Lemma 11.4.7. Let $A$, $A'$ be two simple $k$-algebras one of which is finite and central over $k$. Then $A \otimes _ k A'$ is simple.

Proof. Suppose that $A'$ is finite and central over $k$. Write $A' = \text{Mat}(n \times n, K')$, see Theorem 11.3.3. Then the center of $K'$ is $k$ and we conclude that $A \otimes _ k K'$ is simple by Lemma 11.4.4. Hence $A \otimes _ k A' = \text{Mat}(n \times n, A \otimes _ k K')$ is simple by Lemma 11.4.5. $\square$

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