In this question F is a field and all algebras are finite dimensional F algebras.
Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is the center of D and the tensor product of A and B over F is isomorphic to M_n(D) for some n. Can we find all the elements of X?
(M_n(D) is the algebra of all n-by-n matrices with entries from D.)
It is Obvious that every central simple F algebra is in X. Are there some interesting elements of X?