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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. :(

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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

2 Answers 2

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

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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: http://www.lohar.com/researchpdf/on_a_property_of_pre_schreier_domains.pdf

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 www.lohar.com

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