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.

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