= Semisimple quotient of a module endomorphism algebra
{title2=$\operatorname{End}(X)/J\cong\prod_aM_{m_a}(k)$}
For a finite-dimensional module over an <algebraically closed field>, write a <Krull-Schmidt decomposition> $X=\bigoplus_aM_a^{\oplus m_a}$ with distinct indecomposable types. Each endomorphism ring of $M_a$ is a <local endomorphism ring> with residue division algebra $k$, by the <finite-dimensional division algebra over an algebraically closed field> result. Modulo the <Jacobson radical> of $\operatorname{End}(X)$, the blocks of one type become $M_{m_a}(k)$ and all maps between different types vanish. A composite through a different indecomposable type cannot be invertible, since that would make one type a direct summand of the other. This yields the displayed product and the <Levi decomposition of a quiver automorphism group>.
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