Mathieu Subspaces of Associative Algebras

Details Mathieu Subspaces of Associative Algebras Mathieu Subspaces of Associative Algebras

2010-05-24

arxiv.org/abs/1005.4260v2

Mathematics

Rings and Algebras

A new case of Mathieu subspaces has been added; some mistakes and misprints have been corrected. Latex, 42 pages

Scientific paper

Motivated by the Mathieu conjecture [Ma], the image conjecture [Z3] and the well-known Jacobian conjecture [K] (see also [BCW] and [E]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in [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 spaces) 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.

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