Lemma 11.4.4 (074C)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 11: Brauer groups
Section 11.4: Lemmas on algebras
Lemma 11.4.4 (cite)

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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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