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## Main Question or Discussion Point

Notations:

R denotes a commutative ring with identity

Terms:

R-module: a module whose base ring is R

Question:

Prove that if a nonzero commutative ring R with identity has the property that every finitely generated R-module is free then R is a field.

Idon't know how to complete the proof. The only thing I know is that if every finitely generated R-module is free then R-module R is also free, since it's generated by {1}.

Thanks for any help!

R denotes a commutative ring with identity

Terms:

R-module: a module whose base ring is R

Question:

Prove that if a nonzero commutative ring R with identity has the property that every finitely generated R-module is free then R is a field.

Idon't know how to complete the proof. The only thing I know is that if every finitely generated R-module is free then R-module R is also free, since it's generated by {1}.

Thanks for any help!