Mathieu Subspaces of Associative Algebras
Mathieu Subspaces of Associative Algebras
复制标题
DOI：
10.1016/j.jalgebra.2011.09.036
复制
发表时间：
2010-05
期刊：
影响因子：
--
通讯作者：
Wenhua Zhao
中科院分区：
文献类型：
--
作者：
Wenhua Zhao
在Mathieu猜想（Mathieu, 1997 [M]）、像猜想（Zhao, 2010 [Z3]）和著名的雅可比猜想（Keller, 1939 [K]；另见Bass et al.， 1982 [BCW]和van den Essen， 2000 [E1]）的推动下，最近在Zhao (2010) [Z4]中引入了matthieu子空间的概念，作为理想概念的自然推广。本文首先研究了结合代数在域上的Mathieu子空间的根中的代数元素，并证明了具有代数根的Mathieu子空间的一些性质和刻画。然后，我们给出了任意交换环上的强简单代数（没有非平凡Mathieu子空间的代数）和任意域上的准稳定代数（其所有子空间不包含代数的单位元的代数都是Mathieu子空间）的一些刻画或分类。此外，还完全确定了域上矩阵代数的协维一Mathieu子空间和极小非平凡Mathieu子空间。
Motivated by the Mathieu conjecture (Mathieu, 1997 [M]), the image conjecture (Zhao, 2010 [Z3]) and the well-known Jacobian conjecture (Keller, 1939 [K]; see also Bass et al., 1982 [BCW] and van den Essen, 2000 [E1]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in Zhao (2010) [Z4] for associative algebras. In this paper, we first study algebraic elements in the radicals of Mathieu subspaces of associative algebras over fields and prove some properties and characterizations of Mathieu subspaces with algebraic radicals. We then give some characterizations or classifications for strongly simple algebras (the algebras with no non-trivial Mathieu subspaces) over arbitrary commutative rings, and for quasi-stable algebras (the algebras all of whose subspaces that do not contain the identity element of the algebra are Mathieu subspaces) over arbitrary fields. Furthermore, co-dimension one Mathieu subspaces and the minimal non-trivial Mathieu subspaces of the matrix algebras over fields are also completely determined.