2

I saw the proof of this proposition in here, but I have a question about this.

Definition of Noetherian ring is that ring is commutative, and every ideal of R is finitely generated, right? Principal ring is that ring's every ideal generated by single element, so it is clear. But I curious about isn't that not only PID, but also Principal ring is Noetherian?

Thank you!

Fawkes4494d3
  • 2,984
종민이
  • 95
  • 1
    In the question you have linked, the proof by OP is not correct, as pointed out in the comments. Check this post which is hyperlinked as the original in your link, since the linked one has been marked duplicate. – Fawkes4494d3 Sep 04 '20 at 07:00
  • What is a principal ring? – pancini Sep 04 '20 at 07:11
  • Our professor says, A commutative ring R is called a principal ring(principal ideal ring) if every ideal of R is principal (i.e., every ideal R is generated by a single element). – 종민이 Sep 04 '20 at 09:19
  • Principal ring is just generated by a single element which is also commutative, so we can say it is also Noetherian. Am I wrong? – 종민이 Sep 04 '20 at 09:20

1 Answers1

3

You're right, it has nothing to do with being a domain.

In a principal ideal ring, all ideals are finitely generated because a fortiori they are singly generated. That is, a one-element generating set is a finite generating set!

The thing is that in some contexts authors are just sticking to domains. So there is no big mystery about including the domain condition, it's just a context thing.

rschwieb
  • 153,510