Let $R$ be a one-dimensional, noetherian and reduced ring. Let $M$ be a finitely generated and torsion-free $R$-module. I am looking for sufficient conditions on $M$ such that
$$\bigcap_{\mathfrak{m}} \mathfrak{m} M = 0$$ whenever $\mathfrak{m}$ runs through an infinite set of maximal ideals of $R$.
I was wondering if there is a well-known type of modules that satisfy this type of condition. This could be interpreted as follows: For every $m \in M$ the set of zeros of $m$ on $\operatorname{Spec}(R)$ is finite. Here the zeros of $m$ are given by those $\mathfrak{m}$ such that $m \in \mathfrak{m} M$.