Let $R$ be a commutative ring with unity and $M$ be a free $R$-module. Let $S$ be a finite basis of $M$. If $T\subset M$ has cardinality greater than $S$, I am wondering whether $T$ is linear dependent over $R$.
I know that this is true when $R$ is field. But I am not sure if $R$ is not field.
If it is not true when $R$ is not field, could you suggest me an example such that $T$ is linear independent over $R$?