TIỂU LUẬN VÀNH VỚI ĐIỀU KIỆN HỮU HẠN Hạng đều (uniform rank)

TIỂU LUẬN VÀNH VỚI ĐIỀU KIỆN HỮU HẠN Hạng đều (uniform rank) HẠNG ĐỀU (UNIFORM RANK) Ta đã biết, môđun A có hạng hữu hạn với điều kiện E(A) là tổng trực tiếp của các môđun con không phân tích được. Bổ đề 5.16: Nếu môđun A là tổng trực tiếp hữu hạn n môđun con đều không phân tích được thì A không chứa bất kì tổng trực tiếp của n + 1 các môđun con khác không nào.

         N A.   -  B:     .    k(A). -  1 ,A 2  n  A 1  A 2    A 1 A 2    = rank(A 1 ) + rank(A 2  rank(A n ).         a  :E 1  E 2    i      E 1  2  E n  c  : cho A 1 ,A 2  n   i  i   i  A = A 1  A 2               A   A 1  A 2        A n + 1 A 1  A 2     n+1 = 0 suy ra A  A 1  A 2     A n+1  A 1  A 2       . b  :    a, rank(B).  rank(B)  B  e A.  rank(A)  rank(B) + rank(A/B).     rank(A) + 1  rank(B)  rank(A).    rank(B)  B  e A: ( e  e  e b  :   rank(B). (B    C  e A  B  e  e B).   A/B c  c + B    } = C/B   A/B. rank(C)  rank(A/B).  B  C  e A.  rank(A)  rank(B) + rank(C)            B.  1: , n A  e A.   rank(f(A))   e A. 2:  a, r.ann(A) = l.ann(A)   d, R A R   a, r.ann(A) = {n  R: Xn = 0} l.ann(A) = {m  R: mX = 0}  (XA) 2 = 0   l.ann(A)  n  r.ann(A) hay l.ann(A)r.ann(A)  (AX) 2 = 0   r.ann(A)  m  l.ann(A) hay r.ann(A)l.ann(A) r.ann(A) = l.ann(A).  ann(A)   2 = 0   ann(A) . c.B  ann(A)   B. d, R A R    ) A   A.   A    ann(A) = ann(A  ).  A 1  A 2 A 1 , A 2 ann(A 1 )  ann(A).  R A R  3:   R A R    A 1 ,A 2  n sao cho A 1  A 2     e R R R .  i = ann(A i  i   P i 