摘要
模的包络与覆盖理论在研究环模理论、同调代数、代数表示论中有着重要的作用。I(m,n)-内射与I(m,n)-平坦模可通过(m,n)-内射覆盖与(m,n)-平坦包络来研究。若R是一个环,左R-模M称为I(m,n)-内射的(右R-模N称为I(m,n)-平坦的),如果对任意(m,n)-内射左R-模G,Ext1(G,M)=0(Tor1(N,G)=0)。文中证明:若M是左R-模,则M是I(m,n)-内射的当且仅当M是一个左R-模I(m,n)-预覆盖的核;进而证明了在(m,n)-凝聚环上M是I(m,n)-内射左R-模当且仅当M=KL,其中K是内射左R-模,L是约化I(m,n)-内射左R-模;有限表现右R-模C是I(m,n)-平坦的当且仅当C是一个右R-模F(m,n)-预包络的上核。
The theory of envelopes and covers plays an important role in the theory of rings and modules,homological algebra,representation theory of algebras and so on.I(m,n)-injective and I(m,n)-flat modules are discussed by (m,n)-injective covers and (m,n)-flat envelopes.Let R be a ring.A left R-module M (respectively right R-module N) is called I(m,n)-injective (respectively I(m,n)-flat) if Ext1(G,M) =0(respectively Tor1(N,G) =0) for any (m,n)-injective left R-module G.In this paper,it is proved that a left R-module M is I(m,n)-injective if and only if M is a kernel of an (m,n)-injective precover of a left R-module.Suppose that R is a left (m,n)-coherent ring.It is shown that a left R-module M is I(m,n)-injective if and only if M is a direct sum of an injective left R-module and a reduced I(m,n)-injective left R-module; a finitely presented right R-module N is I(m,n)-flat if and only if N is a cokernel of a I(m,n)-flat preenvelope of a right R-module.
出处
《江南大学学报(自然科学版)》
CAS
2014年第2期218-221,共4页
Joural of Jiangnan University (Natural Science Edition)
基金
国家自然科学基金项目(11201063)
福建省科技厅基金项目(2013J05013)
福建省教育厅基金项目(JA11209)
莆田学院教改项目(JG201316)