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? 


cheers



Special subalgebras of central simple algebras
You want to know which algebras $A$ are such that $A\otimes B$ is central
simple for some algebra $B$. 
(All algebras and tensor products being over $F$.) If $Z(A)$
and $Z(B)$ are the centres of $A$ and $B$ then $Z(A)\otimes Z(B)$
is contained in the centre of $A\otimes B$. Hence $A$ and $B$ must both have
centre $F$. If $I$ is a two-sided ideal of $A$ then $I\otimes B$ is a two-sided
ideal of $A\otimes B$. As $A\otimes B$ is simple, then $I$ is zero or $A$,
that is $A$ is simple.


We conclude: the only algebras in $X$ are the central simple algebras over $F$.


