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.
It is always true that $n\le m$ when $R$ is an integral domain. If one has $n>m$ elements in $R^m$ one can regard them as the rows of an $n\times m$ matrix $A$. Then there is a nonzero vector $v\in K^n$ with $vA=0$ where $K$ is the field of fractions of $R$. Then we can multiply $v$ by a common denominator to bring in into $R$. It is clear then that the rows of $A$ are not the basis of a free module of rank $n$.