= Minimal primes are associated primes
{title2=$\operatorname{Min}_R(I)\subseteq\operatorname{Ass}_R(R/I)$}
For a <Noetherian ring> and a proper <ideal> $I$, every minimal prime $P$ over $I$ is an <associated prime of a module> $R/I$. The localized quotient has one prime, so its finitely generated <maximal ideal> is nilpotent and its <socle> is nonzero. An element with <annihilator> $PR_P$ can be lifted to the quotient. Clearing denominators for a finite generating set of $P$ gives a nonzero element with <annihilator> exactly $P$.
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