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A generalization of Mathieu subspaces to modules of associative algebras

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Zhao W.

Open Mathematics

2010

tom 8

nr 6

1132-1155

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $$ \mathcal{A} $$, which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], 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 σ(N) and τ(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 σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.

Ograniczanie wyników

1 Open Mathematics

1 Zhao W.

1 2010

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A generalization of Mathieu subspaces to modules of associative algebras