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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