A Generalization of Mathieu Subspaces to Modules of Associative Algebras

TL;DR

This paper extends the concept of Mathieu subspaces from associative algebras to modules, introduces stable and quasi-stable element sets, and explores their properties through examples involving matrix and polynomial algebras.

Contribution

It generalizes Mathieu subspaces to modules of associative algebras and defines related stable element sets, broadening the theoretical framework.

Findings

Properties of stable and quasi-stable element sets established

Examples from matrix and polynomial algebras analyzed

Generalization links Mathieu subspaces to module theory

Abstract

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $\mathcal A$, which was introduced recently in [Z4] and [Z6], to $\mathcal A$-modules $\mathcal M$. The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets $\sigma(N)$ and $\tau(N)$ of stable elements and quasi-stable elements, respectively, for all $R$-subspaces $N$ of $\mathcal A$-modules $\mathcal M$, where $R$ is the base ring of $\mathcal A$. We then prove some general properties of the sets $\sigma(N)$ and $\tau(N)$. Furthermore, examples from certain modules of the quasi-stable algebras [Z6], matrix algebras over fields and polynomial algebras are also studied.