I am not sure I understand the difference between free modules and finitely generated modules. I know that a free module is a module with a basis, and that a finitely generated module has a finite set of generating elements (ie any element of the ring can be expressed as a linear combination of those generators).
But then, is the difference just that the generators are not linearly independent ?
In that case, why do we specify sometimes "finitely generated free module" ? Because surely if the above is right (which I do not think it is), free would imply finitely generated so there is no need to specify finitely generated...
Thank you for your help !