# Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR

Let $$H$$ be a (commutative or non-commutative) monoid. We say that $$H$$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ideals of $$H$$ that is strictly increasing with respect to inclusion, where a principal left ideal of $$H$$ is a set of the form $$Ha := \{xa: x \in H\}$$ (with $$a \in H$$). The ACCPR (ascending chain condition on principal right ideals) is defined in a left-right symmetric way.

Now, Proposition 0.9.3 in P.M. Cohn's

• Free Ideal Rings and Localization in General Rings (New Math. Monogr. 3, Cambridge Univ. Press, 2006)

reads as follows (an atom of $$H$$ is a non-unit $$a \in H$$ such that $$a \ne xy$$ for all non-units $$x, y \in H$$):

Theorem 1. If $$H$$ is a cancellative monoid satisfying both the ACCPL and the ACCPR, then $$H$$ is atomic, meaning that every non-unit of $$H$$ factors as a product of atoms.

I'll refer to Theorem 1 as Cohn's theorem on atomic factorizations in cancellative monoids. I'm aware that the statement is, in its essence, much older than Cohn's book. In the section "Notes and comments on Chapter 0" (loc. cit.), Cohn himself writes, "The results of Section 0.9 are for the most part well known." In practice, I'd like to understand if it's possible to track the history of the theorem better than I've seen done so far. To my knowledge, one of the first occurrences of something of the "same" quality is Proposition 1.1 in

• P.M. Cohn, Bezout rings and their subrings, Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, pp. 251-264,

where Cohn famously wrote:

An element of an integral domain is called an atom if it is a non-unit which cannot be written as a product of two non-units. If every element of a ring $$R$$ which is not a unit or $$0$$ can be written as a product of atoms, $$R$$ is said to be atomic. The following result is easily verified:

PROPOSITION 1.1. An integral domain is atomic if and only if it satisfies the maximum condition on principal ideals.

I say "famously" because the conclusion turned out to be not-so-easy-to-verify after all, as it was proved wrong by A. Grams in

• Atomic rings and the ascending chain condition for principal ideals, Math. Proc. Cambridge Phil. Soc. 75 (1974), No. 3, pp. 321-329.

In Cohn's paper, an integral domain is really a commutative domain: I'm stressing this point because, elsewhere in his work, Cohn is using the term "integral domain" to refer to both commutative and non-commutative domains. In particular, Proposition 2.5 in

• P.M. Cohn, Free Rings and Their Relations, Academic Press, 1985

Theorem 2. Any integral domain with left and right $$ACC_1$$ is atomic.

Here, an "integral domain" is actually a commutative or non-commutative domain, and the left (resp., right) $$ACC_1$$ is nothing else than the ACCPL (resp., ACCPR). So, I'll refer to Theorem 2 as Cohn's theorem on atomic factorizations in domains.

In my view, the standard proof of Theorems 1 and 2 in the commutative case is not really of the same difficulty as Cohn's proof of the result in the non-commutative case (although the latter is still an easy proof by any modern standards, at least in hindsight). Therefore, I'll focus on the non-commutative case and ask the following:

QUESTION. Are you aware of any (published) results that predate Cohn's theorems on atomic factorizations (either in domains or in cancellative monoids)?

By the word "predate", I mean either a result of the form "If $$H$$ is any monoid (resp., domain) in a certain non-trivial class of non-commutative monoids (resp., domains) satisfying the ACCPL and the ACCPR, then $$H$$ is atomic". I count on your common sense for the actual meaning of "non-trivial class"; in particular, a group does not count as non-trivial, and neither does a monoid $$H$$ that comes by with a length function, that is, a function $$\phi: H \to \mathbf N \cup \{\infty\}$$ such that $$\phi(x) < \phi(y)$$ whenever $$x$$ divides $$y$$ (i.e., $$y \in HxH$$) but not the other way around.

Edit #1. I had a look at M.L. Dubreil-Jacotin's paper

• Sur l'immersion d'un semi-groupe dans un groupe, C. R. Acad. Sci. Paris 225 (1947), 787-788,

which was suggested by Benjamin Steinberg in the comments. Here is the main result:

Let $$S$$ be a cancellative semigroup with no identity such that (1) if two elements $$a, b \in S$$ have a common right multiple (i.e., $$aH \cap bH \ne \emptyset$$), then one is a left divisor of the other (i.e., $$a \in bH$$ or $$b \in aH$$); (2) each element has only a finite number of left divisors. Then every element of $$S$$ can be uniquely written as a product of indecomposable elements of $$S$$ (i.e., elements having no left divisors).

Honestly, I don't see much of a resemblance to Proposition 0.9.3 in Cohn's 2006 book. To me, the result looks much closer to the Fundamental Theorem of Arithmetic.

To start with, there is no (implicit or explicit) reference to the ACC on principal left and on principal right ideals: Instead, there is a much stronger condition on the number of left divisors of an arbitrary element (i.e., there are only finitely many of them). Moreover, the conclusion is much stronger than the existence of an atomic factorization, because an "indecomposable factorization" à la Dubreil-Jacotin is unique (in the strongest possible sense). It follows that $$S$$ is a free semigroup and the "indecomposable elements" are "prime elements" (and again in a very strong sense). But this is not usually the case with atoms, let alone that atomic factorizations are, in general, all but unique in any sensible way.

• I don't think this should be attributed to Cohn. This seems like folklore and variations on this theme can be found in the work of Levi, Dubreil-Jacotin, Grillet and others to axiomatize free semigroups and free commutative semigroups. Probably the commutative case is much older. Jun 23, 2021 at 12:30
• I agree with Benjamin Steinberg, but would also point out that much of Cohn's work in his 2006 book appeared earlier in his "Free rings and their relations" book. (By the way, I've found that one should be very careful about Cohn's claims, terminology, etc., and your post reinforces that belief.) Jun 23, 2021 at 15:19
• The works I refer to of Levi and Dubreil-Jacotin characterize free monoids in terms of the principal left/right ideal structure and certainly Cohn knew their work since he cited the latter in ams.org/journals/proc/1962-013-03/S0002-9939-1962-0139680-3/… Jun 23, 2021 at 16:43
• This theorem really shouldn't get any name in my opinion Jun 23, 2021 at 16:43
• Levi, F. W. (1944), "On semigroups", Bulletin of the Calcutta Mathematical Society, 36: 141–146 it uses a different formulation. The other paper is closer. Jun 23, 2021 at 18:31

After recalling (or introducing?) the notions of "atom" and "atomic ring" (the latter definition makes explicit reference to the monoid of regular elements, as can be seen from loc. cit., p. 587), Cohn proves that, if a ring $$R$$ is Morita-equivalent to a FIR, then the monoid of regular elements of $$R$$ is atomic: The proof relies on (i) Proposition 4.3 from the same paper (where Cohn shows that a FIR satisfies the ACCPR) and (ii) the observation that a FIR satisfies the ACCPR if and only if it satisfies the ACCPL.
It is perhaps worth noting that, in the 1964 paper (cited above), Cohn was not yet explicitly "thinking in monoids", to the contrary of the 1967 paper; and none of these papers includes any reference to the work of other people building on the idea that "ACCPL & ACCPR $$\implies$$ atomicity".