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- Feb 5, 2012

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Want to confirm my understanding about Free and Finitely Generated modules. I want to know whether the following ideas are correct. Thank you for all your help.

1) Is every free module a finitely generated module?

No. Because a free module may have an infinite basis. So we cannot say it's finitely generated. However if the free module has a finite basis it's finitely generated.

2) Is every finitely generated module a free module?

No again. If \(M\) is a \(R\)-module which is finitely generated by a set \(S=\{x_1,\,x_2,\,\cdots,\,x_n\}\subset M\) then for each element \(x\in M\) we have,

\[x=r_1 x_1+\cdots+r_n x_n\]

where \(r_1,\cdots,r_n\in R\).

However we don't know whether \(S\) is linearly independent. So generally \(M\) is not free.

Am I correct?