# Proof of every PID is Noetherian

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!

• 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. Commented Sep 4, 2020 at 7:00
• What is a principal ring? Commented Sep 4, 2020 at 7:11
• 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
Commented Sep 4, 2020 at 9:19
• 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
Commented Sep 4, 2020 at 9:20