Example of strict inclusion for the localization of associated primes

Question

Let $A$ be a commutative ring and $M$ an $A$-module. It is well known that $$\operatorname{Ass} M\cap\operatorname{Spec}S^{-1}A\subset\operatorname{Ass}S^{-1}M,$$ and that equality holds if $A$ (or $M$) is Noetherian. Presumably strict inclusion is possible if $A$ is not Noetherian, but I couldn't think of an example (I couldn't find any reference mentioning this, either.) So naturally I turn to my trustworthy math stack exchange.

Answer 1

Let $R=\mathbb F_2^{\mathbb N}$ be a countable direct product of copies of the field with two elements. This is a von Neumann regular ring and thus $\operatorname{Ann}(a)$ is principal generated by an idempotent for all $a\in R$. If $P$ is a prime ideal of $R$ containing the ideal $\mathbb F_2^{(\mathbb N)}$, then $P$ is not associated. Otherwise, $P=Re$ with $e\in R$. Since $\mathbb F_2^{(\mathbb N)}\subseteq P$ we get $e=1$, a contradiction. Set $S=R-P$. Then $\operatorname{Ass}R\cap\operatorname{Spec}S^{-1}R=\emptyset$. On the other hand, $S^{-1}R$ is a field, and thus $\operatorname{Ass}S^{-1}R\ne\emptyset$.