Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes.

Over non-noetherian rings the notion of associated primes does not behave very well, and it is sometimes appropriate to rather consider the so-called weakly associated primes of a module. (A prime ideal of $R$ is _weakly associated_ with an $R$-module $L$ if it is a minimal prime of the annihilator of an element of $L$; see Bourbaki, AC.IV.1 Exer. 17, for basic facts about this notion.)

Unfortunately, if $M\rightarrow N$ is an essential morphism of $R$-modules then $M$ and $N$ need not have the same weakly associated primes. (This happens for example over every non-noetherian valuation ring with maximal ideal of finite type.)

This leads to the following question:

> What are examples of (classes of) non-noetherian rings over which weakly associated primes do not change along essential morphisms?