Let $A$ be a simple central algebra and $B$ a commutative algebra -- what can be said about the 2-sided ideals of $A\otimes_k B$? (I am searching for a situation where the ideals of $A\otimes_k B$ are in 1:1 correspondence with the ideals of $B$)
McConnell/Robson Lemma 9.6.9 says the following:
Let $U$ be a simple ring with centre $k$ and $V$ any $k$-algebra. Then
(i) the map $A \mapsto A \otimes U$ provides a 1-1 correspondence between ideals of $V$ and ideals of $V \otimes_k U$;
(ii) if $V$ is a prime ring then $V \otimes U$ is a prime ring
so your 1:1 correspondence appears even if $B$ is not commutative.