$A\otimes B$ simple implies $A$ and $B$ simple

Question

Let $A$ and $B$ associative algebras (unital) over a field $\mathbb F$. Suppose that $A\otimes_{\mathbb F}B$ is a simple algebra. Are $A$ and $B$ simple algebras?

I know to prove this result when $B$ is a finite dimensional central simple algebra. But this case I don't whether it is true.

I appreciate any hint. Thank you.

Answer 1

If $J\subset A$ is an ideal, then so is $J\otimes_{\mathbb F} B$.