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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
Please don't add "thank you" as an answer. If you are the same person as the original question poster, please visit this page for help consolidating your two user profiles. That will allow you to add comments, vote up, and accept answers for this question. – Willie Wong Feb 15 '13 at 15:22

Here are very simple examples :

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

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

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$\mathbb{Z}$ is finitely generated – Chen M Ling Jun 3 at 17:09

The basis of a free module need not be finite, so free does not imply finitely generated. You are correct that for a general finitely-generated module, there may be relations between the generators, i.e. two different linear combinations of generators (with coefficients in the base ring) can have represent the same element of the module.

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