Let $R$ be an integral domain.
Let $M$ be a free $R$-module of finite rank, say $m$.
Let $N$ be a free submodule of $M$ of finite rank say $n$.
Q. Under what conditions on $R$ among Noetherian/UFD/Dedekind/local, it is always true that $n\leq m$?
When the inequality holds for specific assumption on domain mentioned above, then please suggest reference for proof.
When $R$ is PID, we always have inequality $n\leq m$; I know its proof. I am considering some non-PID's which are not so bad, namely the four mentioned above.
Note also that submodules of free modules over noetherian domain are not necessarily free; in this regard, I am already considering in the question that $N$ is a submodule which is also free.