FP-投射模與強(qiáng)GFP-內(nèi)射模
發(fā)布時(shí)間：2018-08-03 11:57
【摘要】：本文主要研究了FP-投射模和強(qiáng)GFP-內(nèi)射模.R-模M稱為FP-投射模是指對(duì)所有的有限表現(xiàn)模N,都有ExtR1(M,N)=0；E是強(qiáng)GFP-內(nèi)射模是指任意R-模B的任意有限表現(xiàn)子模A,任何A到E的同態(tài)能提升為B到E的同態(tài).在第二章中給出了FP-投射模及強(qiáng)GFP-內(nèi)射模的等價(jià)刻畫及其基本性質(zhì).在第三章中證明了每個(gè)模是FP-投射模當(dāng)且僅當(dāng)每個(gè)模是強(qiáng)GFP-內(nèi)射模；當(dāng)且僅當(dāng)每個(gè)有限表現(xiàn)模是內(nèi)射模.也證明了當(dāng)R是左Noether環(huán)時(shí),則每個(gè)模是FP-投射模(強(qiáng)GFP-內(nèi)射模)當(dāng)且僅當(dāng)R是半單環(huán).而當(dāng)R是左凝聚環(huán)時(shí),每個(gè)模是FP-投射模(強(qiáng)GFP-內(nèi)射模)當(dāng)且僅當(dāng)R是VN-正則環(huán)且是左自內(nèi)射環(huán).然后引入了左FP-遺傳環(huán)的概念.證明了R是左FP-遺傳環(huán)當(dāng)且僅當(dāng)每個(gè)有限表現(xiàn)模的內(nèi)射維數(shù)至多為1.最后定義了模的強(qiáng)左FP-投射維數(shù)及環(huán)的強(qiáng)左FP-投射維數(shù),證明了R的強(qiáng)左FP-投射維數(shù)為0當(dāng)且僅當(dāng)每個(gè)模是FP-投射模.R的強(qiáng)左FP-投射維數(shù)至多為1當(dāng)且僅當(dāng)R是左FP-遺傳環(huán).
[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.
【學(xué)位授予單位】：四川師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O153.3
本文編號(hào)：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.
【學(xué)位授予單位】：四川師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O153.3
【參考文獻(xiàn)】
相關(guān)期刊論文 前6條
1 楊同海 ,陳建龍 ,唐懷鼎;fp~*-內(nèi)射,fp-平坦模和 Coherent 環(huán)[J];安徽師大學(xué)報(bào)(自然科學(xué)版);1988年03期
2 蔣方明;;f-投射模的直積和正向極限[J];南京理工大學(xué)學(xué)報(bào)(自然科學(xué)版);1993年06期
3 徐龍玉;汪明義;;關(guān)于零化子凝聚環(huán)[J];四川師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2006年02期
4 苗佳晶;;關(guān)于P-投射模[J];吉林師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2007年03期
5 俱鵬岳;王欣欣;;關(guān)于FP-內(nèi)射覆蓋的幾點(diǎn)注記[J];紹興文理學(xué)院學(xué)報(bào)(自然科學(xué));2010年01期
6 黃影;;關(guān)于FP-投射維數(shù)的若干性質(zhì)[J];沈陽師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2008年01期
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