . . . 

        T-             R           F     2  ,          ,    e,   eRe = eF .

Abstract A. E. Pentus, T-ideal of generalized identities for a class of primitive algebras with involution, Fundamentalnaya i prikladnaya matematika 1(1995), 255{262.

The T-ideal of generalized polynomial identities of any primitive algebra R with involution is found, provided R is an algebra over an algebraically closed *eld F of characteristic di+erent from 2, the ring R contains no primitive symmetric idempotents, and there exists an idempotent e such that eRe = eF .

          T-                 ,   .   7], #. $. %  1]   . '(  8]. *  ,          ,  +    ,   +    ,    ,    ( ,  ,  +  4]. . , F |  , (R ) |       F     , X = fx1 x2 : : :g, X = fx1 x2  : : :g, Y = X  X |     +     . .  + , RF hY i,  

  (       R  F hY i |        F,       Y . 3    + , RF hY i |       |   , (  

X ri0 yj1 ri1 : : : yjm rim i  i 2 F , rik 2 R  yjk 2 Y . 4     ,   + , R    + , RF hY i       (: r 7! r   r 2 R, xi 7! xi , xi 7! xi, i = 1 2 : : :. *    f(x1  x2 : : : xk  xl1  xl2  : : : xlm ) 2 RF hY i        (   ) + , R,   f(a) = 0      a (    A = f(r1 r2 : : : rn r1  r2  : : : rn) j ri 2 Rg,  n = maxfk mg.  +   ,,               + , R (   + , RF hY i, +     ( ,   ( G (R).            1995, 1, N 1, 255{262. c 1995      !, "#    \%     "

256

. . 

3:( ; + , RF hY i       ( ,    ,

  ;(r) = r    <    r 2 R  RF hY i  ;(f ) = ;(f)]     f 2 RF hY i. $  I + , RF hY i (  T -  ,   ;(I)  I      <:( ; + , RF hY i. . , f(Y ) |     ( + , RF hY i. . T(R= f(Y ))    ,        T -    + , RF hY i,      . $  , T (R= f(Y )) |  , T -   ,      f(Y ). >        ,    . $    e  e0 + , R (    ,,   ee0 = e0 e = 0. $   e

  ( ),       ,              ,    . ?+ ,               (s)  + , Mn (C) (n = 2m) n  n {   +   + , C. '(,  +  n  n {      m  m {       (. A    (  :        A B s = u;1 A B t u = Dt ;B t  C D C D ; C t At   0 ;I  u = I 0   I |  m  m {  . .   R |                F !       2, |     R.      R   ! 

   !    ,    %  e = e2 ,  ,  eRe = eF . &  G (R R) = T (R= S2 (e= Y )) + T(R= Invs (e= Y ))  S2 = ex1 ex2e ; ex2 ex1 e, Invs = e x1e + e x1e.    . $(           2.13.19 9,  . 302]   ,   x x = 0     <    x 2 Re     <    r ( R e r e + e re = 0, <  Invs          + , R. $(     + ,    S2 |      , <  T(R= S2 (e= Y ))+T (R= Invs (e= Y ))  G (R R). .  + +( ,    +  . %    ,  ,    L = Re ++   + 

   U    D = EndR L = eRe = eF . 4    +( ,     , +    U  +      D, + ,+ +( ,    +     F,        ( ,. ?   + , R <    b  b0, + ,   e = eb0 e be 9,   2.13.18,  . 301],  (     U      ,   -   :,    hr1 e r2ei = eb0 e r1 r2e. H:+   +        + , R h(Yh ). * ,         +   +<::   (   F  fi 

T-     

257

a1yk1 : : :am ykm am+1 ,  yki 2 Yh   i = 1 : : :m, aj 2 R   i = 1 : : :m + 1. . , Vh | F-      R,    <    1, e, e , eb0 e , e be,  +   +<::   ai     h  <    ai . I   1.4.2 5,  . 46{ 47],   <   v +  + , R,     v = v2 = v  vxv = x    <    x    Vh \ Soc(R). . <  + , vRv    + , (t  t)-     F    +  t > 0,   <   v |    . 3   e   ,  +   ,  ,     :( + , vRv,    , ,   <  <            e11 . $   + , R   (     + ,

 . 4    , +  :( F  + ,      ,       + +  +  +,      e  -

      ,    e11. A+ (, vRv | + ,   (  t  t,  t = 2m   +  +   . . , E = feij j 1 6 i j 6 tgP|      F  + ,  , +,   e = e11  v = tk=1 ekk . J,   + Pt ++ vxv = x    <    x    Vh \ Soc(R),  Vh \ Soc(R)  ij =1 eij F. A  ,     <    + , R S = f1=s1  s2 : : : sq g, +,   1) <      F- (       Vh \ Soc(R)= 2) +       ,  +  = P ( <    3) Vh = qk=1 sk F + V \ Soc(R). h P P A Vh  qk=1 sk F + tij =1 eij F, <      h(Yh ) P   ,     ++   + pi=1 fi i,  i 2 F,  fi  i = 1 : : : p |     +<::   (    S  E. *(   ( Mon       fi    h. '   (    Mon ++ <          1,       S  E  Yh . .  fi 2 Mon (  E-,       <     ,       S  Yh . *(      E-   ( Mon   ( G0. %    , 1 ++ E-     ( Mon. . , f | ( ,  (    Mon. * ,  

 (   :. I. f(Yh ) = g,  g | E-  (    G0. *(   ( GI    + E- . II. f(Yh ) = g0 eij g1 ,  g0, g1 | E-  (    G0. *(   ( GbII       g0 ( , ),    ( GeII |       g1 (+  )   II. III. f(Yh ) = g0ei0 j1 g1ei1 j2 g2 : : :ein;1 jn gn ein jn+1 gn+1. *(   ( GbIII       g0 ( , ),   ( GeIII |       gn+1 (+  ),    ( GmIII |         E-  g1 g2 : : : gn   III. F    +  E-     GI  GbII  GeII  b e GIII  Gm III  GIII  +  , +         (    Yh . .   +  +  , 

258

. . 

    , +      , +   (   <         . *(   ,  

(   +   +(      <         (    GI , GbII , GeII , GbIII , GmIII , GeIII ,        ( GfI , GbfII , GefII , GbfIII , GmfIII , GefIII .       : ,    : L0c = f g] j g 2 GI g  f i g] j g 2 GeII  GeIII  1 6 i 6 tg f g j] j g 2 GbII  GbIII  1 6 j 6 tg 0 Ls = f i g] j g 2 GfI  GefII  GefIII  1 6 i 6 tg  f i g] j g  hN sf 

s 2 S h 2 GbfII  GbfIII  GmfIII  f 2 GfI  GefII  GefIII  1 6 i 6 tg 00 Lc = f i g j] j g 2 GmIII  1 6 i j 6 tg L00s = f i g j] j g 2 GbfII  GbfIII  GmfIII  1 6 i j 6 tg  f i g j] j g  Nhsf  s 2 S h 2 GbfII  GbfIII  GmfIII  f 2 GbfII  GbfIII  GmfIII  1 6 i j 6 tg L0 = L0c  L0s L00 = L00c  L00s  L = L0  L00: H ,  

 gN       g = sk0 yl1 sk1 : : :yln skn   (: gN = skn yln : : :sk1 yl1 sk0 . H:+  ( , +     L00. I  +      i g j] ( L00  <    e1i gej m+1 + , RF hY i.    (   ; m   m + 1 6 i 6 2m i = ii + m   1 6 i 6 m   1   m + 1 6 i 6 2m "(i) = ; 1   1 6 i 6 m: .   (i j) = "(i)"(j).  +   ,,   (e1i gej m+1 ) = (j m + 1)(1 i)  e1j  g ei m+1 . H  + ,   g = gN"(g), 

8 < 1    g      +  + <    "(g) = : s 2 S ;1  : . <    e1j  gNei m+1   + j  gN i ]. >   +

     L00,     i g j]   j  gN i ]. H+   <         (, <      , +  ( (+.   (    L00    L000   (:   F   L000 

+, +(F   ( = ( +  ,  <    (  , (,   ,F <   (      F +     L00)=  (,  +  + i  g i],  g = ;g. . , F L]  F L0] | + ,    : ,     (    L  L0 = L0  L000     . *    +   (    Mon   ( F L]  F L0].   I    Pf (L) = Pf0(L0) = f] = g],    II    Pf (L) = Pf0(L0 ) = g0 i] j g1],  Pf (L) = g0 i0 ] j1 g1 i1 ] : : : jn  gn in ] 

T-     

259

 jn+1gn+1 ] 2

F L]. >   ,   Pf (L) 

 +  j  g j],  g = g ,    Pf0(L0 ) = 0.         Pf0(L0)   (: Pf0(L0 ) = g0 i0 ] j1 g1 i1 ]0 : : : jn gn in]0 jn+1gn+1], 

 jk  gk  ik ] 2 L0 0 jk  gk  ik ] = ;"(g )(i  m + 1)(1 j ) ijk gNgk jik ]]    . k k k k k k Pp P   h = i=1 fi i    Ph0(L0) = pi=1 Pf0i (L0)i . ? +( (+( ,      4]),     Ph0(L0) = 0,  h(Yh ) 2 T (R= S2 (e= Y )) + T(R= Invs (e= Y )), <    ,   (+ , +( , ,   +( ,,   0 (Ph0(L0 )) = 0     :( 0 : F L0] ! F F -   2,  . 143]. H:+  ( , + :( 0 .           <    i g j] 2 L00 n L000       (: 8 0 i g j]

  i g j] 2 L000 > > < "(g)(i j)0 j  gN i ]   i g j] 2 L00 n L000  i g j] = >  i g j] 

  L000 : 0

  i = j  g = g : . ,  , Ind |    (     +     ( Yh  n = jIndj.  ,         <   : 1) a 2 R= 2) a1 : : :an  n = jIndj,  ai 2 R= 3) <   z10 2 Re  <    zgi 2 Re,  1 6 i 6 t, g      GbfII  GbfIII  GmfIII  zg0 2 Re,   + g      GfI  GefII  GefIII =  1){3)           . 3    ( 3)    ( ,  z-<   ,      (g i),   +     z-<   |     +   z-<   . ZR1 :   sg 2 GbII  GbIII  1 6 i 6 t,  aszgi = 0 sg i]e11 ZR2 :   sg 2 GI ,  aszg0 = 0 sg]e11 ZR3 : aei1 = 0 1 i]e11   i = 1 : : : t ZR4 :   xr sg 2 GfI  GefII  GefIII   xr sg 2 GfI  GefII  GefIII ,  ar szg0 =zxr sg0 ar szg0 =zxr sg0

  xr sg 2 GbfII  GbfIII  GmfIII   xr sg 2 GbfII  GbfIII  GmfIII , ar szgi =zxr sgi   i = 1 : : : t ar szgi =zxr sgi   i = 1 : : : t ZR5 : ar ei1 = zxr i   i = 1 : : : t ar ei1 = zxr i   i = 1 : : : t ZR6 :   g 2 GmIII ,  e1j zgi =  j g i]e11   j = 1 : : : t, i = 1 : : : t ZR7 :   g 2 GeII  GeIII ,  e1j zg0 = 0 j g]e11.

260

. . 

(  ZR1{ZR7 s 2 S, xr  xr |     (    Yh , g | E- , ei1, i = 1 : : :t |      E. $ ,(   ,   ,    ,,     +   fi 2 Mon    afi (a1 : : : an)z10 = 0 (Pf0i (L0))e11 . .<  ah(a1  : : : an)z10 = 0 (Ph0(L0 ))e11 , , + ,+ h(Yh )         , 

 0 = 0 (Ph0(L0 ))e11  <  0 (Ph0(L0 )) = 0. A+ (,    :( 0 : F L0] ! F   ,   0 (Ph0 (L0)) = 0,     +(.      <    ai  i = 1 : : : n,     ZR4  ZR5,      ,   O 6, P     3], <  (+   ,   + z-<   .     (      (           + z-<   : Z1 = f(ysg i) j y |     (    Yh  s 2 S  g | E- g, Z2 = f(y i) j y |     (    Yh g. ? ,   LIz :   <     szgi ,  (ysg i) 2 Z1    +  y 2 Yh   sg 2 GbII  GbIII  1 6 i 6 t, <     szg0 ,  sg 2 GI , <      + ( Z2  <    fej 1 j 1 6 j 6 tg   eF-    (= CH1 : hzxr i ej 1i = hei1  zxr j i      zxr i ,  (xr  i) 2 Z2 , zxr j ,  (xr  i) 2 Z2 = CH2A :hzxr sgi ej 1i = hszgi  zxr j i      zxr sgi,  (xr sg i) 2 Z1, zxr j , 

(xr  j) 2 Z2 = CH2B :hzxr i szgj i = hei1  zxr sgj i      zxr sgj ,  (xr sg j) 2 Z1 , zxr i, 

(xr  i) 2 Z2 = CH3 : hzxr sgi pzhj i = hszgi  zxr phj i      zxr sgi,  (xr sg i) 2 Z1 , zxr phj ,  (xr ph j) 2 Z1 .  ,       <    wgi (    Re,  (g i) |   +   z-<    (   ( ,  w-<   ), + ,   LIw :   <     swgi ,  (ysg i) 2 Z1    +  y 2 Yh   sg 2 GbII  GbIII  1 6 i 6 t, <     swg0 ,  sg 2 GI , w-<      + P ( PZ2   eF-    (       U0 = s2S tj =1 sej 1 F  U= WR1 : hwgi  sek1i = hswgi  ek1i = hsek1  wgii = hek1  swgii = 0    w<    wgi ,  s 2 S      ek1, 1 6 k 6 t (   E= WR2A :   s 2 S  s = s ,  hwgi swgi i = hswgi  wgii = 0    w-<    wgi = WR2B : hwgi  sw Phj i P= ;hswhj  wgii = "(s)hswgi  whj i = ;"(s)hwhj  swgii = = ; tk=1 tl=1 hek1 sel1 i k g i] l h j] + e11"(g)"(i) i  gNsh j]     w-<   ,  + i  j +      ,   s 2 S= WR3 : hwg0 P swhj iP= ;hswhj  wg0i = "(s)hswg0  whj i = ;"(s)hwhj  swg0i = = ; tk=1 tl=1 hek1 sel1 i k g] l h j] ; e11"(hs)"(j) j  Nhsg]     w-<   ,  + j  ( +       ,   |    ,   s 2 S=

T-     

261

WR4 : hwg0Pswh0 iP= ;hswh0  wg0i = "(s)hswg0  wh0i = ;"(s)hwh0  swg0 i = = ; tk=1 tl=1 hek1  sel1 i k g] l h]     w-<   ,  + i  j +    . >  w-<       ,    z-<      (: P zg0 = wg0 + tj =1 ej 1 0 j g] P zgi = wgi + tj =1 ej 1  j g i] i = 1 : : : t:

. +    z-<    (   ,   ZR6  ZR7    +   . ?      +     ,,   (  LIw  WR1 {WR4  +   LIz , CH1 {CH3 . .  +    w-<   . P ++-,     +   w-<     (     +.      w1,      LIw , WR1  WR2 (WR4),  ,(   , : + +      +  . . , w-<      i < P k 

P   . J,   wk     ,   1) wk 2 ( tj =1 s2S sej 1 F)? = 2)     sm 2 S hwk  sm wk i    ,  +  +  ( eF,      WR2A , WR2B   WR4, +   ( mkk = 3)     <    wj , j < k  sm 2 S hwj  sm wk i    ,  +  +  ( eF,      WR2B   WR3 , +   ( mjk = 4)  <      fswj js 2 S 1 6 i 6 kg    , eF -    (    P P    U0 = s2S tj =1 sej 1 F . 4   <   wk ++  <    wk0  wk0 ,  wk0  wk0 2 Re  0 wk    ,+   1)  3).      <    wk0   ,(         + 9, A   2.1.6,  . 151],    +     e R. .   , 

P R-  P U = Re, X = eRe = eF , U0 = H = tj =1 s2S sej 1 F + T w1 + + T wk;1 + Twk0 , T = s1 F + sq F,   <   wk0 2 Re, +,   wk0 2 H ? , s1 wk0  : : : sq wk0 eF- (    U0  hwk0  sm wk0 i = mkk ; hwk0  sm wk0 i   m = 1 : : : q.  +   ,,   <   wk = wk0 + wk0       1) { 4). .+ ,+         CH1 {CH3  LIz ,     ,   O    , <    a1 : : : an.      <    a 2 R,      ZR1{ZR3  ,        . A+ (,       <   ,   ( F +( ,   .

 .

  X |        !        '   h i  U   .   H U0 |             U .   t1  : : : tn |   X-     ! !    h i  !         U  ,  !,     T = t1 X + + tn X    ! %       ti = ;ti  i = 1 : : : n0, ti = ti  i = n0 +1 : : : n,  1 6 n0 6 n.   1  : : :n | %     X,  ,  n0 +1 = 0 : : : n = 0.

262

. . 

&    %  v 2 H ?,  ,  (1) t1 v : : :tn v X-              (2) hv ti vi = i  i = 1 : : : n.

U0 )



1]  . . 14-     .   . | 1977. | $% . | &. 2. | (.8. 2] * (. + . | ,.: , , 1965. 3] 234 +. 5.   6 373% 8 (n n)-;3   ; <;  % // ><6 ;3;. 4 | 1992. | . 47. | N 2. | (. 187{188. 4] 234 +. 5. (3  T-  6 373% < ;3%6   %  <; // ><6 ;3;. 4 | 1992. | . 47. | N 6. | (. 227{228. 5] Beidar K. I., Mikhalev A. V., Salavova C. Generalized Identities and Semiprime Rings with Involution // Mathematische Zeitschrift | 1981. | B.178. | S. 37{62. 6] Chuang Chen-Lian. *-DiEerential Identities of Prime Rings with Involution // Trans. Amer. Math. Soc. | 1989. | V.316, N 1. | P. 251{279. 7] Littlewood D. E.| Identical relations satisHed in an algebra // Proc. London Math. Soc. | 1931. | V.32, N 2. | P. 312{320. 8] Rosen J. D. | -Generalized polynomial identities of Hnite-dimensional central simple algebras // Israel J. Math. | 1983. | V.46, N 1{2. | P. 97{101. 9] Rowen L. H. Ring Theory Vol. 1. | Academic Press, New York, 1988. '    (: *  1994.

        Q-       

. . 

       

   Q-        |     X     Q-  

 ,      ' : X ! S      S ,  ' OX = OS , (X=S ) = 1  ;KX '- . & '( )    *.    , . .  '  F   ( (  j;KX + ' hj. , ,    S   - )  

,    X=S       ,  

 S |    An . &        .       

 X  S .

Abstract Yu. G. Prokhorov, On general elephant problem for three-dimensional Q-Fano ber spaces over a surface, Fundamentalnaya i prikladnaya matematika 1(1995), 263{280.

We consider Q-Fano 7ber spaces X=S over a surface, i. e., a three-dimensional variety X with terminal Q-factorial singularities and a projective morphism ' : X ! S onto a normal surface S such that ' OX = OS , (X=S ) = 1 and ;KX '-ample. In this situation we discuss Reid's conjecture on general elephants, i. e. on general members of the linear system j;KX + ' hj. We prove that the surface S has only cyclic quotient singularities, besides if for X=S the elephants conjecture is true, then singularities of S are Du Val singularities of the type An . In the last case some conditions on singularities of X and S are obtained.



                                     .          5], 8], 14]. $ %&     & 

%    

 '     :  1 ( , Reid.)  ' : X ! S |                      X     Q-        ,  ' OX = OS ;KX '-    (. . ' : X ! S           ). " # # #      h 2 Pic(S) $  #           j ; KX + ' hj 

  % #& 

 (. .   #  )     .       ( . (        ( (  93{11{1539)  *.       (  M-90 000)            1995, 1, N 1, 263{280. c 1995       , !"    \$     "

264

. . 

/  %, &     1       0  1     ( . 12], 10]      ' : X ! S   0

').   &     ,  0       0   0     %  = ;1 '   3  "Q-5  

 %", . .  0   % "1"  

   0%&  60           1     . 7   Q-'     S        0        X     Q-30   

 , %&  0  3 ' : X ! S    S     6 2 0,  ' OX = OS ,   ;KX      '-  (X=S) = 1 (. . ' : X ! S |    60     dimS < dimX). 9 Q-5  

    

   

     1 ( . 1]  dimS = 0):  2.  ' : X ! S | Q-'   . " # $    Q-'   '0 : X 0 ! S #   (   S , #            1  ,     #  X ; ; ! X0 #' # '0 S = S

# X ; ; ! X 0 |     (  .   &    

   Q-5  

      . :  ,  % 0 

  '   60        

 %  0 0 16]. ; 0  0   '     

     % 0  Q-5  

 .  ,     %&  3 (  .)  ' : X ! S | Q-'   # #      S . " # S 

  % #& 

     . 7  0  ( .    2.4),   % Q-5  

  ' : X ! S dimS = 2,  S     1     30 

 . 9 ,    $0   17], 15], 6], x3    ' 0 0 "6   " '       X=S ; ; ! X1 =S1 ; ; !    Xn =Sn 0,   Xn =Sn     1. 0 ',        (x4) 

 %  

  Q-5      ,  0     1. 90 ,        1 %  0  

   An . 7 ,  s 2 S |   0  An , n > 2,    ';1 (s)        0   0  > 1 | 6 '0 0 30 

    0  n + 1. >     

  Q-5  0, 00 (ii), (iii).

           Q-   

265

1         .

 %            C.       . 9 % Q- 1      0   (   0  

 ) S 0 0        I(S) := minf m 2 NjmKS {   / g

(1)

     0     0 S. >  (S s) |  0          

 . :   ( .,  , 5], 6.11)  (S s)      30    0 (S 0  s0 )  (C2  0)  0    G  GL(2 C),

     %&  0    1. 0 (S 0  s0) ! (S s)                       

 ,  0  G  GL(2 C2) |        #   (S s) (    Itop (S s)).  6    0  I(S s)   0  det : G ! C . ; 

  (S s)     %  0   0 , 0 G  SL(2 C. *          &        

  s 2 S,   0 3  Cn =G,  G = Zn | '0 0 ,       %&  Cn .   0

0'%     1      '0 0 30 

   ( .,  , 2]). / 0      

    0 3  30 C2 =Zn ,      Zn : C2   '   1 exp( 2iq 0 n )  (q1 n) = (q2 n) = 1: 2 0 exp( 2iq n ) >  q10  q20  q { 0 '   ,  0 6 qi < n 0 6 q < n qiqi0  1 modn  q  q1 q20 modn. @   0   

  S = C2 =Zn   %         (n q),  6  

         & P   Anq . >  Se ! S |     1   

  S  E = Ei  Se |  0%    . 9  

 3 ;     "  ":

i i i

:::

i i

c1 c2 c3 cr;1 cr    ci ( ci = ;Ei2 )   %     nq '  % : n =c ; q 1 c2 ;

1

: 1 c3 ; : : : ; c1 r

!  1.1. (i) +       #& 

       An 

   An+1n. (ii) , q0 |      ,  0 6 q0 < n q0  q10 q2 modn,  0 qq  1 modn  (  qn  & #  #      cr  : : : c2  c1. 0

. . 

266

 "#  "     .  ,  %  0  

  0       %&   C3 :  

  (S s)  C3 An An Dn  n > 4 E6 E7 E8

  uv + yn+1 ,    z 2 + x2 + yn+1 z 2 + x(y2 + xn;2) z 2 + x3 + y4 z 2 + x(y3 + x2) z 2 + x3 + y5

(2)

$   /  3],     0

30'%  %' (. .  3 0 2) : S ! S,    %& 0    %  0  

  (S s).  0    0   

  (S s)        (2) 0 ,        ,  



 . 7 , 30 

     %  %  0:  

   %' (S s) An  Dn  En E6 E6 Dn Dn A2k+1 A2k+1 An A2k A2k+1

30 (S s)=

(x y z) ! (x y ;z) (x y z) ! (x ;y z) (x y z) ! (x ;y ;z) (x y z) ! (x ;y z) (x y z) ! (x ;y ;z) (x y z) ! (x ;y z) (x y z) ! (x ;y ;z) (x y z) ! (;x y ;z) (u v y) ! (;u v ;y) (u v y) ! (;u ;v ;y)

  A2 E7 A1 D2n;2 Ak Dk+3 A2n+1 A2k+1k A4k+42k+1

(3)

  # $  %    &. 7'    0  3   : S 0 ! S Q- 1     

       ,  KS = KS . >  : S 0 ! S | 0    3       %  0  

   0 : Se0 ! S 0 |     1   

  . @ 0'

 0 : Se0 ! S         1   S. B0 , 0    0 s 2 S 3     '   3 ;,  : 0

y y i y y

C  3 ; |   

 3   1   0 : Se0 ! S, 3 ;0,

 &     1 | 6   

 3  0 : Se0 ! S 0 , a   1     %     & 3 0 .  1 

           Q-   

267

      S    

   A5 s, 3    



% ('  %) 0 %      S 0    ;1 (s)     0  A2 . !  1.2. +   ,     % ,  # # ,  

(s0 ) = s,  Itop (S s) > Itop (S 0  s0). - 

    #      ,  # : (S 0  s0) ! (S s) |    . D   

  0 0 '                

 . $ %&      

    . ' 1.3.  : S 0 ! S |                  %         .  #  ( ,  (S 0 =S) = 1.  #

C 0  S 0 | (  # ) &     #    s = (C 0 ) |     Anq . .     & #  e Se Se0 ;! 0 # #   0 S ;! S #  : Se ! S 0 : Se0 ! S 0 |     %        S S 0 ,     . " #  (  %  #&$  : (i) e |    , . .  0 : Se0 ! S |     %       S ,    C 0  KS > 00 (ii) e | #   s0  Se,    C 0  KS < 0 (. . : S 0 ! S      #      ). D3 : S 0 ! S  (i)             %  . @0 3  0%,   ,        '    1

 3 ,      % 0    3 . ( ) #   $   &.      0

30'          

  . ;     , 

    , | 6   0 ( . (1)). * 1.4 (,13], ,14]) (i) "          #  1 |         cDV -     , . .    

          $          & #& 

     . (ii) 1(#           (X x) #  I = = I(X x) > 1             (X 0  x0) #  1         # I , #  &$   X 0   #  x0 (              ). (iii) , (X x) |       ,  $  #                 j ; KX j 

 x #& 

&    . 0

0

. . 

268

!  1.5. (i) 211]  (X x) |         cDV -     . " # X nfxg #   . 4&#  # ,  # &          (X x) #   1 (X nfxg) |      # I(X x). (ii)  (X x) |            F 2 j ; KX j | #    &$  x #& 

&    . " #      (X 0  x0) ! (X x) #    (F 0 x0) ! (F x),   x F 0 2 j ; KX j. 4&#   ,  Itop (F x) = I(X x)  Itop (F 0 x0). - 

  , Itop (F x) = I(X x)  #    #,  # (X 0  x0)    , . . (X x) |          . 0

2      Q-          

/ % 2.1 (,4])  ' : X ! S |   Q-'  #      S .  #  ( ,  X 

  %     #  1. " # S |       ' : X ! S | (  (   )       . ' 2.2.  ' : X ! S |   Q-'  #     . " # (#   ';1(s), s 2 S    # 

    . 0   $ . >  Z |        ';1 (s)  IZ | 0  -

 Z. >  ' 0          0 ! IZ ! OX ! OZ ! 0   ,  R1' (OX ) (0 00 ;KX '-   8], 1-2-5). > H 1 (Z OZ ) = (R1' (OZ ))s = 0, . . pa (Z) = 0. >  ' : X ! S | 

  Q-5      . $ %&      ( . 9], Proposition 3.1) 0  ,   

  S    :   s 2 S |   0, 

 &   0   H  S,      H    

     0   ';1 (s) ,

    , H ! S | 0   3 0    ';1 (s). $  5], 6.7, s 2 S -      

 . D 00   6       2.4. / % 2.3.  ' : X ! S 3 s |       Q-' # #     . " # 

     #  X X 0 ;! 0 #' #' (4) # 0 S ;! S # # : S 0 ! S |        ,  '0 : X 0 ! S 0 | (       Q-' . 5 G = Gal(S 0 =S)    #     X 0 ,

           Q-   

269

    '0 : X 0 ! S 0 | G-    , X = X 0 =G  : X 0 ! X = X 0 =G |       . - 

  , #  G  X 0   #       (      |      #  > 1  X .  6 ' X 0 =S 0          X=S. 0   $ . >  s 2 S |   0,   0 0 # : (S 0  s0) ! (S s)   . > G = Gal(S 0 =S). >  X 0 | ' X S S 0 . @     (4)  X = X 0 =G. > 00     G = Gal(S 0 =S)  S 0   s0 = #;1(s),   ' 

     G : X 0 ,     '    '0;1(s0 ). ; %  ,  X 0   1      

       G  X 0   0        0 x01 x02 : : : x0m 2 '0;1(s0 )  X 0 ( .,  , 5], 6.7). @0 ,

1    0 . / % 2.4.  X=S | Q-'   # #     . " #    S  (     %           . 0   $ .  0   S, 6    , ' : X ! S 3 s |  0 

  Q-5 . F

 0   0 0 X 0 =S 0 3 s0 ! X=S 3 s            2.3. >  x01  x02 : : : x0m 2 X 0 | 0 0     G         xi := (x0i). 9 0 0 xi 

    %   0% 0    xi 2 Ui  X. @ 3     Ui nfxig     '0 0: 1 (Ui nfxig) = ZI (Xx )  (5)  I(X x) -   0 0 x 2 X ( .    1.5). >    ,  (S s) | '0 0 30 

 ,  G - '0 0 . :    2.3  (5)   ,  &    G-     0  C 0 := '0;1(s0 ). 9 C 0 |    '   0  (   2.2). > ,  0 0  Cj0  C 0 , Cj0 ' P1               G. @ G  PGL(2) ,     ,         %& ( .,  , 19]): G = Dn |  6 0 2n, n > 3, G = A4, G = S4  G = A5. ,   , G  GL(2).    ,   A4 , S4  A5  %           ,   Dn               0    1      (6   0 2 Dn  %    ). @0 ,  G    C 0     0      0 . F

    3 ;  C 0. 9    G : ;        1    . D        %&     . i

' 2.5.  ; | ,  &$  # 

.  #  ( ,  Aut(;) 



    %     ;. " # $    Aut(;)-     # ;0  ;  

. . 

270

. .

@@ ;; @ s; @@ @ .

1   2.6 (    $  .)  ' : X ! S | Q-'  

# #      s 2 S |    . " #   ';1 (s) 

      #   #  rItop (S s) #      r 2 N. - 

  ,

 #  G  C 0  

   # (       ,  r > 2. 0   $ .        2.4   ,   G = Zn    C 0    % 0     % 0  Ci0 ' P1,      Zn : P1         0, 6 %  Zn : X 0    0         % 0. 1     0       1.5. 1   2.7.  #  ( ,     #  2.6   ';1(s)  (    #   x #  I(X x), # $    Itop (S s). " # Itop (S s)   . - 

  , $        Ci  ';1 (s),  # ($ x,  (

$ #   x1 #  r I 2(Ss) #      r 2 N. 0   $ . : 0        2.6  ,      G = = Zn : '0;1 (s0 )       0 , . .     G : ;    top

   1 . @    2.5 &    G-      3 ;. ; %   ,   G         0%&  0  C10  C20  '0;1 (s0 ),     ,      0. 9 0 x0 = C10 \ C20        0 x = (x0 ), 0        0   G0  G   0  2         C10  C20 . I G0 = Zn=2 | 0 '0 0,     , &    &   G0-    0 x01 2 C1, x01 6= x0. @ x1 = (x01 ) |  0 0   0  r I 2(Ss) . top

1   2.8.  X 

 #  ';1(s)  %        #  6 2. " #   s 2 S     #& 

   A1 . ' 2.9.  ' : X ! S |   Q-'  #          S h 2 Pic(S) |   #   1 . " # # n  0       j ; KX + ' (nh)j  . 1    , '(Bs j ; KX + ' (nh)j) |   

      (      S . 0   $ . 7     &     ,    h    .    &% 0 % C 2 jhj   F = ';1 (C). @ F | 0    ,     2.2  3 ' : F ! C | '   0  .

           Q-   

271

90 ,  , %    . 9   0,  C \ '(Bs j ; KX + ' (nh)j) = ? , 60    ,

F \ Bs j ; KX + ' (nh)j = ?: :   %       

H 0(OX (;KX + ' (nh)) ! H 0(OF (;KX + ' (nh)) ! H 1 (OX (;KX + ' ((n ; 1)h)):

@0 00  n  1   ;2KX + (n ; 1)' h  ,      / {5  H 1 (OX (;KX + ' ((n ; 1)h) = 0. > 3     KF = (KX + ' h)jF . @0 ,   H 0(OX (;KX + ' (nh)) ! H 0(OF (;KF + ' ((n + 1)h)) ! 0:

; %   F \ Bs j ; KX + ' (nh)j = Bs j ; KF + ' ((n + 1)h)jF j. >  L  F | % 0   'jF : F ! C.     0, 0 ,  L \ Bs j ; KF + ' ((n + 1)h)jF j = ?. >   % 2.1  0 s 2 (S n'(Sing(X)))  ';1 (s)     0 0 (      0 ). ; % 'jF : F ! C | 

   0 0  0  L     0

    L2 = 0  ;1. :  6,    ,    ;2KF ; L + ' ((n + 1)h)jF  

 633 0  K   n  0. :  &    %        H 0(OF (;KF + ' ((n + 1)h)) ! H 0(OL (;KF + ' ((n + 1)h)) ! ! H 1 (OF (;KF ; L + ' ((n + 1)h)):

>    F  H 1(OF (;KF ; L + ' ((n + 1)h)) = 0. @0 , L \ Bs j ; KF + ' ((n + 1)h)jF j = Bs j(;KF + ' ((n + 1)h))jL j. @0 00 L ' P1 ,  Bs j(;KF + ' ((n + 1)h))jL j = ?. L 0    .

3 "    j;KX + '(h)j

* 3.1.  ' : X ! S | Q-'    #          S h 2 Pic(S) | #     #  .  #  ( ,  $            j;KX + ' (h)j 

     (  

#& 

 . " # $      #    X0 = X ; ; ; ! X1 = X 0 ; ; ; ! X2 ; ; ; ! : : : ; ; ; ! Xn

# '0 ='

; =1

# '1 ='

0

2 ;

# '2

3 ;

n ;

# ' (6) n

S0 = S S1 = S S2 ::: Sn # (# 'i : Xi ! Si |   Q-'  #      (#  #   #     (   #   # #    #&$  #  0

. . 

272

# :



; ; ; ! Z0 #p #q Z

Xi;1 # 'i;1 Si;1

 = i

Xi # 'i Si

(7)



; ; ; ! Z0 #p jjq Z

(8) Xi;1 Xi # 'i;1 # 'i  Si;1 ; Si #  p : Z ! Xi;1 , q : Z 0 ! Xi | #        ,  (   Z ; ; ! Z 0 |  &         1,   #  (8) i : Si ! Si;1 |      ,    (Si =Si;1 ) = 1 (. . i |  ,    1.3). i

    0      30         8],20]. >  X |    0    , H |           X (. . H         0 )  c 2 Q, c > 0 | 0 0

 .    1  1  f : Y ! X  

   X  H. >  HY |        ,  

        H  Y . > ,  KX + cH     Q-/  . @ X KY + cHY = f (KX + cH) + ai Ei (9) i

 ai 2 Q  Ei |  0%    . I ,  (X cH)      (        ,     ,      )     ,    i    ai > 0 (   

 ai > 0, ai > ;1, ai > ;1). :  ,  6          1   1  f ( . 20]). M    X   Q-30   

 ,  KX + cH     Q-/  % c    ,     6        (X cH),   ,       (X c0H)  % c0 6 c. M  (X cH)   1      

 ,  '  '     (10) e = e(X cH) = #fi jai 6 0g 0       1  f, 6            #   (X cH). ; ,  e(X cH) = 0   0 , 0 (X cH)        

 . 9  (X H)     0 '     c(X H) = max fc j (X cH) { 0   0  g

(11)

           Q-   

273

            (X H). O                   clog (X H). D   

   (X cH)  %&  : (*) X   1     Q-30   

 R (**) H |        X      0 R (***)  0 c 2 Q  (X cH)   1 0   0  

 . :    20],   0,   0 , 00 (*{***)  0%        ( .,  , 1]).     ,     %& / % 3.2.  X H |  ,  (*,**) c = c(X H) |         (. (11)). " # $       p : Z ! X  ,  (i)  (Z cHZ ) #       (*{***), # HZ |      H  Z 0 (ii) p : Z ! X |   #      (  <  ),

  (Z=X) = 10 (iii) KZ + cHZ = p (KX + cH)0 (iv) e(Z cHZ ) = e(X cH) ; 1. D3 p : Z ! X    3.2         #   (X H). 9   ' : X ! S | 

  Q-5      . 9     h 2 Pic(S)  H = j ; KX + ' (h)j. >   2.9,  h     ,  jHj         0 . ' 3.3 (,1], ,7], 8.8) =    #&$  : (i) $  #   H 2 H 

  % #& 

     0 (ii) $  #   H 2 H 

  %         0 (iii) (X H) 

         0 (iv) (X H) 

         . $ %&    ,     0   . ' 3.4.  ' : X ! S |   Q-'  #      h(1) h(2) h(0) = h(1) + h(2) 2 Pic(X) |   #    ,    

   H(k) = j ; KX + ' (h(k))j, k = 0 1 2   &   # (    . " #  ,    (9), X KY + cH(k) Y = f (KX + cH(k) ) + a(ik)Ei k = 0 1 2 i

(1) (0) (2) 

 a(0) i > ai , ai > ai . !  3.5.     ai      X c  &       ,    (   ,        #      #   h 2 Pic(S)    ,      h  nh, n 2 N.

. . 

274

( #. > ,  & 6   H 2 H    

   ,   %  0 . @ (X H) | 0   0 , . . c = c(X H) < 1. F

 0     p : (Z HZ ! (X H). >  E  Z |   0%    . > 00 KZ + cHZ = p (KX + cH) = p ((1 ; c)KX ) + p ' (h)

(12)

   KZ +cHZ  

 633 0  S. 7 , (KZ +cHZ ) = ;1 >  0 (Z cHZ )         S. >  0     (KZ + cHZ )-3   60           60           1     . D  , 0 ,      :

Z ; ; ! Z0 #p #q (13) X X0 #' # '0 S = S 0 0 C  q : Z ! X |       60      0   0   KX + cHX ,  '0 : X 0 ! S { 

  Q-5 . 0

0



Z ;; ! #p X #'  S ;

Z0 jjq X0 # '0 S0

(14)

C  '0 : X 0 ! S 0 | 

  Q-5 ,  : S 0 ! S | '   3,   (S 0 =S) = 1 (. . | 0, 00   1.3).  %  (X 0  cHX ) | 0   0 , . . c(X 0  HX ) > c(X H): (15) >  E  Z | p- 0%    ,  E 0  E 00 |  

  E  Z 0  X 0   

 (   E 00 = 0,  dimq(E 0 ) < 2). @    2.9 '(p(E)) = s 2 S | 0  ('0 (E 00)) = ('0 (q(E 0 ))) = s. $ 3 '0 : X 0 ! S 0  %     1, 6   (13) dimq(E 0 ) < 2, . . E 0 | q- 0%    .     (14),   ,  dimE 00 = 2, '0 (E 00) = C 0 |   0  0,  (C 0 ) = s.    (13)        % 3 0

0

KZ + cHZ = q (KX + cHX ) + a0E 00 a0 > 0 0

0

0

(16)

0

( .,  , 8], 0    5-1-6). ; % e(X 0  cHX ) = e(Z 0  cHZ ). >    %   0      (10)  0 00   ai  3 (9) 0

0

           Q-   

275

 %  3 ( . 20], th. 2.23)    %   (13), (14): e(X 0  cHX ) = e(Z 0  cHZ ) 6 e(Z cHZ ) 6 e(X cH) ; 1: (17) 7 , 0 00 c | 0   0  (X H)  c < 1,  KZ + HZ = p (KX + H) ; bE = p ' h ; bE b > 0 b 2 Q   h  0   b     h. ; % ;KZ + q '0 h = HZ + bE 0 ;KX + '0 h = HX + bE 00 :    (14)    '0 (E 00) = C 0, E 00 = '0;1 (C 0). >6 0

0

0

0

0

0

HX = ;KX + '0 ( h ; bC 0 ): 0

0

9  h  S 0  

 633 0  K  , 6  h     ,  h0 := h ; bC 0 0  . >     1 0

0'% 0 (X 0  c0HX )  . . D         

  Q-5       'i : Xi ! Si ,       Hi = j; KX +'ihi j      0 ,  hi 2 Pic(Si )     , '        c0 = c c1 = c0 c2 : : : ci = c(Xi  Hi) ci+1  : : : (18)    (13), (14): X0 = X ; ; ! X1 = X 0 ; ; ! X2 ; ; ! : : : # # # (19) S0 = S ; S1 = S 0 ; S2 ; : : : > (15)        (18) |  %&. 7 ,  ci = ci+1 ,   (17)   e(Xi  ci Hi) > e(Xi+1  ci+1 Hi+1). @0 ,  0     1      0 j > i 0,  e(Xj  ci Hj ) = 0   cj > ci . ;   0,         (19)     . .,   0 1   c(Xn  Hn) > 1. 9 6    0,  limci > 1. ' 3.6.  #    (19)          %

'n : Xn ! Sn ,    $            Hn = j ; KX + 'n hn j 

  % #& 

     .       $  . >  %    0          6].      := limci 6 1 0   ,  (Xi  Hi) | 0   0   i  0. > 6 & 

      %& / % 3.7 (,20], ch. 18, ,18])  A  R |  (            (X H), # X |           Q-       ,  H |  # (        X . " # A #      &   &$    #    . 0

i

n

276

. . 

9 0   ,  (Xi  Hi) |         i  0.  6    0          %&   ai 6 0 3    (9).

!  3.8. ?    Q-'  #      ' : X ! S      ;4KS  ' (KX2 )+T' , # T'  S |  #   ,  

 (#   ' (. 29], (3.10.1)). A    ,   ,  (  #      i  (6). B  ,  #  , 

 i |   (ii)   1.3,      S = S0  S1  : : : Sn  & #& 

     (.  # 4.2).

4 $   

   6 3     ,  ' : X ! S 3 s |  0 Q-5  

      ,  0        1. * 4.1.  ' : X ! S 3 s |    Q-'     #     .  #  ( ,  # ' : X ! S          , . . $     F 2 j;KX j  # ,   #  

  % #& 

     . " #  &  %  #&$  (  : (i)   s 2 S   0 (ii)   s 2 S | #& 

   A1 0 (iii)    X 

  ';1 (s)  % #   x1 x2 #  > 1,   & #  n > 3  &          ,   (S s), (F x1), (F x2) | #& 

   An;1.

1   4.2.  ' : X ! S |  #     , #       

    2. " # S 

  % #& 

       An. 0   $ . >  n = Itop (S s) |  0   0 . D  

,  n > 2,        (i), (ii). @0 00 ;KX  f = 2,  f | &  ',     'F : F ! S         &  0 3. >  '1 0 '2 'F : F ;! F ;! S | 30' U  . @ '1 : F ! F 0 | '  ,  '2 : F 0 ! S | 0       3. > 3     KF = 0. ; %   ,  3 '1 : F ! F 0 0    F 0 0   %  0  

  0 ';2 1 (s). 9 0 00 deg '2 = 2,  #(';2 1 (s)) 6 2. F

   #(';2 1(s)) = 2, . . ';2 1 (s) = fx01 x02g |   0. @ '1     0         

  s 3   0  

   (S s), (F 0 x01), (F 0 x02) 3 . :    2.4  ,  6  

  | %  0  An;1. :   0    3      ( .    1.2)   ,      F     0 ';1 1(x0i ), i = 1 2 0  

   Ak k 6 n;1,  , 

           Q-   

277

k = n ; 1  0   0 xi  x0i,  '1 : (F xi) ! (F 0 x0i) | 3.      % 2.6    % 1.5 n = Itop (S s) 6 I(X x) 6 Itop (F x)  % 0 x 2 X   0  > 1. ; %  ,   

 1 

   %      

    x01, x02   ';1 1 (x0i ) = xi | 0, n = Itop (S s) = I(X xi ) = Itop (F xi)        % 1.5 (X xi ), i = 1 2 | '0 0 30 

    0  n. > 00 F \ ';1(s) = fx1 x2g,  0   0  > 1  X | 6 0 x1, x2. D    (iii)    . F

     #(';2 1(s)) = 1, . . ';2 1 (s) = fx0g | 



 0. > ';1 1 (x0) = ';1(s) \ F ,   ,   6 | 6  0,    0 . M  ';1 1 (x0) = ';1 (s) \ F = fx1g | 0,  '1 | 3  Itop (F 0  x0) = Itop (F x1),  x1 | 



 0   0  > 1

 X      % 2.7 6   0  Itop (S s) = 2. D  

 (ii)    . @0 ,    ,  ';1 1(x0 ) = ';1 (s) \ F |   0 ,  '1 | 3  Itop (F 0 x0) > Itop (F x1). D3 (F 0 x0) ! (S s)     3 30'   f1 g 0 2. > 00 (S s) | '0 0 30 

    0  n = Itop (S s) > 2,   F 0  S  %  1  %&    ( . (3)) : N=o 1 2 3 4 5

(F 0 x0) Itop (F 0 x0)

(S s)

n = Itop (S s)

E6 A2k+1 Am A2k A2k+1

A2 Ak A2m+1 A2k+1k A4k+42k+1

3 k + 1 k > 1 2m + 2 2k + 1 4k + 4

24 2k + 2 m+1 2k + 1 2k + 2

$     % 2.6  C = ';1 (s) &     0      0   0  > n,   x1 . 7 ,   ,  I(X x1 ) = r1Itop (S s) = r1 n Itop (F x1) = m1 I(X x1 ) Itop (F x1) 6 Itop (F 0 x0) (20)  0 r1 m1 2 N.     , Itop (S s) 6 Itop (F 0 x0): (21) F

            '. 1) @0 00 Itop (S s) = 3,  I(X x1 ) = 3r1,   X, 0 x1 , &    &  0      0 x2   0  3r2, r2 2 N ( .     2.7). ;  ,  x1 x2 2 F, 6 F 6' F 0 . . 3 '1 : F ! F 0 | 3. F

    %& '1 3 ; ( . . 1). >3 ;00  ;  &    1 | 6   

 3 0 0  ';1 (s) \ F 6= ?  6 

    0      0 . $   ,  00 Itop (F x1), Itop (F x2),  3 ;0  ;    1 ,    %&   0 x1 2 F, x2 2 F    0  ;01, ;02 0,  j;01j > 3r1 ; 1, j;02j > 3r2 ; 1. V ,  ;    E6   ,  ;01 ;02 |  A2   ;    1  %&   :

. . 

278

y y i y y i . .  F  %       0 x1, x2  A2 ,  %    % 1.5  X  %     0   0  > 1 | 0 x1 , x2,  %&  '0 0 30 

    0  3. D    (iii) n = 3. 2) > 1    % '1 : F ! F 0 | 3, 6 Itop (F x1) < Itop (F 0  x0) = 2k+2 : (21), (20)    Itop (F x1) = I(X x1 ) = = Itop (S s) = k + 1. F

 0    3 '1 : F ! F 0     %&  3 ; ( . . 1). >3 ;00  ;  &    1 | 6   

 3 0   0  ';1 1 (x0 )  6      0      0 . ; % 3 ;0  ;    1 ,    %&  

  F        0 . M  ;0  ,  F   1    % 0,     , X   1   0   0  > 1. @     % 2.7 Itop (S s) = 2        (ii). 9   ,  ;0        0  ;01, ;02, 0

   %   0 x1 2 F , x2 2 F. @0 , ;   

y y ;01

:::

i i i ;00

:::

y y ;02

@0 00 Itop (F x1) = k + 1,  j;01j = k. ;  ,  2k + 1 = j;j > j;01j + j;02j. M  j;02j = k,  Itop (F x2) = k + 1  Itop (F x1) = Itop (F x2) = Itop (S s) = = I(X x1) = I(X x2) = k + 1,    % 1)    (iii). M   j;02j < k,  Itop (F x2) < k + 1       2.7   k + 1   , , j;02j = k;2 1 . 7 ,  0 x1 , x2      Itop (F x2) = k+1 2 0  C0  ';1(s). L   ,  3 ;00      1 .  j;00j = j;j ; j;01j ; j;02j = 2k + 1 ; k ; k;2 1 = k+3 2 > 1. >   . 3) L  0 00   5)   (21). 4) > 1    % '1 | 3,    (20), 6

 2k + 1 = Itop (S s) 6 I(X x1) 6 Itop (F x1) 6 Itop (F 0  x0) = 2k + 1. ; % Itop (F x1) = Itop (F 0  x0), . . '1 | 3,    . @   0 . $ %&   0 %,  

  Q-5  0 00 (ii)  (iii)     4.1       &  %. / 4.3.  P1  C2 ! C2 { #   . D##  # 

2  #&$   : (u v) ! ("k u ";kv)  G = Zn  C2uv P1xy  Cuv k k ; k (x y u v) ! (x " y " u " v), # " = exp(2 i=n), k 2 N, (n k) = 1.  (  X = (P1  C2 )=G, S = C2 =G. " #    &$    : X ! S       Q-' . <   X 

   ';1 (0)   #

         n1 (1 k ;k),    S 

 0 #& 

&       An.

           Q-   

279

/ 4.4.  X 0 |      P2xyz  C2uv , #     x2 +y2 +z 2 f(u v) = 0, # f(u v) 2 m4 f(u v)                . 4      '0 : X 0 ! C2   &   &. <  

X 0 

 #  &    &  &   x0   '0;1 (0)   #  (x y z u v) = (0 0 1 0 0),    

   cA. 4 #   # 

G = Z2  X 0 C2  #&$   : (x y z u v) ! (x ;y z ;u ;v):  X = X 0 =G, S = C2 =G. " #     ' : X ! S       Q' . ,#   G-       X 0 -   x0 .    #

# ,  X 

  % #     x   cAx=2   ';1 (0). - S 

 0 #& 

&       A1 .

"  

Q

1] Alexeev V. General elephants of -Fano 3-folds // Compositio Math. | 1994. | V. 91. | P. 91{116. 2] Barth W., Peters C., Van de Ven A. Compact complex surfaces. | Springer-Verlag, Berlin{ Heidelberg{New York{Tokyo, 1984. 3] Catanese F. Automorphosms of rational double points and moduli spaces of surfaces of general type // Compositio Math. | 1987. | V. 61. | N 1. |P. 81{102. 4] Cutkosky S. Elementary contractions of Gorenstein threefolds // Math. Ann. | 1988. | V. 280. | P. 521{525. 5]  .,  .,  . !" #$#" !%" | .: , 1993. 6] Corti A. Factoring birational maps of threefolds after Sarkisov | Preprint, 1992. 7] Kawamata Y. Crepant blowing-ups of 3-dimensional canonical singularities and its application to degenerations of surfaces // Ann. Math. | 1988. | V. 127. | P. 93{163. 8] Kawamata Y., Matsuda K., Matsuki K. Introduction to the minimal model program // In \Algebraic Geometry, Sendai, 1985", Adv. Stud. in Pure Math. | 1987. | V. 10. | P. 283{360. 9] Koll)ar J., Miyaoka Y., Mori S. Rationally connected varieties // J. Algebraic Geometry | 1992. | V. 1. | P. 429{448. 10] Koll)ar J., Mori S. Classi*cation of three-dimensional +ips // J. Amer. Math. Soc. | 1992. | V. 5. | N 3. | P. 533{703. 11]  ,-. ./0 %1# #$#02 !$$32%4 | .: . 1971. 12] Mori S. Flip theorem and the existence of minimal models for 3-folds // J. Amer. Math. Soc. | 1988. | V. 1. | N 1. | P. 117{253. 13] Reid M. Minimal models of canonical threefolds // In \Algebraic Varieties and Analitic Varieties (S. Iitaka, ed.)", Adv. Stud. in Pure Math., vol. 1. | Kinokunya, Tokyo and North-Holland, Amsterdam, 1983. | P. 131{180. 14] Reid M. Young persons guide to canonical singularities // In \Algebraic Geometry, Bowdoin, 1985", Proc. Symp. Pure Math. | 1987. | V. 46. | P. 345{414. 15] Reid M. Birational geometry of 3-folds according to Sarkisov. | Preprint, 1991. 16] #3 7.8. . %9#%92 4  ## // :;3. <= >. . %. | 1982. | ?. 46. | . 371{408.

280 17] 18] 19] 20]

. . 

Q

Sarkisov V.G. Birational maps of standard -Fano *berings. | Preprint, 1989. Shokurov V.V. 3-fold log models. | Preprint, 1994. $! ?. ?" 3%3. | .: , 1981. Koll)ar J. et al. Flips and abundance for threefolds // Ast)erisque 211 | 1992. &    ':   1995.

       ,   

    

 ,        . .  

        . . . 

  ,     S ,         S          ,     S  .          ,  ,     ,  !"     ,        ,           ".

Abstract I. Kh. Sabitov, Quasi-conformal mappings of a surface generated by its isometric transformation, and bendings of the surface onto itself, Fundamentalnaya i prikladnaya matematika 1(1995), 281{288.

It is proved that any surface S isometric to a given compact surface S and disposed su+ciently close to S generates a quasi-conformal mapping of S onto itself. On the base of this result it is proved that a compact surface admitting sliding bendings onto itself is topologically a sphere or a torus and its intrinsic metric is of rotation type.

   :       

       . 1.   |                    ,    ,                      !          ,           . #,                           ,             ,             ,     ,        ,          . % ,   M |             ,              ds2 = 2 (x y)(dx2 + dy2 ): (1) ,          ,

  -    

,  N 93-01-00154,   \2    ,

",  N 1.4.15.            1995, 1, N 1, 281{288. c 1995      !, "#    \%     "

282

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-,   S S |       M  R3  C 2   !,   S           S ,    ,   S            S    S . .    

 -  r   S     : r = r ; hn (2) r = r + U (3)  r |  -    S , n |      S , h |   ( )     S      S ,  U |   1

,           

  S S . .     , ,                S . %,  P 2 S  (2)      P 2 S ,        S ,    S   P . 2   P  (3)      P~ 2 S ,   M   ,   P . 4 ,     z : M ! M ,    z (P ) = P~ . 5,               ,         6  . 5  (x y) |       P~ ,  ( ) |    P          M . 4         z . 7          ,   !         ,    ,      S S       C n, n > 2, 0 6  6 1,      . % (2) (3)  r(x( ) y( )) + U (x( ) y( )) = r( ) ; h( )n( ): (4) :    !   r1 = @r=@ , r2 = @r=@,          Fi (x y;  )  (r(x y) + U (x y))ri ( ) ; r( )ri ( ) = 0 i = 1 2      det (@ (F1  F2)=@ (x y)) 6= 0 (   1

U    )       x y   1      C n;1. 2  (2) ,   1  h( ) 2 C n;1. 2             S     ( ). 5  ds 2 = (2 + 2hL + (2HL ; K2 )h2 + h2 )d 2 + (5) + 2(2hM + 2HMh2 + h h )dd + (2 + 2hN + (2HN ; K2 )h2 + h2 )d2  ,   , H K |      , L, M N |  1         S . 4    ,    ds 2,        (x y),    (1). 4 , (1)          (5),       1  z = x + iy = z ( ), =  + i,     6  @z @z (6) @ = = q( ) @ 

       

283

  @ +i @ , @ = 1 @ ;i @ ,  @@ = = 21 @ @ @ 2 @ @ (L ; N + 2iM )h + H (L ; N + 2iM )h2p+ 2(@h=@ =)2  q( ) = ?1 + 2Hh + (2H 2 ; K )h2 + 2j@h=@ j2;2 + @( )]2( ) @ = 1 + 4Hh + (4H 2 + 2K )h2 + 4j@h=@ j2;2 + + 4KHh3 + K 2 h4 + 2(Lh2 ; 2Mh h + Nh2 )h;4 + + ((2HL ; K2 )h2 ; 4HMh h + (2HN ; K2 )h2 )h2 ;4 : (B  ,       @,            h    ,       1 .) : ,      (5)     ( )  (x y)      (1), ,   1  h( )    z ( )         2 (x( ) y( )) = p 2 ; K )h2 + 2j@h=@ j2;2 + 1 + 2 Hh + (2 H @( ) : (7) 2 =  ( ) 2 2j@z=@ j

2  ,    S 3 P ( ) ! P ( ) 2 S S 3 P ( ) ! ! P~ (x y) 2 S    !  z = z ( )    6  (6). .           P ! P~     S   .  1.  S  S |    ,   ,    R3      C n, n > 2, 0 6  6 1.  S          S ,         (2)  @ > 0.          S                 S    ,     !"

        ! (6)    (7). D    . 1. B   S       !  ,  ,       C m , m > 1, 0 6 6 1. 2   1

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U (x y; t), t |   1

,   S (t)     t           

     S               S  ,        . E        P       P~ (t) 2 S ,      

    S (t)  

    S   P . 6               (z ),            z ( ):    P (t) 2 S  P~   

284

. . 

  1   S \ "  

 P~      1      S . 2. .           C 1 ,       ,      1             C 1,  !      S    C 2   !. 3. . ,   S S (t)  

   t   ( S (t)    C 2),         

M           M        ,     S (t)    . 2. 5        !    :          ,     ,           (  , -,  1           

, 1         )? .     ,   ,  ?2],                      ,                (              ?3]). . ?2]     ,          \ 1"       ,         . I           .  2.     R3              g           .   g 6 1,              ,       "  .

   . .         S    1  h  (2)    ,    (6)  1  g = 0. D ,    z = z ( )       ,       ,      g > 2     z ( )     . J    g = 0. 2    S     ,    z (  t),  t |   1

,   -            . 4   ,   z (  t)    t = t0      . .   M   ,               = 0;  z = a =(a + b ). .   (7)  1  K = 2      4 a

(8) K a + b = K( )jjaaj+4 b j : 5 ! 1      1     K( )  1=j j4: 2  (8)  ! ;a=b  K(;a=b) = jb=aj4:

       

285

5   (8) = a ~=(a + b ~),   4 K a +a 2b = K( )jaja+j4 2b j : (80) :    ~  ;a=2b,  K(;a=2b) = 16jb=aj4: 5 n-         (80 )     

! ;a=nb    K(;a=nb) = n4 jb=aj4: D ,  b(t0) 6= 0,   n ! 1  K(0) = 1,  ,        . 5    z (  t) = ca((tt))

++ db((tt))    t       . 2  a(0)=d(0) = 1, b(0) = c(0) = 0,              z = a(ct()t ) ++b(1t) . 5  

           ,             

   ,  ,  t,                  S   M . D     t = t0

   S     1  ,              = 0 = 1,                z = a . D    t = t0   , ,  ,      1   ,           . L      1(t) 2 (t)    z (  t). 5   c(t)  0. 2      1 = 1    S ,  

2 (t) = b(t)=(1 ; a(t)),   a(t) 6= 1  t 6= 0,           ,  ,      !, . 4  M   ,   ~ = ; b=(1 ; a). 4       z       : z~ = z ; b=(1 ; a). .      

~ ! z~      z~ = a(t) ~. 2      1   K  K(a ~) = K( ~)=jaj2 (800) ( ,       K~ ,            1  ). 5 ~ ! 1  K( ~)  C=j ~j4,  C = Const. B   (800) ,   ja(t)j = 1, . . a(t) = ei(t) . 2  a(t) 6= 1  t 6= 0 a(0) = 1,      t = t0 ,    (t0 ) = 2,   |

1 . 5 ,      S    ! ~  1  a(t), b(t)      

 . 2  (800)

 K(e2i ~) = K( ~),      2n         ,      1        K ,     ' 2 ?0 2)    K(ei' ~) = K( ~), . .                  .

286

. . 

J      c(t) 6 0. 2    ,    c(t) 6=p0. L     z (  t)  p      1(t) = = (a ; 1 + (a ; 1)2 + 4bc)=2c 2 (t) = (a ; 1 ; (a ; 1)2 + 4bc)=2c. 4  S   ~ = ( ; 1 )=( ; 2 )     z~ = (z ; 1 )=(z ; 2 ),      z~ = A ~,  p p A = (a + 1 ; (a ; 1)2 + 4bc)=(a + 1 + (a ; 1)2 + 4bc): 2  1 (t) 6= 2(t),  A(t) 6= 1,  A(0) = 1,   !        ,       (800). J    g = 1. 2   M     ,       z ( )       z = + B . 4 , (7)      K: K( + B ) = K( ): (9) # , K( ) = K( + n!1 + m!2 ) (10)  n m | 1   ,  !1 , !2 |   ,    !     . 5  !1 !2      ( 0) ( )    ,   > 0. 5,        B (t) = B1 + iB2 , B (0) = 0,           . -  ,      

B (t), 0 6 t 6 ",     " > 0        ;,    . .       z = x + B           ;,       . .   (9)     

 1  K  . J  ,    ,     = 0. L  K(B (t)) = K(0) = Const. 2      ,    "     1 = B (t1 ) 2 = B (t2 ), 0 < t1 < t2 < ",       T1 T2    = 0  1  1 2   

K( 1 ) = K( 2) = K(0). 2     1 2 ,   ,      ;,        K( )   K(0),  1   T1 T2 ,    1 2 , K     K(0). 5     ,         1  !     T1 T2 . .  !  K = K(0). 6 t1 t2        ,     T1 T2     ,  

 !        "-             ". 2     ,        K = Const, . .     ,     R3    . % ,          . 5  B = B1 + iB2 , B1 (t) = ac(t), B2 (t) = bc(t), a b |   , a2 + b2 = 1. . 

b ; a = Const  1  K = Const,  , K    K( ) = f (x), x = b ; a. 4   u = b ; a, v = a + b,       ds2 = 2 (u)(du2 +dv2 ), . .    S         .

       

287

:             1. I  ,    ,    K = Const,    ,   ,          ,         . J      . 1. b = 0. 2 K( ) = K(), . . K        = Const,        ,    . O  

   . 2. b ; a = 0  (a b) k ( ). # 1  K    

b ;a = Const             

   , . . K( ) = f (x), x =  ; a=b. 2  (10)  K( ) = f ( ; a=b) = f ( + m + n ; a( + m )=b): (11) . !  f ( ; a=b) = f ( ; a=b + n ): 4 , K     ,        ,           . B   b ; a = Const       . 3. 5  b ; a = Const          . % (11)  m = 0  f (x) = f (x + n ), . . f    T1 =  . 4    ,  n = 0  f (x) = f (x + m(b ; a )=b), . .    T2 = (b ; a )=b. O          ,  K  Const. D ,   m0  n0 2 Z,    m0 (b ; a ) = n0 ,   a=b = = ; (n0 )=( m0 ): (12) 4 ,     a : b      !    ,       !  (12). .      |  

K = Const |      = a=b,   

      ,         (         ), . .  1     

       ,       1 . D ,           R3    3        . 6 ,               ,         . D . %      ,       !        :    S     C 1,       1     ,      1

      !    . 3. :    2         C 1 ,   1        ,     . .

         . -         ,     2:    ,    

h = th1 + o(t), t ! 0, 1  h1  0,   ,      z = z ( )      z = z0 ( ),           2.

288

. . 

  

1] . .  .        | .: , 1959. | 628 . 2] M. Spivak. A comprehensive introduction to Di&erential Geometry, v. 5. | Berkeley: Publish or Perish. | 1979. | 661 p. 3] E. Rembs. In)nitesimal Verbiegungen von Fl*achen in sich // Math. Nachr. | 1957. | B. 16. | S. 134{136. '    (:   1995.

          . .   

  70-           (14.02.1924{26.05.1989)

  

     

  A |              ,  ,       a 2 A      n,     an A  Aa.   A |        .

Abstract

A. A. Tuganbaev, On left distributivity of some right distributive rings, Fundamentalnaya i prikladnaya matematika 1(1995), 289{300.

Let A be a right distributive right nonsingular ring. Assume that for every element a 2 A there exists a natural number n such that an A  Aa. Then A is a left distributive ring.

           ,   |  .              .   MA     ,   M  ! ",  !  #        A. $1]   , '  A |       ,      " a         ' n, ' an A Aa,  A |   . (      ,  #   ,    $2]. *        1.  1. %  A |  '%  (  )%  (  *,   ,   ') - ,  a .    %    n,   anA Aa. /) A |  '%     *.     1     . +  !  ! "       . +          ( ),       ()        . .      ,      

   .   ' (    ')       ' (  ')   ,   / #  ! '! (  '!)    "   . (0        ',  1       .)       ,            '.           1995, 1, N 1, 289{300. c 1995          !,  "#   \%    "

290

. .   

      ,      !      '.      ' (  4 ),     / '! !  !   . 5 r(B)  l(B)  '        / B  A.    M ' J(M), End(M), max(M), Z(M)  '      / , " 1, / !   !   ,       . ( / T  A   : ( )   ,  T |       /  A,  / #      / #   A7 ()  '    ,   ! " a 2 A, t 2 T      " b 2 A, u 2 T , ' au = tb7 ()     ,   !  ! " a 2 A, t 2 T, ' ta = 0,      " u 2 T, ' au = 07 ( )       ,   !  ! " a 2 A, t 2 T, ' a2 = 0, ta = 0,      " u 2 T , ' au = 0. ( / T  A    /  !        #! ! " ! (. $3, . 51]) : ( ) T |  '            /  A7 () #    AT   1 fT  f : A ! AT , '  "  f(T )    AT , AT = ff(a)f(t);1 j a 2 A t 2 T g, Ker(f) = fa 2 A j at = 0    t 2 T g. "! ! AT   

   ' !  A  T, fT |  ' 1,  a 2 A, B |       A,   aT , BT  fT (a), fT (B)AT . ; T = A n M,  M |       A,   AM , fM , aM , BM  AT , fT , aT , BT . <  '    / !    T ,   ' ! TA,  ' 1 Tf : A ! TA, Ta,     TB  TA, /  / Tf(B),  B |      A, ',  T = A n M,  M |      A,     T    M. + A         ( ),              ( )    M #  AM (MA).

 1. %  T | (  ( ,%  (   (,-

   * A. /): ()   T  '  (,  AT % % 1 (')     * A 2. - ,  % 2,  , ,,% % 

'2, - , , 3 T ,  AT % % 1 ()     ') t 2 T % %   %  n,   r(tn ) = = r(tn+1 ),  AT % % 1 ())   A |  *  % , ,,     (24 %  ,  AT % % .

   . =     ()  ( )     ( )  (),        ( )  ().  /  ( ). ( a 2 A, t 2 T , b  at,

' ta = 0. = 0 = tb = b2    bx = 0    x 2 T. ( u  tx 2 T. = au = 0  T     .  /  (). ( ta = 0,  a 2 A, t 2 T. >        n, ' r(tn ) = r(tn+1). = 

         

291

  T  '    ,       u 2 T, b 2 A, ' tnb = au. (" tn+1b = tau = 0, b 2 r(tn+1 ) = r(tn ). = au = 0, T     

 ( ).

 2. %  A | %*  *. /):

()   * A (2.  2. %  2 ') (,  (      ,, (,,   (3  24 - ,   ,-  (  % ,  (3 - 4 - ,   ', %), (   % 1 (')  ,, 2 (2  2  * A (  (2 (,    ,   * A ,  4 ,, 24 (24   (  , , 4 ,, 24 (  (24  )1 ()  * A  )% 1 ())   A , (,  A | (2. (  , (,, (3  1 ()   A (,  A | '  1 ()   A , (,  A | (2. (   1 ()   A , (  (,  A | , ( '  1 (3) r(a) = r(an )   ') a 2 A  4  % 24 n1 () '. ,, 2. 2.   (2.  M 3 A  (  (2.  ,  . ,, 2, (2,  ,1 ()   T | (  ( ,%  (   (,   A,  AT % %     %*2,  *,.

   . ( ( ), (), (), ( )     $4, . 286, 288, 316, 321]. ( ( )     ( ). ( ()     ()    , '             #  

    . ( (/)     ( )       '      $4, . 319].  /  (). B, ' r(a) r(an). ( b 2 r(an). = an bn = 0. (  ( ) (ab)n = 0. (" ab = 0, r(an) r(a).  /  (). =     /    (   )    ', ' M  /   (   )   ,  M  /   '   , #   ()   '   .  /  (). (  1() AT #. ( f : A ! AT |  ' 1, a 2 A, t 2 T, q = f(a)f(t);1 ,

' q2 = 0. =    T  '    ,       u 2 T , b 2 A, ' au = tb. = 0 = q2 f(tu) = f(a)f(t);1 f(au) = f(ab), ab 2 Ker(f). (" abs = 0    s 2 T. =      ( ) r(a)   , ' bs 2 r(a),  0 = atbs = a2 us = a2 x,  x  us 2 T. (  () r(a) = r(a2 ),  ax = 0, a 2 Ker(f), q = 0.  3. 5  ,%  MA  2 % : () M |  '% 2. ,% 1 (')  (6 2 ,%  M  '% 21 ()  2-(2 (,%  ,%  M  '% 21

292

. .   

())   '24 - ,  m n 2 M .%   - , 2 a b  * A,   1 = a + b, maA + nbA mA \ nA1 ()   '24 - ,  m n 2 M .%   - , 2 a b c d 2 A,   1 = a + b, ma = nc, nb = md.

   . 0   ( ) ) () ) ()  ( ) ) ( ) ' . () ) ( ). ( f  m+n, T  mA \ nA. =    fA = fA \ mA+fA \ nA,       " b d 2 A, ' fb 2 mA, fd 2 nA, f = fb + fd. = nb = fb ; mb 2 T, md = fd ; nd 2 T . ( a  1 ; b, z  a ; d = 1 ; b ; d. = 1 = a + b, fz = f ; fb ; fd = 0, ma = md + mz = md + fz ; nz = md ; nz, nz = ;mz 2 T, ma 2 T7 a b |  ". ( ) ) ( ). ( F, G, H |     M, f 2 F \ (G + H), f = m + n, m 2 G, n 2 H.  '    ' f 2 F \ G + F \ H. (       a b 2 A, ' 1 = a + b, ma 2 nA, nb 2 mA. = fb = mb + nb 2 F \ G, fa = ma + na 2 F \ H, f = fb + fa 2 F \ G + + F \ H.  4. %  T | ,  (24 3,  .   * A, Q  AT , f : A ! Q | . ),,63,. /): ()   '24 - ,  q1 : : : qn 2 Q .%   - , 2 t 2 T , a1  : : : an 2 A,   qi = f(ai )f(t);1 ( i = 1 : : : n1 (')   B | (2.   * A,  BT = ff(b)f(t);1 j b 2 B t 2 T g1 ()   a 2 A, B | (2.   A  f(a) 2 BT ,  at 2 B    ) t 2 T1 ()) (B + D)T = BT + DT , (B \ D)T = BT \ DT   '24 (24   B D  * A1 ()   N | (2.   * Q, E  f ;1 (N),  E | (2.   A  N = ET 1 ()   G | (2. (,%  ) ,%  f(A)Q,  .  . - , t 2 T ,   Gf(t) f(A) ,    , 2. ,%  f(A)G 3,6 (,% ,%  % Gf(t)  * f(A)1 ()    * A  '%   ,   * Q  f(A)  '% 2  , (, f(A)Q |  '% 2. 2. ,% 1 (3)    * A  '%  (,  Q |  '%  (  *1 ()    * Q  '%  (,  (B \ (D +E))T = (B \ D +B \ E)T   '24 (24   B D E  * A1 ()    * A= Ker(f) , ( (, (),  Q | , ( (, ()  *1 ( )   T     , , 24 3,  .,  Q  = TA, (,  4 ),,63, A ! Q, A ! TA ( 1 (,)   T = A n M , ) M | (2.   A,  Q |    *  J(Q) = MT  MM 1 ()   L N | (2  2  * Q  L  )  N ,  f ;1 (L), ; 1 f (N) | (2  2  * A  f ;1 (L)  )  f ;1 (N)1

         

293

()   A |   ( (  ()  *,  Q |   ( (  ()  *1 (()   Q |   (  *,  (DE)T = DT ET   '24 (24   D E  * A.

   . ( ). 0  /   / ' , ' n = 2. ( qi = f(di )f(ti );1, di 2 A, ti 2 T , i = 1 2. =    / T  '    ,       u1 2 T, u2 2 A, ' t1 u1 = t2u2  t. =    u1  t1 2 T,  t 2 T. = f(ti );1 = f(ui )f(t);1 , qi = f(ai )f(t);1 ,  ai  di ui 2 A. ( ()     ( )7  () |   (). ( ( )     ( ), (), (). ( ( )   #  ( ), (). ( ()     ( ). ( (/)     (),  3     1!   !    . ( ()     ( ), ( ). ( ()     ( ). ( ()   #  ( ), ( ).  () 1 Q  = TA     $3, . 51],      '! 1         / T.  /  ().  '   , '  "  MT  ,  "  Q n MT  . ; MT  /   ",  1 2 MT , '   () #   " t 2 T , ' 1 t = t 2 M \ T,  '  '. (   a 2 A, t 2 T , f(a)f(t);1 2 Q n MT . = a 2 A n M = T , f(a) |   ",  f(a)f(t);1 |   ". ( ()     ( ). ( ()     ().  /  ( ). ( x 2 D, y 2 E, t 2 T.  ;1 f(y) 2 (DE)T . =     Q  '    ' f(x)f(t) P 1    ,         i=0 f(t);i f(y)Q ' / . ("  ;n;1 f(y) = Pn f(t);i f(y)qi . =    " q  : : : q 2 Q, ' f(t) 0 n i=0 P P f(t);1 f(y) = ni=0 f(t)n;i f(y)qi , f(x)f(t);1 f(y) = ni=0 f(xtn;iy)qi 2 (DE)T .  5. %  A |  3%, (  *. /): ()   a 2 A, B | (2.   A  aM 2 BM   4 M 2 max(AA ),  a 2 B1 (')   a 2 A  aM = 0   4 M 2 max(AA ),  a = 01 ()   B D | (2  2  * A  BM = DM   4 M 2 max(AA ),  B = D1 ())   %*   * A '4,    ,  '2   ') M 2 max(AA )  * AM '2  %*2,1 ()   B . (2.   A,   BM |   * AM   ') M 2 max(AA ),  B |   * A1 ()     ') M 2 max(AA )  * AM   (,  A |   (  *1 () 2. ,, 2. (2.  M  * A    (  (2,  ,, (, A=M |  .

   . ( ). ( H  fh 2 A j ah 2 B g. >    , ' H = A.   . = H  /   M 2 max(AA ). ( T  A n M. =      aM 2 BM ,    4() at 2 B    t 2 T. = t 2 T \ M  ' '. ( (), ()

294

. .   

    ( ).

 ( ) !      2(),  ' |   (). ( ( )     ()   , ' BM (AB)M AM BM = BM . ( ()     ( ).  /

 (/). (  ( )  '   , ' MN |     AN    N 2 max(AA ). ; M = N,    4() MN = J(AN ) |     AN . ( M 6= N. = M + N = A, AN = NN + MN = J(AN ) + MN ,  MN = AN |     AN .  6. 5  ,%  MA   *, A  2 % : () M |  '% 2. ,% 1 (')   '24 - ,  m n (3 ) (6  N ,%  M , 4,   mA \ nA = 0, .%   - , 2 a b 2 A,   1 = a + b, ma = nb = 01 () M  , (6   F  G, ) G | % . ),,62. '3 ,%  F 1 ()) M  , 2-(24 (,% . H , ' 4 6 ,% , F  G, ) F G | 3,62 ( 2 ,% .

   . 0   ( ) ) ()      N  3. 0   () ) ( ) '  . () ) ().   . = # " 1 h : F ! G   " n = h(m) 2 G,  m 2 F. =   #   " a b 2 A, ' 1 = a + b, ma = nb = 0,  n = na + nb = na = h(ma) = h(0) = 0  '  '. ( ) ) ( ). ( m n 2 M, H  mA + nA, f : H ! H=(mA \ nA) |  " 1, B  r(f(m)) + r(f(n)).  , ' B = A. =      " a 2 r(f(m)), b 2 r(f(n)), ' 1 = a + b. =    ma 2 nA, nb 2 mA,    3   M .    , ' B 6= A. = B D 2 max(AA ). ("  /     f(mA), f(nA)   1  , 1

   A=B  T. = H  1  , 1   T  T, ' ' .  7. %  MA |  '% 2. ,%    *, A. /): () 0 = Hom((N + T )=T (N + T)=N) = Hom(N=(N \ T) T=(N \ T)),   '24 (,% . N T ,%  M (,    ,   N + T = M ,  0 = Hom(M=T M=N))1 (')  ,(  2  * End(M) *  21 ()   ,  4 ((  3,624 ( 24 (6  ,%  M  (-  , (,, ) A ( %  ),   (6 2 ,%  M ,21 ())    ( 2 (6 2 ,%  M 3,62 (-  , (,, ) A=J(A) | (     *),  M | *(. ,% 1 ()   N T |  (,%  ,%  M ,   N \ T = 0,  T \ f(N) = 0   ') ),,63, f : N ! M 1 ()   N | (,%   M , fTj gj 2J | P ,  4 (,% . 3 M , ,4 %  (  N ,  N \ j 2J Tj = 01 ()   N T |  (,%   M ,   N \ T = 0, F  G | ,7 (   2 (,%   M ,      ,%  N  T ,  F \ G = 01

         

L

295

(3)   H | . (,%  ,% LM ,   H = j 2J Hj ,  % % . (,%  L ,%  M ,   L = j 2J Lj , Hj Lj ,  Lj (   2  M 1 ()   M   '24 (,24 %,, % 24 (   24 (,% .,  M | ,2. ,% 1 ()   '2  % 24 (   24 (,%  ,%  M , %  (,  M | ,2. ,% .

   . ( ). ; F  N=(N \ T ), G  T=(N \ T ),  /     1 F  = (N + T )=T, G  = (N + T )=N   , '   6 0 = Hom(F G). ( ()     6. + /         1 , '    6,  F G |     F  G |  1    M,    F G  1. =  (), ( )     , '  /  '  4       1 , 1   '  ' !  ,  /   #        1 , 1   ! !  . ( ( )     6,  ()          . ( (/)     ( ), ()     M. ( ()   #  (/). ( (), ()     ().  8. %  M |  '% 2. ,% , N | (,%   M , f | -,63, ,%  M . /): ()   f ;1 (N) + N = M ,  f(M) N 1 (')   f(M) N + f(N),  f(M) N 1 ()   M = N + f(N),  M = N 1 ())  ,%  M  ,, 2 (,%  (   2,  End(M=J(M)) | %*  *1 ()   )  % )   n ,%  Ker(f n )    % 2, 7, ,%  Ker(f)1 ()   M | %  7 ,%  Ker(f n )    )  % ) n,  M | %  7 ,%  Ker(f).

   . ( ). ( T  f ;1 (N). (  h(m + T ) = f(m) + N    1 h : M=T ! M=N. (  7( ) h = 0. = T = M, f(M) N.  /  (). ( m 2 M. (       n x 2 N, ' f(m) = n + f(x). = m = (m ; x) + x 2 T + N, M = T + N. (

 ( ) f(M) N. ( ()     ()  ' f(N) f(M).  /  ( ). 0  ()  , '    M            . =  End(M=J(M)) 1  

     " 1 ! 1     M   "    .  /  ( ).  '   , ' Ker(f n+1 ) | #     Ker(f n )    n. ( H  Ker(f n ), L |       Ker(f n+1 ), ' L \ H = 0. =    f n+1 (L) = 0,  f(L) H. (  () L Ker(f) H. = L = L \ H = 0,  /    . ( ()     ( ).

296

. .   

 9. %  A |  '%  (  *. /): ()   ) a 2 A  ')  % ) n (2.  r(an)   

% 2, 7, ()   r(a)1 (') 6  * A=J(A), A=Z(AA ), A= (J(A) \ Z(AA ))    %*2, ,    ,   (  2 - , 2  * A     J(A) \ Z(AA )1 ()   M N |  ,2 (   (  (2  2  * A,  A = M + N , M \ N = MN + NM 1 ())   ') ,, ) )   ()   M  * A ,  L 4 4  M (  (24    * A  (% , . %( (       ,, 2. - , 1 ()  ,, 2 (2  2  * A    (% ,) (  (2,  ,, (,   ') (  ()   N ,  A n N (  (1 ()   A   '24 (,24 %,, % 24  ,   * A , (1 ()     * A ( '24 %4 % 24    %  (-  , (,, ) A (),   * A , (.

   . ( ( )     8( ). =          !      ,   ()     8( ), ().  /  (). ( m 2 M n N, n 2 N n M. (  3      " a b 2 A, ' 1 = a + b, ma 2 N, nb 2 M. =       M  N   ',  a 2 N, b 2 M. = A = M + N, M \ N = (M + N)(M \ N) MN + NM.  /  ( ). =    '  #     '!      ',     ()  '   , ' L  . ( h : A ! A=J(A) |  " 1. (  () h(A) |    . (  2() h(M)  /   '    h(B),  B |  / # J(A)     A. = B 2 L.  /  ( ). ; M |        ,    8( ) M |   ,  A=M | , M |   '   . ( T  A n N, x 2 A, t 2 T . >     #     " u 2 T , ' xu 2 tA. (  3      " a b c d 2 A, ' 1 = a + b, xa = tc, tb = xd. ; a 2 T,  / / u  a. ( a 2 A n T = N. = b = 1 ; a 2 A n N = T,  xd = tb 2 T . ; d 2 N,  tb = xd 2 N \ (A n N), 

'  '. (" d 2 T  / / u  d. ( (), (/)     7(), ().  10. %  N | (  (2.   '% ) (  * A, T  A n N . /): ()   ,  T  '  (,  (  *  24 AN % %     *(2, (  *,1 (')     * A 2. - ,  % 2,  , ,,% %  , - , , 3 T ,  AN % %     *(2, (  *,1 ()     *  24 NA % % ,      *(2, (  *,.

         

297

   . ( ). =      9( ) / T  '    ,

   1( ) AN #. (  4()  AN  . ( 4()  7( ) AN |      . ( ()     ( ).  /  (). (  4()  NA  . ( '     4(/)  NA     . (  7( ) NA |      .

 11. 5   * A  2 % :

() A |  '%  (  3%, (  *1 (') A  '%  (, (,   ') ) ,, ) ()   M ,  A n M  '  (1 ()   ') ,, ) ()   M (  *  24 AM % %     *(2, (  *,1 ())   ') ,, ) ()   M  * AM % %      '% 2, (.

   . 0   () ) ()     9( ), 10( ). 0   () ) ( ), ( ) ) () ' . 0   ( ) ) ( )     4(), 5().  12. 8    * A  - , 2  % 2,  ,   * ,   2 % : ()  * A  '%  (1 (')   ') ,, ) ()   M (  *  24 AM % %     *(2, (  *,1 ()   * A  ,, 2 (2  2    (  (2,  ,, (,   ') (  ()   N (  *  24 AN % %     *(2, (  *,.

   . 0   ( ) ) ()     9( ), 10(). 0   () ) () '  . 0   () ) ( )     11.  13. %  A |  '%  ( %*  *, N | (  (2.   * A, E  fa 2 A j at = 0    ) t 2 A n N g, F  fa 2 A j ta = 0    ) t 2 A n N g. /): ()   * A  ,, 2 (2  2 (  (2,  %,,

'24 %4  (4 ,, 24 (24    A1 (') (  *  24 AN % %     *(. ( '  ,   2, ,  N ,, 2, (2,  , H     ) ),,63, f : A ! AN , (, H = E = F 1 ()   M | ,, 2. (2.   * A,  M | (  (2.  , (,  * AM % %     *(. ( '  1 ()) 2. ,, 2. (2.    3 )% 24 - , ,  ,  G 4 )% 24 - ,   * A (  , , L 4 - , ,  4   , ,, , (,  1 ()   D | ,, 2. (2.  , h : A ! A=D |   2. -(,63,,  h(A) | '      ') )% ) - ,  a 2 A - , h(a) )%   h(A).

298

. .   

   . ( ( )     2()  9().  /  ().

(  12  AN #        . (  2(), () AN |  . (" E |   '   . (  9( )  N  /       '    H, #   2()   / #  N   '   . = H |   '   , / #    '    E. *         E  , ' H = E. (  2( ) E = F. ( ()     ()   9 ( ).  /  ( ). ( /      (), ()    , '  /    '    /          . *    , '  " a  /       '     b 2 A,

' ab = 0,  b = 0. =       '     A   ',  b /  ' !  ! '!   . (" b = 0. ( ( )     ( )  ( ).

 14. 5   * A  2 % :

() A |  '%  (  )%  (  *1 (') A |  '%  ( %*  *1 ()   ') ,, ) ()   M  * A (  *  24 AM % %     *(. ( '  1 ())   * A  ,, 2 (2  2    (  (2,  ,, (,   ') (  ()   N (  *  24 AN % %     *(. ( '  .

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299

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1 f : h(A) ! Q  1. (      5(/) h(M) |   '     h(A),  M |   '     A. (  13() H = fa 2 A j ta = 0    t 2 A n M g. (" 1 g  fh : A ! Q   ' 1,

 # # Q    ' !  A  A n M. ( ) ) ( ). ( M |       . (      5(/) M |   '   . (     13() #  ' 1 f : A ! AM , g : A ! MA. ( H  Ker(f). (  13() H |   '   . (  4() H = Ker(g), '    A=H '  . (      4()  MA '  . ( ) ) (). ( M |       . (      5(/) M |   '   . (  14  AM #         . (   = MA. (" MA | MA #  '  . (  4() AM  '          . (  15 MA |    .  17. %    * A  ,, 2 (2  2     ,    ') - ,  x .    %    n,   xnA Ax. /): ()   * A '. ,, 2. 2.  M    (  (2,  ,, (,   ') (  ()   N ,  A n N    ( 2,   ,%  (  2, (, ,,  A=N | ,   '  1 (')     * A  - , 2  % 2,  ,   * ,  A |  3%,    *,   ,   ') (  ()   N

  *  24 NA % %      2,  *,.

   . ( ).  , ' M |    . = MA = A  P 1 = ni=1 mi ai ,  mi 2 M, ai 2 A. (     P      ' d, ' mdi Ami M  i = 1 : : : n. ( H  ni=1 mdi A. = H 6= A. (" H /           B. =      B |   ,  A=B |    ' mdi 2 H B  , ' mi 2 B  P n i = 1 : : : n. = 1 = i=1 mi ai 2 B,  '  '. (" M |   , A=M | , M |   '   . ( T  A n N, t 2 T , a 2 A.

300

. .   

(       " b 2 A, ' tna = bt. =    N |  

'   ,  / T   , tn 2 T . (" T  '  . ; a 2 T ,  b 2 T . ("   A=N    . (). F/      ( )       1().

     1. (  14 A |    ,                . (  17() A |    . G  R  A     '       2()  ,   17( ) |     . (  16  A   .

 



1] Grater J. Strong right D-domains // Monatsh. Math. | 1989. | V. 107. | P. 189{205. 2]  . .   !"#  $ %! // &$. '$. |1990. | . 47. | N 2. | ). 115{123. 3] Stenstrom B. Rings of quotients | Berlin e. a.: Springer, 1975. 4] %- .. ., /01 2. &. /%! !  0  0. | &.: 3, 1979. '    (:   1994.

     

. .  

     

. .   

  70-           (14.02.1924{26.05.1989) 

                 .               : 1)       "# $  # # R-      #   #  -% # (##)  #    %       ' 2) R      V - .

Abstract G. M. Brodski, A. G. Grigorjan, Ring properties of endomorphism rings of modules, Fundamentalnaya i prikladnaya matematika 1(1995), 301{304.

A certain method of studying ring properties of endomorphism rings of modules is justi0ed. As an example of its applications the equivalence of the following conditions is proved: 1) the right annihilator of every proper 0nitely generated (principal) left ideal in any endomorphism ring of an injective right R-module contains a nonzero idempotent' 2) the ring R is a semiartinian right V -ring.

               ,           .    f : X ! Y  Dom f = X , Cod f = Y , Coim f = X= Ker f  ! coim f " ! 

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*   X ' l(A)  r(A) |    

    A   . +   ,1]    . / X    (! ),    (  )  0  "!  R . 1 ,  ,  |   ,  2    R 



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  |   (*  * )  X 2  ,   , !  (   ) X "   . 5         D( ), C ( ), S ( ), T ( ), 

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 "       . 1  |  ,  1)   "*  X   Y  X  (Y ) = \PfKer f j f 2  ' Dom f = X ' Ker f  Y g 

(Y ) = fIm f j f 2  ' Cod f = X ' Im f  Y g' 2)  (  ),   |  * ,   | "   ,    (  ,  )  -#,    "*   R,  X 2 D( )   " ( ! -

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  1  1 !         1. 1) ) 2). $   S ( )    U 2 C ( ) = S ( )   X 2 D( )  # f : X ! U  g : X ! Ker f , ! f 2  ,  , (Ker f ) = Ker f . $#  X= Ker f = X= Im(ig) = X= Im('E) = X= (Im('E)),  i : Ker f ! X | U= 

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# X ! X= (Im I )    ,  ! X= (Im I ) 2 C ( ) . +    , ! (  )     (  ,  )  -#, , ! I 2  . : FR , PG , PR , SF , FT , TL , CIC , IN , CC , CT * " *, *  * " 2*, *  *, * *  "  , * ! *  *, *  *, * ! *  0 * " 2*, *  0 *, * ! *  *, *  * * R-   "    FR, PG, PR  . . $ 1 = feR j e2 = e 2 Rg, 2 = 10 = fRg, 3 = fr(J ) j J 2 L( R)g, 4 = fr(J ) j J 2 L ( R)g, 5 = 7 = fI 2 L(R ) j l(I ) = 0g, 6 = fI 2 L(R ) j (9e = e2 2 R n f0g) eI = 0g, 8 = fI 2 L(R ) j (9e = e2 2 R) l(I ) = Reg, 9 = fRe j e2 = e 2 Rg, 11 = fl(I ) j I 2 L(R )g, 12 = 14 = fJ 2 L( R) j r(J ) = 0g, 13 = fJ 2 L( R) j (9e = e2 2 R n f0g) Je = 0g, 15 = fJ 2 L( R) j (9e = e2 2 R) r(J ) = eRg, 1 = 8 = PR, 2 = 7 = 10 = 14 = f0g, 3 = TL, 4 = SF , 5 = fU j Hom (U R) = 0g, 6 = fU j (9K 2 L(U ) n fU g) U=K 2 PR g, 9 = 15 = IN , 11 = CT , 12 = fU j (8Q 2 IN ) Hom (Q U ) = 0g, 13 = fU j (9K 2 L(U )) 0 6= K 2 IN g, 1 = 2 = 6 = ff 2 Ep(R) j Dom f 2 PR g, 3 = ff 2 Ep(R) j Dom f 2 PG g, 4 = ff 2 Ep(R) j Dom f 2 FR g, 5 = ff 2 Ep(R) j Dom f 2 FT g, 7 = ff 2 Ep(R) j Dom f  Cod f g, 8 = 1 \ 7 , 9 = 10 = 13 = ff 2 Mon(R) j Cod f 2 IN g, 11 = ff 2 Mon(R) j Cod f 2 CIC g, 12 = ff 2 Mon(R) j Cod f 2 CC g, 14 = ff 2 Mon(R) j Cod f  Dom f g, 15 = 9 \ 14. ;  





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 (   )    & .  -/&.. 5   1, 1    |       ,     *, ,  2. 6 ( 1 : 1)   * ,/0'/ 3- 7 - &- R-/1  &(. 1   - 3 - -&- (- -) -    1 . /&  4 2) R    & 1 (/ &(/ V - */.     . !     1       10     10 -  (  ) = :(10 10) ) ) (13 13),      = 10 13   = 10 13      S ,  !  13  ,2]. ? "  ?. 8. /*      ". i

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1]  .   : ,    . . 1. | .: , 1977. 2] Dung N. V., Smith P.E. // J. Pure and Appl. Algebra. | 1992. | V. 82. | P. 27{37. %      &:  1994.

             E-  . .  

        . . . 

 511.36



             jP (f1 () : : :  fs ())j

  !!    " E-!" f1 (z) : : :  fs(z)  $%  "    &%&',  P (f1 (): : :  fs ()) 6= 0.

Abstract

A. I. Galochkin, Lower bounds of polynomials on values of algebraically dependent E-functions, Fundamentalnaya i prikladnaya matematika 1(1995), 305{309.

In the paper a lower bound of the modulus of a polynomial jP (f1 () : : :  fs())j with integer coe-cients on the values of E-functions f1 (z) : : :, fs(z) at an algebraic point  is obtained, provided P (f1 (): : :  fs ()) 6= 0.

 I |        , K |            I, ZK |       K.  f1 (z) : : : fs (z) | 

  KE-   (      x 1   3   #1]),   %&'  (            yi0 = Qi0 (z) +

s

X

j =1

Qij (z)yj  i = 1 s Qij (z) 2 K(z):

(1)

+ E-  f1 (z) : : : fs (z)   ,      C(z)          %   &

    Qij (z), 

   - 

  (#1]   3)  , % f1 () : : : fs ()   ,         , %   

   ,   

  .     ,  (.,  , #2], #3], #4]). 1% ,

      %,  E-    ,   (. #1]   11 12),     &%    

%       %,  %, &'  E-  . 2  %'    %       ,  E-     ,      .  , %  '% .  .     %    

     %,   E-  % . .   "    & &%  " &%' /' %   ! . 0%  N MHS000.            1995, 1, N 1, 305{309. c 1995      !, "#    \%     "

306

. .  

.  f1 (z) : : : fs (z) |    KE-  ,     

  (1),  |    ,         !

 Qij (z)  (1), { = #K() : I] > 2. #     P (x1 : : : xs ) 2 ZI #x1 : : : xs]  P (f1 () : : : fs ()) = 0,     l+1 l jP (f1 () : : : fs ())j > CH ; { d  (2)  d  H |        P , l |      $   C(z)     f1 (z) : : : fs (z) (1 6 l 6 s), C   | ' ,   C $    d,     f1 (z) : : : fs (z),   |      f1 (z) : : : fs (z) ( :       $ ' !   ). 6   , ,  ,   % '     , 

 2 K |       ,      K ,   K(). 2  &'   , %. 7%     Fi(z) =

1 X

 =0

Fi z   Fi 2 K i = 1 v

(3)

   A(x1  : : : xv ) 2 K#x1 : : : xv ]  ,  , A k] (F1 () : : : Fv ()), k = 1 {, , %   

,   ,  .  

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  &'   %      K. 8  (2)   , x 6   12   #1],  ,          , 

P k] (f1 () : : : fs ()) 6= 0 k = 1 {: 9   ,      %         %    ,

  6 %. :  %% &'%    . (#1],   11, x 2).     KE-   f1 (z) : : : fs (z)     

  (1) ( Qi0(z)  0, i = 1 s)   $   C(z),   2 K           Qij (z), i j = 1 s.   L(x1  : : : xs ) = h1 x1 +    + hs xs hj 2 ZK  H = maxj jhj j 6= 0 j = 1 s: #   " > 0 



max L k] (f1 () : : : fs ()) > C1 H 1;s;"

k=1{

  C1 > 0, $    ",     f1 (z) : : : fs (z).

(4)

             E- 

307

<  %   &'% .  kj , k = 2 {) j = 1 v |    . #    X > 1   h1  : : : hv $  ZK ,  ,    



h 1k] k1 +    + h vk] kv  < C2X 1; {{;v1  k = 2 {= 0 < max jh jk]j 6 X k=1{ j =1v

(5)

 h jk] | ,  ' hj   K,   C2 > 0 $     K   kj .   . ? , , 

#K : Q] = #K : I]  #I : Q] = 2{:

.  ,   ZK    Z    , 2{ .

. ?

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   .   , Z. @ & ,           (5) ,    &  & . 2   ,  %      ( %        , m = 2({ ; 1)        n = 2{v  , , %      5.1 ,   #5]      , ,(       (5)    '    7  .

   .  M N |   ,   

 (. 1    ,  %  f11 : : : fss  j 2 Z+  1 +    + s 6 M (6)    f1 (z) : : : fs (z). 2         K(z), %  .  ,  % ,  ,  F1(z) : : : Fv (z)  M = N (7) E1(z) : : : Em (z)  M = N + d (8)  d = deg P .      (  N m = g(N + d) v = g(N) g(x) 2 R#x] deg g(x) = l (9) ( ,      %    x 11   4   #1]). 2  1 2   4   #1]  , ,  ,  (7) (8)     ,    % ,    (6)  M = N  M = N+d     , 

  &'  ,   .   ,  % K(z), %   z =   % %%      ,  .    , , 

  % (                 (1),       % %%  &     , .  

 .

308

. .  

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k] 0 < max jh j 6 X kj j

(10)

   X > 1   (,  , C3 C4  : : :   ,      %,  , %'   X H.      (  X  

, 

{v ; ({ ; 1)m > 0 (11)   L (F1 () : : : Fv ()) = L 1] (F1() : : : Fv ()) 6= 0 | (12)

k ]

    ,

 (10),   (4) %     L (F1() : : : Fv()), k = 1 {,       " s = v    %%.  P(x1 : : : xs ) |   ,   &'  

%  . C  ,  ,  ( , 

R(z) = L (F1 (z) : : : Fv (z)) P (f1 (z) : : : fs (z)) = = a1(z)E1 (z) +    + am (z)Em (z) aj (z) 2 K(z) (13)       % %%  &     a1 (z) : : : am (z). 2 

         %       ,  P (f1 () : : : fs ()) 6= 0. C ,   (12), R() 6= 0. .          T (z) 2 ZK #z], 

S(z) = T (z)R(z) = b1(z)E1 (z) +    + bm (z)Em (z) (14) bj (z) 2 ZK #z] bj () 2 ZK  j = 1 m       bj () = 0  . ,

 (13), jb jk]()j < C4 XH k = 1 {= j = 1 m:

.      (   - 

   &  " > 0 max jS k] ()j > C5(XH)1;m;"  C5 = C5 ("): k=1{

2   (10), (13) (14)

(15)

jS 1] ()j < C6 X jP j {v

jS k] ()j < C7HX 1; {;1  k = 2 {

 P = P (f1 () : : : fs ()),    (15)  , %      X jP j + HX 1; {{;v1 > 2C8(XH)1;m;" :

(16)

             E- 

309

2   X  ,    %   

HX 1; {{;v1 = C8(XH)1;m;" :

(17)

 .     ,    H,  ,  X,     ,  

 %    (12), |         P   ,   aP      (     a. C  , (16) (17)       "    

% (11)  : {mv

jP j > C9H ; {v;({;1)m :

(18)

D, (9) , 

{v ; ({ ; 1)m = {g(N) ; ({ ; 1)g(N + d) =

= g(N + d) ; {dg0 ( ) 2 (N N + d): (19) 2 N = {ld      (     , , %'        g(x),  , ,          %,     % f1 (z) : : :  fs (z). C  , (19)  : {v ; ({ ; 1)m > 05c({ld)l 

 c > 0 | (  .      g(x). C         (11) {mv < 4c{({ld)l  {v ; ({ ; 1)m , (18)       . F  

   

    , 

 

#6].

 

1] . .   .      | .: , 1987. 2] Lang S. A transcendence measure for E-functions // Matematika. | 1962. | V. 9. | P. 157{161. 3] . .   . )* + ,     - E-. // ,. -, | 1967. | . 2. | N 1. | /. 33{44. 4] . 1. 2 . ) , -,     - E-. // ,. -, | 1968. | . 3. | N 4. | /. 377{386. 5] . 1. 3 4 ,. 5* 67  8* +   | .: 1- -  29, 1981. 6] . 1. 2 . On some equations connected with E-function // Diophantische Approximationen 26.09 bis 02.10.1993, Tagungsbericht 43. | Math. Forschungsinstitut Oberwolfach, 1993. | S. 20. '    (:   1995.

           . . 

  70-           (14.02.1924{26.05.1989)

     



 R |            Q, 1 2 R, n > 3, H |         GLn (R),         En (R),  |   ! "      P H ,    P En (R). $    ! "        P H .

Abstract

I. Z. Golubchik, Isomorphisms of projective groups over associative rings, Fundamentalnaya i prikladnaya matematika 1(1995), 311{314.

Let R be a two-sided order in a regular ring Q, 1 2 R, n > 3, H a subgroup of the linear group GLn (R) containing the elementary subgroup En (R),  an automorphism of the projective group P H which is identical on P En (R). Then  is identical on the group P H .

 . .    . .   1]      1. %  R S | ' (  '  '), 1=2 2 R, 1=2 2 S , n > 3, m > 3  ' : GLn (R) ! GLm (S ) | +,-+, .%//. 0. % % '  ( ,/  ( e  f  ' , ' Rn  Sm    ,  ') +,-+, 1 : e  Rn ! f  Sm   ')  +,-+, 2 : (1 ; e)  Rn ! ! (1 ; f )  Sm , ,     1 A 2 En(R) ;  '(A) = 1 (e  A) + 2 (1 ; e)  A;1 :  '. .  () 2] ) 1  * m > 2. + ( PGLn(R) | -. ** GLn (R) *  / , 1 : GLn(R) ! PGLn(R), 2 : GLm (S ) ! P GLm (S ) |  .))-)1.  . .   3] *      )1 1.  2. %  R  S | ' (  '  1  1=2, n > 3, m > 2, En(R)  G  GLn (R), Em (S )  H  GLm (S ), H | , ()      GLm (S )  ' : PG ! PH | +,-+, .%//. 0. % % '  ( ,/  ( e  f  ' , ' Rn  Sm ,  ') +,-+, 1 : e  Rn ! f  Sm   ')  +,-+, 2 : (1 ; e)  Rn ! (1 ; f )  Sm , ,   ; ;  '1(A) = 2 1 (e  A) + 2 (1 ; e)  A;1 *+  +       INTAS.         1995, 1, N 1, 311{314. c 1995             ,  !"  

 \$  "

312

. .  

  1 A 2 En(R). ; ;  + ( A 2 GLn(R)  1 (A) = ';12 1 (e  A) + 2 (1 ; e)  A;1 . 3.  | )-) . **1 PG, 415  P En(R).  1 . 6. ) 4]   ,     /1  (/ R,   1 )4 *( 415 )-) . **1 PGLn(R), 415  PEn(R). 3)  )   8  / 9  .  . .    . .   5]    

 3. %  R | PI- '  1, n > 3, H | /.%//  GLn (R),  En(R),   |  ,-+, / ) .%//( PH ,  ()  PEn(R). 0.      ) .%// PH . 6*)),   (/ Q 1 . 1)  )1  65),     . a 2 Q    x 2 Q,   . a  x  a = a. + (/ R  Q 1  ) *),  

(8a 2 Q)(9c b t s 2 R)t;1 s;1 2 Q  a = b  t;1 = s;1  c]: ;1)  () 5 1     

 4. %  R | % ) /  .% ,  ' Q, 1 2 R, n > 3, H | /.%//  GLn(R),  En(R),   |  ,-+, / ) .%//( PH ,  ()  P En(R). 0.      ) .%// PH . < ( )1 4 *     * 4.

  5. %  R | % ) /  .% ,  ' Q, 1 2 R, n > 3, a 2 GLn(R). 0. % % t 2 R, ),   t;1 2 Q    2(1 r 2 R, 1 6 i 6= j 6 n 3 , a;1  (1 + t  r  t  eji )  a   En(R). . 3   (/ Q . ,  aii  Q = ei  Q . e2i = ei :

(1)

+ ( t1 2 R, t;1 1 2Q

t1  ei  aik 2 R t1(1 ; ei )  aik 2 R    i k:  (1) aii  bi = ei , . bi 2 Q. + ( t2 2 R, t;1 2 2Q 

(2)

bi  aij  (a;1 )jj  t2 2 R: (3) +4),  c = 1 + a;1  (t2  r  t2  eji )a 2 En (R)    r 2 R  1 6 i 6= j 6 n. 6) *  6. 6]. %  n > 2, w 2 Rn  ejj , v 2 ejj  Rn, v  w = 0  2 ekk  w = 0,

2 v  ekk = 0    (1 j k. 0. 1 + w  v 2 En (R).

           

313

+ 4)

v1 = t1  ei  eji  a v2 = t1  (1 ; ei )  eji  a (4) w = a;1  t2  r  ejj : 3.  (2)   , , v1 v2  w 2 Rn  v1  w = v2  w = 0: (5) >  (, c = (1 + w  v1)(1 + w  v2 ). ?) .,  (1) v2  eii = 0  * )) 6, 1 + w  v2 2 En (R). + 4) w1 = ;bi  aij  (a;1 )jj  t2  reij + (a;1 )jj  t2  r  ejj (6)  w2 = w ; w1 .  (3), w1  w2 2 R,   *  bi   ,  v1  w1 = 0.  (5)   ,  v1  w2 = v1  (w ; w1) = 0  1 + w  v1 = (1 + w1  v1)(1 + w2  v1). <   (6)  (4) * ): ekk  w1 = 0, ejj  w2 = 0, . k 6= i j ,  * )) 6, c = (1 + w1  v1 )  (1 + w2  v1 )  (1 + w  v2 ) 2 En(R). 3) )1),   t = t2  t1 * ),  a;1  (1 + t  r  t  eji)  a 2 En (R). + 4 5 .   4. + * 4 5 ;  ;    1 a;1  (1 + t  r  t  eji )  a = 1 a;1  (1 + t  r  t  eji)  a  1 (1 + t  r  t  eji ) = 1(1 + t  r  t  eji): + 4) 1(b) = 1 (a)  c = a  b;1 . 3. c  (1 + t  r  t  eji)  c;1 =  (1 + t  r  t  eji) 4  /  (/ Rn. +5  )) ), * ),  1 + t2  r  t2  eji = c  (1 + t2  r  t2  eji )  c;1: P + ( T = t8  eii . 3. C  T  C ;1 = T  C  T  S  T  C ;1 = T  S  T T  (C  S  C ;1 ; S )  T = f0g    S 2 Rn. 6 T ;1 2 Qn , , C /   Rn  1(C ) = 1. 3. 1(b) = 1(a)  1 (a) = 1(b) = 1(a)    a 2 H . 3) 4 . ;)),    . .  (/ R   >. . A   7] ,  En(R) | ) (15   ( GLn (R) * n > 2.  14 * (( . .    )  .

   1]   . .,    . .     

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      . . 

        . . . 

 512.55

   ,    A((x))       A       !" : 1) A((x)) | %   & 2) A((x)) | ' '   & 3) A | ' '   .

Abstract K. I. Sonin, Regular rings of Laurent series, Fundamentalnaya i prikladnaya matematika 1(1995), 315{317.

The following conditions for the ring A((x)) of Laurent series over a ring A are equivalent: 1) A((x)) is a regular ring& 2) A((x)) is a semisimpleArtinian ring& 3) A is a semisimple Artinian ring.

           .  A((x))           A (  Pi=;s aix!! i , s > 0, ai 2 A). $,    !     1       

 1.     :

(1) A((x))    (2) A((x))       (3) A      . &     1     () ( . &    ) 1{3      A((x))     R.  1.   R  ,   A  .

   . &   a   A ,    g  R,  aga = a,     R  , A    R. .    !!  x       ),    ag0a = a,  g0 |     g. 0  g0 (  A,   A          .  2.   R  ,  A   ! "# $!    %  $  .

   . &  ,  ,      ,  ei |         A. 0( Pi=0 eei0xie1 :::  z= 1

      R, (  z . 0            1995, 1, N 1, 315{317. c 1995      !, "#    \%     "

316

. .  

P

j R |   , ,      f = 1 j =;s aj x  a;s 6= 0,  zR = fR. .    !!   ) x       )   fz = z , ,  a;s+i es + : : : + a0 ei + : : : + ai e0 = ei  ) i = 0 1 : : :. &(     ei ,  < 1. 3    f (  zR,  P ab0jexij = be;it 06=60,i  ,    g = 1 zg = f . 4    !j =;P t t 0 !  x  : P a0 = i=0 eibi.   -  ek,  k > t,  ek = a0 ek = ti=0 ei bi ek . &(     ek  ,   

,     ek    0. 6 ,  A    (   )   .

 3.  e |  $   A  eA | $$ '  '    A,  eR | $$ '  '    R.

   . 0 f |    eR. 6 ,  m R      m,  (,   ,,  fR = fx P 1  ,P f = i=0 ai xi  a0 6= 0. 0 (,   R ,    j 1 g= 1 j =0 bj x ,  fg = e,   ,   fbj 2 Agj =0 ,  a0 b0 = e  a0 bk + a1 bk;1 + : : : + ak b0 = 0  k > 1. & ( ,  bj   . 6     ,  ef = f , ,   , eai = ai  ) i. 3    eA   ,  b0,     a0b0 = e. 0 , b0  : : : bj ;1    ,  ,  ;a1 bj ;1 ; : : : ; aj b0 (  eA,       eA  ,  , bj ,    a0 bj = ;a1bj ;1 ; : : :; aj b0. 3   ,   ,     f 2 eR   fR = eR,      eR.

     1. (1))(3). 71]   ,            , : (a)    (   (   )   9 (b)    . 4  1      A,   2 |       (b). (3))(2). A = ni=1 ei A,  fei gni=1 |      , fei Agni=1 |   P      ;. 3   f  R       f = ni=1 fi ,  fi = ei f ,   , R = ni=1 ei R,  fei Rgni=1 |  e (   3)  e    R,   R |    . (2))(1).    )      ,  ,  72]. < ,    $       )  ) . =        $,        a       b,  a = a2 b. .       ,         .   1.     : (1) A((x))      (2) A((x)) # $ (   (3) A # $ (  .

      

317

   . 4   (1))(3)     1    !  , 

             . &       (3))(2)  ,   A |    ,  A((x)) |    . 3  A |     ,    1 A((x))        ,       . 4   (2))(1)      . ;    ;. . ?)   ;. ;. 3         .

 1]  .   : ,    . .2. | .: , 1979. 2] !  ".   . | .: , 1971.

'    (:   1995.