Register to reply

Ideal/Submodule Query

by Ad123q
Tags: ideal or submodule, query
Share this thread:
Ad123q
#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!
Phys.Org News Partner Science news on Phys.org
Physical constant is constant even in strong gravitational fields
Montreal VR headset team turns to crowdfunding for Totem
Researchers study vital 'on/off switches' that control when bacteria turn deadly
mathwonk
#2
Mar5-11, 04:12 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,499
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.


Register to reply

Related Discussions
Real (non-ideal) op-amps - textbook query Engineering, Comp Sci, & Technology Homework 1
Module and submodule Calculus & Beyond Homework 2
Module and submodule Calculus & Beyond Homework 1
Basis for a submodule? Calculus & Beyond Homework 1
Ideal Gases. Query Introductory Physics Homework 2