Ideal/Submodule Query

  • Thread starter Ad123q
  • Start date
  • #1
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!
 

Answers and Replies

  • #2
mathwonk
Science Advisor
Homework Helper
2020 Award
11,093
1,295
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.
 

Related Threads on Ideal/Submodule Query

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
893
Replies
3
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
15
Views
1K
  • Last Post
Replies
1
Views
793
Replies
13
Views
985
Replies
5
Views
791
Replies
1
Views
2K
Top