Prove the colums are linearly dependent over an integral domain

Question

Let $R$ be an integral domain, and let $ M ∈\mathcal{M}_{m,n}(R)$, with $m<n$. I want to prove that the columns of $M$ are linearly dependent over $R$ but don’t know how to do. Thanks for any hint in advance.

Answer 1

Edit: Apologies, I've just seen you asked for a hint and may not have wanted a full solution. In case you hadn't read my previous answer yet:

Do you know any results from linear algebra that might let you deduce this over a field? Then given that an integral domain embeds into its field of fractions, can you extend this from there?

If you really get stuck this was my original answer:

Over a field, a proof that the row rank is equal to the column rank is given on Wikipedia here. Then since $m<n$ we must have a dependence. Then for an integral domain $R$, in its field of fractions we must have a dependence in the columns $c_i$, say $$\frac{a_1}{b_1}c_1+\cdots+\frac{a_n}{b_n}c_n=0$$ for some $a_i,b_i\in R$. Then multiplying through by $b_1\cdots b_n$ we have $$b_2\cdots b_na_1c_1+\cdots+b_1\cdots b_{n-1}a_nc_n=0$$