Suppose $M$ is a finitely generated left module over a ring $R.$

We define the rank of $M$ as the minimal number of generators of $M.$

If in addition $M$ is free, then we define the free-rank of $M$ as minimal cardinality of a basis of $M.$

It is clear that $\text{rank}(M)\leq\text{free-rank}(M)$.

I want to know an example when $\text{rank}(M)<\text{free-rank}(M)$.

If $R$ is commutative, then they are equal; so $R$ must be non-commutative.