MODULES IN WHICH EVERY INJECTIVE ENDOMORPHISM HAS STRONGLY ESSENTIAL IMAGE

Abstract

A module $M$ is called co-Hopfian if every injective $R$-endomorphism of $M$ is an isomorphism. This article introduces the concept s-cH module which is a generalized co-Hopfian. If $M$ is an $R$-module and every injective $R$-endomorphism of $M$ has a strongly essential image, then $M$ is said to be s-cH. We provide some properties of these module and we provide an example of a module which is not s-cH. Also we provide an example of a module which is s-cH, but not co-Hopfian. We show that if an $R$-module $M$ has a descending chain on non strongly essential modules, then $M$ is s-cH. Also we investigate the behavior of s-cH modules under a direct product.