A Generalization of Mathieu Subspaces to Modules of Associative Algebras

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.

A new case has been added; some mistakes and misprints have been corrected. Latex, 31 pages

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