Weakly-morphic modules

Abstract

Let R be a commutative ring, M an R-module and $\varphi _a$ be the endomorphism of M given by right multiplication by $a\in R$. We say that M is weakly-morphic if $M/\varphi _a(M)\cong \ker (\varphi _a)$ as R-modules for every a. We study these modules and use them to characterise the rings $R/\text {Ann}_R(M)$, where $\text {Ann}_R(M)$ is the right annihilator of M. A kernel-direct or image-direct module M is weakly-morphic if and only if each element of $R/\text {Ann}_R(M)$ is regular as an endomorphism element of M. If M is a weakly-morphic module over an integral domain R, then M is torsion-free if and only if it is divisible if and only if $R/\text {Ann}_R(M)$ is a field. A finitely generated $\mathbb {Z}$-module is weakly-morphic if and only if it is finite; and it is morphic if and only if it is weakly-morphic and each of its primary components is of the form $(\mathbb {Z}_{p^k})^n$ for some non-negative integers n and k.