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!
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.