强单投射模、强单内射模及其同调方法

发布时间：2018-02-09 10:40

本文关键词： 强单投射模强单内射模左单投射遗传环左单内射遗传环　出处：《四川师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文主要研究强单投射模与强单内射模及由它们确定的同调维数.设R是任何环,M,N是R-模,若对任意单R-模S及任意正整数i,有ExtRi(M,S =0,则M被称为强单投射模.对应地,若有ExtRi(S,M)= 0,则N被称为强单内射模.本文证明了在交换Noether环R中,有限生成强单投射模是投射模.在Krull维数为1的Noether整环R上,强单内射模是内射模.接着本文引入了环的左整体单投射维数(l.s.p.dim(R))和环的左整体单内射维数(l.s.i.dim(R))的概念,证明了l.s.p.dim(R)= 0当且仅当R是左V-环.以及s.i= dim(R)0当且仅当R是半单环.最后引入了左单投射遗传环与左单内射遗传环的概念,并给出了它们的等价刻画,证明了交换半遗传环是单投射遗传环;极大理想是有限生成的局部交换半遗传环(R,m)是单内射遗传环.
[Abstract]:In this paper, we mainly study strongly simple projective modules and strongly simple injective modules and the homological dimension determined by them. Let R be any ring Mnn is a R-module. If for any simple R-module S and any positive integer I, there is ExtRige MN _ S _ 0, then M is called strongly simple projective module. In this paper, it is proved that a finitely generated strongly simple projective module is a projective module in a commutative Noether ring R, and on a Noether domain R with a Krull dimension of 1, N is called a strongly simple injective module. In this paper, we introduce the concepts of left global projective dimension of rings and left global injective dimension of rings. In this paper, we prove that L.s.p.dimn RG = 0 if and only if R is a left V-ring and s.i = dim(R)0 if and only if R is a semi-simple ring. Finally, the concepts of left simple projective hereditary ring and left simple injective hereditary ring are introduced, and their equivalent characterizations are given. It is proved that the commutative semihereditary ring is a single projective hereditary ring, and the maximal ideal is a finitely generated locally commutative semihereditary ring RJ m) is a simple injective hereditary ring.
【学位授予单位】：四川师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.3

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