Consider the dual module V*. It can be identified with the mappings $M\to F$ with right FM-module structure given by (fm)(x)=f(mx). In particular, if m is not the identity map, the then fm is a constant map. Thus V*/constants is an n-dimensional module annihilated by all non-zero elements of M and hence cannot be generated by fewer than n elements (that is, a basis). Thus V* cannot be generated by fewer than n elements.
Consider the dual module V*. It can be identified with the mappings $M\to F$ with right FM-module structure given by (fm)(x)=f(mx). In particular, if m is not the identity map, the fm is a constant map. Thus V*/constants is an n-dimensional module annihilated by all non-zero elements of M and hence cannot be generated by fewer than n elements (that is, a basis). Thus V* cannot be generated by fewer than n elements.