= Localization of associated primes
{title2=$\operatorname{Ass}_{R_P}(M_P)=\{QR_P:Q\in\operatorname{Ass}_R(M),\ Q\subseteq P\}$}
For a commutative <Noetherian ring>, an <annihilator> $Q=\operatorname{Ann}_R(m)$ contained in $P$ localizes to the <annihilator> of the nonzero $m/1$. Conversely, contract an associated prime of $M_P$ represented by $m/1$ to an <ideal> $Q$. Choose finitely many generators of $Q$ and one denominator outside $P$ clearing all their annihilation equations. For this denominator $s$, the element $sm$ is nonzero and has <annihilator> exactly $Q$. This proves the formula; finite generation of the contracted <ideal> is the key use of <Noetherianity>.
// Destination: noncommutative-algebra.bigb
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