Ideal/Submodule Query - Proving R is Principal Ideal Domain

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In summary, the conversation discusses a proof involving a principal ideal domain and its submodules. The proposition states that every submodule of a free module is finitely generated. The proof uses induction, starting with the rank one case where the submodule is isomorphic to an ideal in the principal ideal domain. The conversation ends with a request for help and clarification on how this proves the proposition.
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Ad123q
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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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  • #2
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.
 

1. What is a principal ideal domain (PID)?

A principal ideal domain is a commutative ring in which every ideal can be generated by a single element. This means that every ideal in a PID is of the form (a) = {ra | r ∈ R}, where a is a fixed element in the ring R and r ranges over all elements of R.

2. How do you prove that a ring is a principal ideal domain?

In order to prove that a ring R is a PID, you need to show that every ideal in R can be generated by a single element. This can be done by considering a general form of an ideal (a) and showing that it is equal to some other ideal in R. Alternatively, you can also show that every irreducible element in R is prime.

3. What are the properties of a principal ideal domain?

Some of the key properties of a PID include:

  • Every ideal is principal, i.e. generated by a single element.
  • Every irreducible element is prime.
  • Every PID is a unique factorization domain (UFD).
  • Every PID is also a Noetherian ring, meaning that every ascending chain of ideals terminates.

4. Can a principal ideal domain have zero divisors?

Yes, a PID can have zero divisors. This is because the definition of a PID only requires that every ideal be generated by a single element, not that the ring be an integral domain. However, in a PID, every irreducible element is prime, so the presence of zero divisors is limited.

5. Are all principal ideal domains Euclidean domains?

No, not all PIDs are Euclidean domains. A Euclidean domain is a PID with an additional property that allows for a division algorithm, which is not guaranteed in a general PID. However, every Euclidean domain is a PID.

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