有限生成模唯一性定理的新证明

有限生成模唯一性定理的新证明 A NEW PROOF OF THE INVARIANCE THEOREM OF THE FINITELY GENERATED MODULE 靳平 Jin Ping first-author 主理想整环上的有限生成模可唯一地分解为循环子模的直和,这是模论中的一个经典结果【1】,但其唯一性部分的证明很复杂。本文利用初等矩阵知识直接给出此唯一性定理的一个简洁证明,并将其中的主理想整环减弱为交换环。 In this paper, we give a new proof of the invariance theorem of the finitely generated module over a p. i. d. In fact, our proof is in the case in which the module is over a commutative ring. INVARIANCE THEOREM. Let M be a finitely generated module over a commutative ring D, and M=Dz 1 Dz 2 …Dz 8 =Dw 1 Dw 2 …Dw t where ann z 1 ann z 2 …ann z 8 and ann w 1 ann w2…ann w t and none of the components are 0. Then s=t and ann z t =ann w i , 1≤i≤s. 1986-04-01 2021-04-01 4