Localization of associated primes (${\mathrm{Ass}}_{R_P}(M_P)=\{Q{R}_P:Q\in {\mathrm{Ass}}_R(M),Q\subseteq P\}$)

For a commutative Noetherian ring, an annihilator $Q={\mathrm{Ann}}_R(m)$ 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