I would be grateful if anyone could give me a reference regarding the following question.
Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar multiples of identity (so called central algebras). Assume that both $A$ and $B$ have a unique non-trivial two-sided ideal, denote it respectively by $J_A$ and $J_B$. I'm interested about the ideal structure of a tensor product $A \otimes B$ (over $F$). The obvious non-trivial ideals of $A \otimes B$ are $J_A \otimes J_B$, $A \otimes J_B$, $J_A \otimes B$ and $J_A\otimes B + A \otimes J_B$. Are there some conditions on $A$ and $B$ that guarantee these are the only non-trivial ideals of $A \otimes B$?
I'm particularly interested in a case when $A=B$ is the algebra of $F$-linear maps on some vector space over $F$ of infinite countable dimension. Then the only non-trivial ideal of $A$ is the ideal $J_A$ of finite rank maps. Are all ideals of $A \otimes A$ of the above form?