Generating set for finitely generated submodule of finitely generated module

Let $R$ be a ring,  $M$ a finitely generated $R$-module, and $N \subseteq M$ a f.g. submodule. Suppose $M$ has a minimal generating set of $m$ elements, then under what conditions may we assume that $N$ has a (not necessarily minimal) generating set of $n \leq m$ generators? I know the result holds, for example, when $R$ is a PID, but are there any other sufficient (or even necessary) conditions on $R$ (and/or $M,N$) to provide a complete characterization? A reference would be welcome.