Philly Ivan Kimuli | Searcharxiv

arXiv2023

Characterizations of regular modules

Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.

math.AC↗

arXiv2022

Weakly-morphic modules

Let $R$ be a commutative ring, $M$ an $R$-module and $φ_a$ be the endomorphism of $M$ given by right multiplication by $a\in R$. We say that $M$ is {\it weakly-morphic} if $M/φ_a(M)\cong \ker(φ_a)$ 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 $k$.

math.RA↗