= Artinian commutative ring is Noetherian theorem
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= Artinian rings are Noetherian
{c}
{synonym}
Every commutative <Artinian ring> is <Noetherian>. Its <nilradical> $N$ is nilpotent by the <nilradical of a commutative Artinian ring>, and the <Chinese remainder theorem> identifies the reduced quotient $R/N$ with the finite product of the residue fields at the finitely many <maximal ideals>. Each $N^i/N^{i+1}$ is an Artinian module over that product of <fields> and hence a <finite-dimensional vector space>. The finite <filtration> by these layers makes every <ideal> finitely generated.
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