The Stacks project

Lemma 11.7.2. Let $A$ be a finite central simple algebra over $k$, and let $B$ be a simple subalgebra of $A$. If $B$ is a central $k$-algebra, then $A = B \otimes _ k C$ where $C$ is the (central simple) centralizer of $B$ in $A$.

Proof. We have $\dim _ k(A) = \dim _ k(B \otimes _ k C)$ by Theorem 11.7.1. By Lemma 11.4.7 the tensor product is simple. Hence the natural map $B \otimes _ k C \to A$ is injective hence an isomorphism. $\square$

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