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Ideal/Submodule Query 
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#1
Mar511, 01:48 PM

P: 19

Hi,
This has came up in a proof I'm going through, and need some guidance. The proposition is that if R is a principal ideal domain, then every submodule of a free module is finitely generated. The proof starts let F isomorphic to R^n be free, with basis {e1, ... , en}. Let P be a submodule of F. Use induction on n. Case n=1: F isomorphic to R (R is a module over R). Then P is a submodule of F which is isomorphic to R. This then implies that P is an ideal in R. This is where I'm stuck, I'm not sure how P a submodule of F which is isomorphic to R implies that P is an ideal in R. Any help appreciated  just ask if you need more background on the proposition. Thanks! 


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