Lemma 11.3.2 (0746)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 11: Brauer groups
Section 11.3: Wedderburn's theorem
Lemma 11.3.2 (cite)

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Lemma 11.3.2. Let $A$ be a $k$-algebra. If $A$ is finite, then

$A$ has a simple module,
any nonzero module contains a simple submodule,
a simple module over $A$ has finite dimension over $k$, and
if $M$ is a simple $A$-module, then $\text{End}_ A(M)$ is a skew field.

Proof. Of course (1) follows from (2) since $A$ is a nonzero $A$-module. For (2), any submodule of minimal (finite) dimension as a $k$-vector space will be simple. There exists a finite dimensional one because a cyclic submodule is one. If $M$ is simple, then $mA \subset M$ is a sub-module, hence we see (3). Any nonzero element of $\text{End}_ A(M)$ is an isomorphism, hence (4) holds. $\square$

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