Let $A$ be an $F$-algebra that's simple and central, and let $B$ be an $F$-algebra. I want to prove that the ideals of $A\otimes B$ has the form $A\otimes I$ where $I$ is an ideal of $B$.
Let $I$ be an ideal of $B$. Then, it is easy to show that $A \otimes I$ is an ideal of $A \otimes B$. So, let $U$ be an ideal of $A \otimes B$ and consider $I = \{b \in B : 1_A \otimes b \in U\}$. Again, it's easy to see that $I$ is an ideal of $B$ such that $A\otimes I \subset U$. Is it true that $U \subset A\otimes I$?
I'm trying to solve using the same strategy that someone use here
