FP-投射模与强GFP-内射模
发布时间：2018-08-03 11:57
【摘要】：本文主要研究了FP-投射模和强GFP-内射模.R-模M称为FP-投射模是指对所有的有限表现模N,都有ExtR1(M,N)=0；E是强GFP-内射模是指任意R-模B的任意有限表现子模A,任何A到E的同态能提升为B到E的同态.在第二章中给出了FP-投射模及强GFP-内射模的等价刻画及其基本性质.在第三章中证明了每个模是FP-投射模当且仅当每个模是强GFP-内射模；当且仅当每个有限表现模是内射模.也证明了当R是左Noether环时,则每个模是FP-投射模(强GFP-内射模)当且仅当R是半单环.而当R是左凝聚环时,每个模是FP-投射模(强GFP-内射模)当且仅当R是VN-正则环且是左自内射环.然后引入了左FP-遗传环的概念.证明了R是左FP-遗传环当且仅当每个有限表现模的内射维数至多为1.最后定义了模的强左FP-投射维数及环的强左FP-投射维数,证明了R的强左FP-投射维数为0当且仅当每个模是FP-投射模.R的强左FP-投射维数至多为1当且仅当R是左FP-遗传环.
[Abstract]:In this paper, we mainly study FP-projective modules and strong GFP-injective modules. R-module M is called FP-projective module, which means that for all finite representation modules N, there is ExtR1 (Mon N) 0. If E is a strong GFP-injective module, it means any finite representation submodule A of any R-module B, and any homomorphism from A to E can be promoted to the homomorphism of B to E. In chapter 2, the equivalent characterizations and properties of FP-projective modules and strong GFP-injective modules are given. In chapter 3 we prove that every module is FP-projective if and only if every module is a strong GFP-injective module if and only if every finite representation module is an injective module. It is also proved that every module is FP-projective module (strongly GFP-injective module) if and only if R is a semi-simple ring when R is a left Noether ring. If R is a left coherent ring, every module is FP-projective module (strong GFP-injective module) if and only if R is a VN-regular ring and a left self-injective ring. Then the concept of left FP-hereditary ring is introduced. It is proved that R is a left FP-hereditary ring if and only if the injective dimension of every finite representation module is at most 1. Finally, we define the strongly left FP-projective dimension of a module and the strongly left FP-projective dimension of a ring. It is proved that the strongly left FP-projective dimension of R is 0 if and only if every module is a FP-projective module. The strong left FP-projective dimension of every module is at most 1 if and only if R is a left FP-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】：2015
【分类号】：O153.3
本文编号：2161672
[Abstract]:In this paper, we mainly study FP-projective modules and strong GFP-injective modules. R-module M is called FP-projective module, which means that for all finite representation modules N, there is ExtR1 (Mon N) 0. If E is a strong GFP-injective module, it means any finite representation submodule A of any R-module B, and any homomorphism from A to E can be promoted to the homomorphism of B to E. In chapter 2, the equivalent characterizations and properties of FP-projective modules and strong GFP-injective modules are given. In chapter 3 we prove that every module is FP-projective if and only if every module is a strong GFP-injective module if and only if every finite representation module is an injective module. It is also proved that every module is FP-projective module (strongly GFP-injective module) if and only if R is a semi-simple ring when R is a left Noether ring. If R is a left coherent ring, every module is FP-projective module (strong GFP-injective module) if and only if R is a VN-regular ring and a left self-injective ring. Then the concept of left FP-hereditary ring is introduced. It is proved that R is a left FP-hereditary ring if and only if the injective dimension of every finite representation module is at most 1. Finally, we define the strongly left FP-projective dimension of a module and the strongly left FP-projective dimension of a ring. It is proved that the strongly left FP-projective dimension of R is 0 if and only if every module is a FP-projective module. The strong left FP-projective dimension of every module is at most 1 if and only if R is a left FP-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】：2015
【分类号】：O153.3
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