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

Authors : T. Amouzegar, Y. Talebi

Pages : 1307-1313

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Publication Date : 2015-12-01

Article Type : Research

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 ∈ H om R ( 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 Hom R ( 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 Hom R ( N,M ) , where S = End R ( M ) . Some applications are: (1) the , connection between M' s being a lifting module and End R ( 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 End R ( M ) . 
Keywords : M-Cofaithful modules, Coretractable modules, Closed and coclosed submodules