Lemma 11.4.4. Let $A$ be a $k$-algebra. Let $K$ be a central $k$-algebra which is a skew field. Then any two-sided ideal $I \subset A \otimes _ k K$ is of the form $J \otimes _ k K$ for some two-sided ideal $J \subset A$. In particular, if $A$ is simple, then so is $A \otimes _ k K$.

Proof. Set $J = \{ a \in A \mid a \otimes 1 \in I\}$. This is a two-sided ideal of $A$. And $I = J \otimes _ k K$ by Lemma 11.4.3. $\square$

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