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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?


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up vote 5 down vote accepted

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 $F$.

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Thanks. It was much easier than I expected! The center of A and B are both F because $Z(A) \otimes Z(B) = Z(A \otimes B) = F$ and so considering the dimensions of both sides we get $Z(A) = Z(B) = F.$ Thanks again. – carlos Jun 23 '10 at 9:54

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