I asked a similar question a few weeks ago in M.SE but it didn't receive any answers, so I decided to post it here with some modifications.

My motivation comes from a theorem given in Pete L. Clark's notes on factorization. More exactly, I'm referring to the theorem 46 and it says this: for a Bézout domain $R$, the following are equivalent:

i) $R$ is a PID.

ii) $R$ is Noetherian.

iii) $R$ is a UFD.

iv) $R$ is an ACCP domain.

v) $R$ is an atomic domain.

I try to understand how Euclidean domains fit into the above theorem. I mean, EDs are domains that appear naturally in factorization theory and also they are related to PIDs or UFDs, but what else can we say about these domains? Is there a relationship with the other domains encountered in factorization theory? If not, is it known why? Thanks in advance.

  • 1
    $\begingroup$ One thing that comes to my mind is that, from the point of view of FT, Euclidean domains are a rather special subclass of the class of domains, either commutative or not, whose monoid of non-zero elements supports a length fnc, which in turn implies the ACC on principal left and principal right ideals, and hence atomicity (this was recently proved to carry over to monoids s.t. $xy=x$ or $yx=x$ only if $y$ is a unit). Given a monoid $H$, a fnc $\lambda:H\to\bf N$ is a length fnc if $\lambda(x)<\lambda(y)$ whenever $y=uxv$ for some $u,v\in H$ with $u\notin H^\times$ or $v\notin H^\times$. $\endgroup$ – Salvo Tringali Mar 11 '17 at 18:24
  • $\begingroup$ @SalvoTringali could you turn your comments into an answer and if possible could you add a reference to the theorem you claim was recently proved? $\endgroup$ – Xam Mar 11 '17 at 19:08

At the request of the OP, I'm turning my comments above into an answer, though different answers are possible and the question sounds a bit soft to me. Let it be as it may, here are my two cents.

From the point of view of factorization theory, Euclidean domains can be understood as a rather special subclass of the class of domains, either commutative or not, whose monoid of non-zero elements admits a length function, which in turn implies the ACC on principal left and principal right ideals (shortly, ACCP), and hence atomicity. (Given a monoid $H$, a function $\lambda: H \to \mathbf N$ is called a length function if $\lambda(x) < \lambda(y)$ whenever $y=uxv$ for some $u, v \in H$ with $u \notin H^\times$ or $v \notin H^\times$, where $H^\times$ is the group of units of $H$. In particular, it is well known that every Euclidean function $f$ on an integral domain $R$ can be "normalized" to a Euclidean function $f^\ast$ such that $f^\ast(a) \le f^\ast(ab)$ for all non-zero $a, b \in R$, and then it is not difficult to show that $f^\ast$ is a length function.)

On the other hand, a monoid supporting a length function need not be factorial (that is, factorization need not be unique, whatever this means, unless you take a quotient of $\mathscr{F}^\ast(\mathcal A(H))$, the free monoid with basis the atoms of $H$, by a very large congruence...), though it must be BF (i.e., the factorizations of a fixed element cannot be arbitrarily long). So, it is clear that the kind of length functions supported by a Euclidean domain must be quite special, insofar as their existence implies factoriality.

As for the connection among the ACCP, atomicity, and the existence of a length function, we have to separate the commutative and cancellative case from the non-commutative or non-cancellative case, since they have completely different stories.

To begin, you should keep in mind that factorization theory has been developed so far almost entirely in the commutative and cancellative setting (there are many explanations for that, but would be beyond the scope of this thread), and in this setting the relevant results have been part of the folklore for a long time (in case you want a reference, see, e.g., [4, Proposition 1.1.4]). It is hard for me to say where they were first proved, but here is a little nugget: Proposition 1.1 in P.M. Cohn's influential paper [1] reads, "An integral domain is atomic if and only if it satisfies the maximum condition on principal ideals". The maximum condition alluded to by Cohn is equivalent to the ACCP, and he doesn't prove his statement, maintaining that it "is easily verified". But it turns out that the claim is false, as shown by A. Grams in [5]. (By the way, Grams establishes in the same paper that the ACCP is not necessary for atomicity.)

As for the rest, it is only in 2013 that D. Smertnig proved that the ACCP is still a sufficient condition for atomicity in cancellative (though possibly non-commutative) monoids, see [6, Proposition 3.1], and the same criterion has been recently extended in a different direction (namely, to unit-cancellative, commutative monoids) by Y. Fan, A. Geroldinger, F. Kainrath, and myself, see [2, Lemma 3.1(1)]. The general unit-cancellative case, which subsumes all the previous results, has now been settled in [3, Theorem 2.22], where it is shown that, for a unit-cancellattive (but possibly non-commutative) monoid, the ACCP implies atomicity, and the existence of a length function is equivalent to BF-ness. (A monoid $H$ is unit-cancellative if $xy = x$ or $yx = x$ for some $x, y \in H$ implies $y \in H^\times$.)

Of course, this is not the end of the story: There are a bunch of interesting monoids, which are atomic, but not unit-cancellative, for which an analogous general criterion is not yet known.


[1] P.M. Cohn, Bezout rings and their subrings, Proc. Camb. Phil. Soc. 64 (1968), 251-264.

[2] Y. Fan, A. Geroldinger, F. Kainrath, and S.T., Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules, J. Algebra Appl. 16 (2017), No. 11.

[3] Y. Fan and S.T., Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics, preprint.

[4] A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton (FL), 2006.

[5] A. Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Camb. Philos. Soc. 75 (1974), 321-329.

[6] D. Smertnig, Sets of lengths in maximal orders in central simple algebras, J. Algebra 390 (2013), 1-43.


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