Coretractable modules relative to a module

Abstract

In this paper, we introduce and study a generalization of coretractable modules. An [Formula: see text]-module [Formula: see text] is called coretractable relative to an[Formula: see text]-module[Formula: see text] if Hom[Formula: see text], for any proper submodule [Formula: see text] of [Formula: see text]. It is shown that a right [Formula: see text]-module [Formula: see text] is coretractable relative to [Formula: see text] if and only if [Formula: see text] is coretractable relative to [Formula: see text], for some [Formula: see text]. Also we show that if [Formula: see text] is a right quasi-duo ring, then there exists a right [Formula: see text]-module that it is coretractable relative to [Formula: see text], for any nonzero right [Formula: see text]-module [Formula: see text] if and only if [Formula: see text] is a left perfect local ring.