Let $R$ be a Noetherian domain of Krull-dimension $1$ (i.e. every non-zero prime ideal maximal). Let $M$ be a torsion free $R$-module . Let $K$ be the fraction-field of $R$ and let $r=\dim_K S^{-1}M=\dim_K K \otimes_R M $ (where $S=R \setminus \{0\}$ ). Suppose $r$ is finite. Under these conditions, if $M$ is finitely generated, then I can prove that $l(M/aM) \le r \cdot l(R/aR), \forall 0 \ne a \in R$ . My question is :

Is $l(M/aM) \le r \cdot l(R/aR), \forall 0 \ne a \in R$ even for such non-finitely generated modules $M$ with the other conditions remaining same ? If this is not true in general, is it true at least when $M$ is countably generated ?

Here $l(\cdot)$ denotes the "length" of the module.