I_(m,n)-内射与I_(m,n)-平坦模

I_(m,n)-Injective and I_(m,n)-Flat Modules

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=KL,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.