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

Year 2015, Volume: 44 Issue: 6, 1307 - 1313, 01.12.2015

T. Amouzegar Y. Talebi

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). 

Keywords

M-Cofaithful modules, Coretractable modules, Closed and coclosed submodules

References

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