Noetherian Rings - Dummit and Foote - Chapter 15 - exercise 9

[Math Amateur]

I am reading Dummit and Foote Chapter 15, Section 15.1: Noetherian Rings and Affine Algebraic Sets.

Exercise 9 reads as follows:

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For k a field show that any subring of a polynomial ring k[x] containing k is Noetherian.

Give an example to show that such subrings need not be UFDs.

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Can someone please help me get started on this problem?

Peter

[Note: This has also been posted on MHF]

 

[jvicens]

Hi Peter,

Sure, I would be happy to help you get started on this problem.

First, let's define what a Noetherian ring is. A Noetherian ring is a ring where every ascending chain of ideals eventually stabilizes. In other words, if we have a sequence of ideals I1 ⊆ I2 ⊆ I3 ⊆ ... ⊆ In, then there exists some number N such that In = In+1 = ... = IN for all n ≥ N.

Now, let's look at the subring R of k[x] containing k. We know that k[x] is a Noetherian ring, since it is a polynomial ring over a field. Therefore, any subring of k[x] must also be Noetherian, since every ascending chain of ideals in R must also eventually stabilize.

For the second part of the exercise, we need to give an example of a subring of k[x] that is not a UFD (unique factorization domain). One example of such a subring is the ring of all polynomials in k[x,y] where the degree of y is even. This ring is not a UFD, since the polynomial y^2 can be factored as (y)(y), but these two factors are not irreducible in the ring.

I hope this helps you get started on the problem. Let me know if you have any further questions.