Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there any good description of the multiplicative monoid of a commutative ring in general? Or in special cases?

I understand that in a UFD, it is the result of adjoining a zero to the Cartesian product of the unit group with a free commutative monoid, but that's pretty much the only case I understand. :(

share|cite|improve this question
How about the case of a Dedekind domain? You should be able to describe the multiplicative monoid in terms of the unit group and the class group, I think. – Qiaochu Yuan Sep 22 '11 at 4:17
I've been having trouble wrapping my head around it -- it doesn't seem like it gives you any information about the relationships between cancellation monoids $R^*$, $R \setminus \{ 0 \}$, and the free monoid on the prime ideals beyond what's contained in the groups they generate. – Hurkyl Sep 22 '11 at 6:01

Many properties of domains are purely multiplicative so can be described in terms of monoid structure. Let R be a domain with fraction field K. Let R* and K* be the multiplicative groups of units of R and K respectively. Then G(R), the divisibility group of R, is the factor group K*/R*.

  • R is a UFD $\iff$ G(R) is a sum of copies of $\rm\:\mathbb Z\:.$

  • R is a gcd-domain $\iff$ G(R) is lattice-ordered (lub{x,y} exists)

  • R is a valuation domain $\iff$ G(R) is linearly ordered

  • R is a Riesz domain $\iff$ G(R) is a Riesz group, i.e. an ordered group satisfying the Riesz interpolation property: if $\rm\:a,b \le c,d\:$ then $\rm\:a,b \le x \le c,d\:$ for some $\rm\:x\:.\:$ A domain $\rm\:R\:$ is called Riesz if every element is primal, i.e. $\rm\:A\:|\:BC\ \Rightarrow\ A = bc,\ b|B,\ c|C\:,\:$ for some $\rm\:b,c\in R\:.$

For more on divisibility groups see the following surveys:

J.L. Mott, Groups of divisibility: A unifying concept for integral domains and partially ordered groups, Mathematics and its Applications, no. 48, 1989, pp. 80-104.

J.L. Mott, The group of divisibility and its applications, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Springer, Berlin, 1973, pp. 194-208. Lecture Notes in Math., Vol. 311. MR 49 #2712

share|cite|improve this answer
While this is interesting, I was wondering specifically about the monoid structure -- e.g. can we say much about the submonoid of $G(R)$ generated by the nonzero elements of $R$? – Hurkyl Sep 22 '11 at 5:47

"Riesz domains" have been studied extensively as pre-Schreier and Schreier domains, and Bill Dubuque should have known that! In any case, here's a link that might be useful:

The paper and the references therein will be helpful for the monoid question too. There have been quite a few generalizations of the pre-Schreier domains too and references to some of them could show up at places on the page

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.