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.