Generalized rigid modules and their polynomial extensions

TL;DR

This paper explores the properties of $\sigma$-rigid modules over rings with unity, focusing on their transfer to polynomial and triangular matrix extensions, thereby advancing the understanding of module rigidity in algebra.

Contribution

It extends the theory of $\sigma$-rigid modules by analyzing their behavior under module extensions like polynomial and triangular matrix modules.

Findings

$\sigma$-rigidness transfers to polynomial modules.

Results on $\sigma$-rigidness in triangular matrix modules.

New conditions for $\sigma$-rigidness transfer.

Abstract

Let $R$ be a ring with unity, $\sigma$ an endomorphism of $R$ and $M_R$ a right $R$-module. In this paper, we continue studding $\sigma$-rigid modules that were introduced by Gunner et al. \cite{generalized/rigid}. We give some results on $\sigma$-rigid modules and related concepts. Also, we study the transfer of $\sigma$-rigidness from a module $M_R$ to its extensions, as triangular matrix modules and polynomial modules.