Cover for Modules and Injective Modules

Authors : N. AMIRI

Pages : 111-116

View : 21 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let R be a commutative ring with identity and M be an R-module with Specinsert ignore into journalissuearticles values(M); \neq f. A cover of the R-submodule K of M is a subset C of Specinsert ignore into journalissuearticles values(M); satisfying that for any x \in K, x \neq 0, there is N \in C such that anninsert ignore into journalissuearticles values(x); \subset insert ignore into journalissuearticles values(N:M);. If we denote by J = \bigcapN \in C insert ignore into journalissuearticles values(N:M); and assume that M is finitely generated, then JM=M implies that M=0, M is called C-injective provided each R-homomorphism f : insert ignore into journalissuearticles values(N:M); \rightarrow M with N \in C can be lifted to an R-homomorphism l : R \rightarrow M. If R is a commutative Noetherian ring and C`=Specinsert ignore into journalissuearticles values(R);, where C`={insert ignore into journalissuearticles values(N:M);|N \in C}, then every C-injective R-module is injective.
Keywords : Commutative ring, D prime module cover, prime submodule, injective module, quasi injective and injective hull