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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!