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

References in corpus (8)

• A Vanishing Conjecture on Differential Operators with Constant Coefficients
• On rational functions orthogonal to all powers of a given rational function on a curve
• Some Results on the Vanishing Conjecture of Differential Operators with Constant Coefficients
• Generalizations of the Image Conjecture and the Mathieu Conjecture
• The Amazing Image Conjecture
• Mathieu Subspaces of Associative Algebras
• Analogue of the Duistermaat-van der Kallen Theorem for Group Algebras
• Mathieu Subspaces of Univariate Polynomial Algebras

Cited by in corpus (3)

• Mathieu subspaces of codimension less than n of Mat_n(K)
• Mathieu Subspaces of Associative Algebras
• Analogue of the Duistermaat-van der Kallen Theorem for Group Algebras