= Finite length of a commutative Artinian ring
{title2=$\ell_R(R)<\infty$}
A commutative <Artinian ring> has finite <composition length> as a <module> over itself. It has finitely many maximal <ideals>, and its <Jacobson radical> $J$ is nilpotent. The <Chinese remainder theorem> makes $R/J$ a finite product of <fields>; each <Artinian module> $J^i/J^{i+1}$ has finite-dimensional components over those <fields>. Adding their lengths along the finite radical filtration proves the assertion, and in particular the <ascending chain condition>.
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