Mathieu subspaces of associative algebras

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.