Mathieu子空间到结合代数模的推广
A Generalization of Mathieu Subspaces to Modules of Associative Algebras
摘要：
本文首先将最近在[Z4]和[Z6]中引入的结合代数$\mathcal A$的Mathieu子空间的概念推广到$\mathcal A$ -模$\mathcal M$ .新引入的概念在某种意义上也推广了子模的概念。与这个新概念相关，我们还分别对$\mathcal A$ -模$\mathcal M$的所有$R$ -子空间$N$引入了稳定元和拟稳定元的集合$σ(N)$和$τ(N)$，其中$R$是$\mathcal A$的基环.然后我们证明了集合$σ(N)$和$τ(N)$的一些一般性质。此外，还研究了拟稳定代数[Z6]、域上矩阵代数和多项式代数的某些模的例子。
英文摘要：
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 $σ(N)$ 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 [Z6], matrix algebras over fields and polynomial algebras are also studied.
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