Weakly-morphic modules

TL;DR

This paper introduces and studies weakly-morphic modules over commutative rings, characterizing their structure and properties, and explores their connections to ring regularity, torsion-freeness, and module decomposition.

Contribution

It defines weakly-morphic modules, characterizes them in terms of ring regularity, and relates their properties to torsion, divisibility, and module decomposition over various rings.

Findings

Weakly-morphic modules are characterized by regularity of elements in the quotient ring.

Over an integral domain, weakly-morphic modules are torsion-free if and only if they are divisible.

Finitely generated abelian groups are weakly-morphic if and only if they are finite, with specific primary component structures.

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 {\it 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 $\Bbb Z$-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 $(\Bbb Z_{p^k})^n$ for some non-negative integers $n$ and $k$.