Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has the IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=\mathrm{rank}(M)$.
Let $A$ be an $R$-linearly independent subset of $M$. What is an example of $A$ such that $|A|>n$?
If $R$ is a division ring, it must be $|A|âŚn$, but if $R$ is commutative, I think it is possible that $|A|>n$.