Suppose M is a finitelly 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.

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.