Let $R$ be a commutative ring with unity and $M$ be a finitely generated free $R$-module.
Let $S$ be a finite subset of $M$ generating $M$ as $R$-module.
From this, can we say that $M$ has finite basis?
If $R$ is a field, it is the case because the maximal linearly independent set in $S$ gives rise to the basis of $M$.
But when $R$ is just commutative ring not being a field, we cannot apply this argument.
If $M$ also has finite basis, can we say further that the number of basis is equal or less than $|S|$?