  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.
