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    A Generalization of Mathieu Subspaces to Modules of Associative Algebras

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    We first propose a generalization of the notion of Mathieu subspaces of associative algebras A\mathcal A, which was introduced recently in [Z4] and [Z6], to A\mathcal A-modules M\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)\sigma(N) and τ(N)\tau(N) of stable elements and quasi-stable elements, respectively, for all RR-subspaces NN of A\mathcal A-modules M\mathcal M, where RR is the base ring of A\mathcal A. We then prove some general properties of the sets σ(N)\sigma(N) and τ(N)\tau(N). Furthermore, examples from certain modules of the quasi-stable algebras [Z6], matrix algebras over fields and polynomial algebras are also studied.Comment: A new case has been added; some mistakes and misprints have been corrected. Latex, 31 page