M-Cofaithful modules and correspondences of closed submodules with coclosed submodule

Abstract

In this paper we introduce and investigate M-cofaithful modules. A module N ∈ σ[M] is called M-cofaithful if for every $0\neq$ f ∈ HomR(N,X) with X ∈ σ[M], Hom$_R(X,M)f \neq 0$. We show that if N is an M-cofaithful weak supplemented module and HomR(N, M) a noetherian S-module, then there exists an order-preserving correspondence between the coclosed R-submodules of N and the closed S-submodules of HomR(N,M), where S = EndR(M). Some applications are: (1) the ,connection between M's being a lifting module and EndR(M)'s being an extending ring; (2) the equality between the hollow dimension of a quasi-injective coretractable module M and the uniform dimension of EndR(M).