# About a type of integral domains

I was reading the notes about factorization written by Pete L. Clark and I found that he uses the name "EL-domain" to refer to those integral domains in which every irreducible element is prime (page 16).

So my question is: is the name "EL-domain" an accepted terminology? I made some google search and I haven't found anything. Perhaps this terminology has been used in some papers, but I don't know. Any help is apreciated.

• Maybe Pete will answer your question here :) Personally, I've never seen it anywhere else. It seems to be a reference to Euclid's lemma. Either way, in a paper an author would probably define an EL-domain if the term were going to be used. – Alex Macedo Jan 5 '17 at 14:39
• I hope so :). I also noticed that the terminology is related to Euclid's lemma, so why not called it Euclidean domain?... yikes! – Xam Jan 5 '17 at 14:48
• well, Euclidean is a much stronger concept. Euclidean implies EL but the converse is not true. In fact, every UFD is EL but they are not necessarily Euclidean. – Alex Macedo Jan 5 '17 at 14:56

Indeed this is not standard terminology: I made it up when writing the notes. As Bill Dubuque says, it stands for "Euclid's Lemma." I would not have used the term "AP domains" back then, because back then I used the term "irreducible" instead of "atom," even though I knew that the literature on factorization theory went the other way around. (Since then I have changed my mind.)

To be honest, both terms "EL-domain" and "AP-domain" seem pretty bad to me. Let me say a little bit about why things worked out this way. The property that irreducibles are prime is an important one, but in the majority of mathematical usage it is drowned out by the more important property of being a UFD or a factorial domain. If you look at Theorem 15.8 of my notes, you'll see that a domain is a UFD iff all irreducibles are primes and every nonzero nonunit factors as a product of irreducibles ("atomic domain"; I used to write "factorization domain," but I switched that one over eventually). The condition of a domain being atomic is much weaker than it being Noetherian, and the way commutative algebra has developed, Noetherian rings have gotten the lion's share of people's attention.

A good way to think about "EL-domain" is as a meaningful way of keeping the uniqueness of factorizations but not their existence. In this it joins many other classes of rings which are roughly (there is not necessarily a canonical or unique way to do it) "non-Noetherian analogues of more familiar properties that imply Noetherianness (or something like it)." Defining and studying such non-Noetherian analogues is in fact an important theme in contemporary commutative algebra, and it comes up in my notes as well. But the terminology for this is all over the place:

• PID $\mapsto$ Bezout domain
• Dedekind domain $\mapsto$ Prufer domain
• Integrally closed Noetherian domain $\mapsto$ Krull domain

As Serge Lang once said, the terminology should be functorial with respect to the ideas. That is unfortunately not really the case here.

• I really appreciate your answer Mr. Clark. Thanks so much. – Xam Jan 12 '17 at 2:41

It stands for "Euclid's Lemma" as Pete mentions here.. As I mention there, they are more commonly called AP-domains (Atoms are Prime) in the literature on factorization theory.

Remark $\$ See this answer for common names of many closely related properties.