A Generalization of Mathieu Subspaces to Modules of Associative Algebras

Details A Generalization of Mathieu Subspaces to Modules of Associative Algebras A Generalization of Mathieu Subspaces to Modules of Associative Algebras

2010-05-24

arxiv.org/abs/1005.4259v2

Centr. Eur. J. Math. 8, No.6 (2010), pp.1132--1155

Mathematics

Representation Theory

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

Scientific paper

10.2478/s11533-010-0068-6

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.

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