Rank of free module and its free submodule

Question

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.

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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.

Answer 1

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 $n$.