Let $R$ be an integral domain, $M$ an $R$-module and $\{m_1,\dots,m_n\}$ a set of generators for $M$. I want to show that any set of elements of $M$ with cardinality $n+1$ is linearly dependent.
My approach was as follows: After reordering we can always choose a non empty maximal linearly independent set $\{m_1,\dots,m_s\}$ for $1\leq s \leq n$ since otherwise $M$ would be torsion and the result holds trivially. Then we can consider the free module $F$ generated by this set and is easy to show that $M/F$ is torsion. Hence given any $a_1,\dots,a_{n+1}$ in $M$ we can find non zero $x_1,\dots,x_{n+1}$ such that $x_1a_1,\dots,x_{n+1}a_{n+1}$ are in $F$. Then given that we know the result holds for free modules and given that $R$ is a domain we are done.
Now to show this holds for free modules suppose for contradiction that there exists a linearly independent set of $n+1$ elements in $M=R^n$ then we get an injective map $R^{n+1}\rightarrow R^n$ which is impossible by the famous exercise 2.11 in AM.
But exercise 2.11 does not assume the fact that $R$ is a domain so I was wondering whether there is a way around this that takes advantage of this further assumption.
