# Example of a domain where all irreducibles are primes and that is not a GCD domain

One has the following relations for a domain $R$:

$R$ GCD domain $\Rightarrow$ All irreducible elements are prime

$R$ PID $\Rightarrow$ $(R$ GCD domain $\land$ $R$ statisfies ACCP$)$

$R$ UFD $\Leftrightarrow$ $(R$ GCD domain $\land$ $R$ satisfies ACCP$)$

$R$ satisfies ACCP $\Rightarrow$ $(R$ UFD $\Leftrightarrow$ All irreducible elements are prime$)$

$R$ GCD domain $\Rightarrow$ $(R$ UFD $\Leftrightarrow$ $R$ atomic domain$)$

So what is the difference between a GCD domain and a domain where all irreducible elements are prime? What is the "weakest" predicate $P$ such that

$R$ GCD domain $\Leftrightarrow$ $($All irreducible elements are prime $\land$ $P(R))$ ?

I'm very far from being an expert on these things, but I suspect that the answer might be the trivial (and disappointing?) one: the property which distinguishes GCD domains is ... the existence of GCDs.

Compare the chain of (strict) class inclusions

$\qquad$ UFDs $\subset$ bounded factorization domains $\subset$ ACCP domains $\subset$ atomic domains

to the chain

$\qquad$ UFDs $\subset$ GCD domains $\subset$ Schreier domains $\subset$ domains where irreducibles are prime.

I don't think there is any "interaction" (beyond the inclusion) within either chain. The relations that you quote have to do with interactions between the two chains, and basically they boil down to the fact that the only thing they have in common is "UFDs", i.e.,

$\qquad$ any class from the first chain $\cap$ any class from the second chain = UFDs,

plus the fact that PIDs $\subset$ UFDs.

So if you're looking for some class of domains such that

$\qquad$ domains where irreducibles are prime $\cap$ (??? domains) = GCD domains,

then at least it can't be one of those in the first chain. Perhaps there is some completely different class of domains that will do the trick, but I haven't come across any.

(Noetherian/Bézout/Dedekind/Prüfer/Krull/normal domains won't work either, but it's more complicated to describe how they fit into the above picture.)

A GCD domain must be a Schreier domain: It must be integrally closed, and every nonzero element must be primal (which is stronger than the condition that irreducibles are prime).

On the other hand, there are Schreier domains that are not GCD domains. Following Example 2.10 from The Schreier Property and Gauss’ Lemma, let $S$ be the integral closure of $\mathbb{C}[X]$ in $\overline{\mathbb{C}(X)}$, let $M$ be a maximal ideal of $S$, and let $R=\overline{\mathbb{Q}}+MS_M$. Then $R$ is Schreier but not GCD.

There are several other examples in the literature, including a fairly simple rank 2 monoid domain over a field.

There are (somewhat complicated) characterizations in the literature of GCD domains, the most common of which appears to be a condition for a PVMD (Prüfer v-multiplication domain) to be a GCD domain.

• I think that the example is $R=\overline{\mathbb{Q}}+MS_M$. Do you have the reference for the example of the rank 2 monoid domain? What would be some good references for the characterizations of GCD domains? – Watson Feb 8 '17 at 23:42
• +1 anyway. By the way, that question seems to be a duplicate of this one. – Watson Feb 8 '17 at 23:43
• @Watson I do believe you're right. This was some time ago, so I'm very uncertain, but maybe the book "Non-Noetherian Commutative Ring Theory" by Chapman and Glaz has some of what I was talking about. – Slade Feb 8 '17 at 23:48