Ideal/Submodule Query


by Ad123q
Tags: ideal or submodule, query
Ad123q
Ad123q is offline
#1
Mar5-11, 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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mathwonk
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#2
Mar5-11, 04:12 PM
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P: 9,422
the statement is that every submodule of a finitely generated free module is finitely generated and free of same or smaller rank. the proof is by induction, and the rank one case is by the definition of a pid.


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