Given $(R,m)$, a Noetherian local ring, and $M$ a nonzero $R$-module. I was wondering if there is a way to describe the elements of $\displaystyle\bigcap_{P\in Ass_RM} P$. In particular, when $M$ is a finite $R$-module we have that $$\displaystyle\bigcap_{P\in Ass_RM} P=\sqrt{Ann_RM}.$$ I was curious if there was a similar statement in the case that $M$ is not necessarily finitely generated.
Thanks in advance.
I think the general statement is that $\sqrt{\operatorname{Ann}_R M}$ is the intersection of the supporting primes (EDIT: never mind, this is false!), and, when $M$ is finitely generated, the minimal supporting primes are associated primes.
To see what can go wrong when $M$ is not finitely generated, take $R=\mathbb{Z}$, $M = \bigoplus_{k\geq 1}\mathbb{Z}/p^k$ (localize at $(p)$ if you want a local example). Then the annihilator is $(0)$, but the only associated prime is $(p)$.
If I had to describe $\bigcap_{P\in\operatorname{Ass}_R M} P$, I might write it as $\bigcap_{M'} \sqrt{\operatorname{Ann}_R M'}$, where the intersection is over all finitely generated submodules $M'\subset M$. It seems hard to say anything simpler without serious assumptions on the lattice of prime ideals of $R$.