The largest algebraic structure in which every irreducible is prime

We know that in a UFD every irreducible is prime.

But this document rise up in my mind a question that; is there a known algebraic structure (larger than a UFD) in which every irreducible is prime? Is there a name for such integral domains?

Thanks for sharing any knowledge,

• Sure, gcd domains and various generalizations thereof, e.g. domains satisfying variants of Euclid's Lemma or Gauss's Lemma, see this answer for a handful. They are sometimes called AP domains (Atoms are Prime)$\$ – Bill Dubuque Mar 24 '15 at 19:18
• Yes, I have heard of GCD domains before, some of the others are new to me. Thanks, But I would love to read your post you have referred there, which I am not able to open from the Google-groups link given there- most probably because of my browser? Could you please send that post as an answer here with a note maybe where this kind of domains fall exactly into, if you don't mind? – S.B. Mar 24 '15 at 19:29

A domain is called atomic if each non-zero and non-invertible elements is a product of irreducible elements.

It can be shown that for an atomic domain $D$ it is equivalent:

• $D$ is a UFD.

• Each irreducible in $D$ is prime.

That is for atomic domains there is nothing but UFDs. On the other hand a UFD is always atomic; as a prime is always irreducible.

Thus, if one wants to have something beyond UFDs one needs to leave the realm of domains where each element has a factorization into irreducibles. As Bill Dubuque mentions this is well possible and done. The "largest," that is most inclusive, defintion you seek would be just: "a domain where each irreducible is prime." A name for this is AP-domain (see comment by OP).

In such a domain it is true that every non-zero non-identity element has at most one factorization into irreducibles (up to multiplication by units and reordering).

Note that for a UFD one has the characterization each non-zero non-identity element has exactly one factorization into irreducibles (up to multiplication by units and reordering)

Put differently, in an AP-domain it is true that each element that has a factorization into irreducibles at all has a (essentially) unique factorization.

Such a domain is called an unrestricted UFD by Coykendall and Zafrullah. Thus, every AP-domain is an unrestricted UFD. The converse is not true though as shown by Coykendall and Zafrullah (in contrast to the situation for atomic domains).

A relevant paper is: Jim Coykendall, Muhammad Zafrullah, AP-domains and unique factorization, Journal of Pure and Applied Algebra (2004).

• I am not sure that atomic domains should have irreducibles as primes...? – S.B. Mar 24 '15 at 19:55
• I have just found this paper from 1991 ndsu.edu/pubweb/~coykenda/paper30b.pdf in which it refers to these domains as AP-domains but it does not seem to fully characterize them. But it differs the atomic property from AP condition (irreducibles being prime) – S.B. Mar 24 '15 at 20:00
• I am not sure what you mean with your first comment. The paper you mention is very relavant; I was not aware of this. In fact I made an error. I will update the answer. – quid Mar 24 '15 at 20:09
• S.B. is right. An atomic domain in which every atom is a prime is a UFD, and this has been pointed out in the Coykendall-Zafrullah paper, but there are atomic domains in which some atoms are not prime. If some readers need to they may consult: researchgate.net/publication/… – mzafrullah Mar 11 '18 at 23:19

It so happens that I addressed this question in hd1203, in the year 2012. Those looking for a characterization for domains in which atoms are prime may want to look up: http://www.lohar.com/mithelpdesk/hd1203.pdf In the above write up, that ended in 2012, you will find descriptions of domains in which atoms are usually prime, domains such as pre-Schreier domains, PSP domains etc.