# Difference between free and finitely generated modules

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 !

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Thank you very much for your reply everyone ! –  user62423 Feb 15 '13 at 14:58

$$\text{As an } \mathbb Z \text{-module, } \mathbb Z/2\mathbb Z \text{ is finitely generated but not freely generated.}$$
$$\text{As an } \mathbb Z \text{-module, } \bigoplus_{\mathbb N} \mathbb Z \text{ is freely generated but not finitely generated.}$$