What commutative Noetherian rings are locally principal?

It is well-known that the localization (at any multiplicative set) of any principal ideal ring is again a principal ideal ring (PIR). A Dedekind domain localized at any prime ideal is a DVR, so is again a PIR. These domains show that a commutative Noetherian ring need not be a PIR to be locally a PIR. Do we have any characterizing conditions for a general commutative Noetherian ring to be locally a PIR? It seems to me that such conditions should be quite restrictive.

• You essentially gave all examples already: these rings are exactly the finite products of Dedekind domains and Artinian PIRs Apr 19, 2023 at 7:10
• @math54321- Thank you so much. I was unaware of such a characterization. Indeed, it is pretty restrictive. Best regards. Apr 19, 2023 at 15:49
• @math54321-Did you mean Noetherian PIRs? A Noetherian local ring with principal maximal ideal is a PIR. Apr 19, 2023 at 16:01
• I did mean Artinian. There is a classic result (maybe due to Hungerford) that a PIR is a finite product of quotients of PIDs. In particular a local non-Artinian PIR is necessarily a domain, i.e. a DVR Apr 19, 2023 at 16:09
• @math54321-Right. I should probably go back and reread Hungerford's paper a bit more carefully. In any event, if you care to write this up as an answer I would be pleased to accept it. Thanks again. Apr 19, 2023 at 17:05

Let $$R$$ be a Noetherian ring which is locally a PIR. Then $$\dim R \le 1$$, and we can write $$R = \prod_{i=1}^n R_i$$ as a finite product where each $$\operatorname{Spec}(R_i)$$ is connected, i.e. $$R_i$$ has no nontrivial idempotents. There are $$2$$ cases to consider:
(1) $$\dim R_i = 0$$: then $$R_i$$ is Artinian, hence a product of Artinian local rings, hence is Artinian local (since $$R_i$$ has no idempotents), so $$R_i$$ is an Artinian PIR.
(2) $$\dim R_i = 1$$: in this case $$R_i$$ must be a domain. It suffices to show that $$R_i$$ is locally a domain, since a Noetherian ring that is locally a domain is a finite product of domains, and as before since $$R_i$$ has no idempotents, the product can only have one factor. Now localizations of $$R_i$$ are localizations of $$R$$, so this reduces to the statement that a $$1$$-dimensional local PIR is a domain. This can be seen by Nakayama's lemma, or a direct argument: if $$S$$ is a local PIR with maximal ideal $$(x)$$, and $$a, b \in (x)$$ with $$ab = 0$$, then writing $$a = ux^n$$, $$b = vx^m$$ for some units $$u, v \in S \setminus (x)$$ (which is possible since $$\bigcap_{k \ge 0} (x)^k = 0$$) gives $$x^{n+m} = 0$$, i.e. $$\dim S = 0$$.
So each $$1$$-dimensional factor $$R_i$$ is a Noetherian domain of dimension $$1$$ which is locally a PIR, hence locally a DVR, i.e. $$R_i$$ is a Dedekind domain. Thus $$R$$ is a finite product of Artinian PIRs and Dedekind domains.