- #1
Ad123q
- 19
- 0
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!
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!