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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. :(

  • 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 04:17
  • 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. –  Sep 22 '11 at 06:01

2 Answers2

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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 $\rm\:b,c\in R\:.$

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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

Bill Dubuque
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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 $R$? –  Sep 22 '11 at 05:47
  • I think you asked this question when I had lost interest, you should read Franz Halter-Koch's book entitled "Ideals Systems: An introduction to Multiplicative Ideal Theory". – mzafrullah Nov 27 '19 at 05:22
  • The monoid of elements of an integral domain is cancellative i.e. if a is nonzero, then ab=ac implies b=c. Using this as a starting point Halter-Koch studied and translated most of the then known notions in multiplicative ideal theory into the language of ideal systems on monoids. It was a good addition to the literature, though his insistence that monoids were the only way to do ring theory drew resistance from some quarters. – mzafrullah Nov 27 '19 at 05:38
  • @mzafrullah Yes. I mentioned Halter=Koch's work briefly in passing here.. Nice to see you back here. – Bill Dubuque Nov 27 '19 at 11:43
  • Consider it a brief appearance to say "Happy Thanksgiving y'all". About Halter-Koch, he apparently translated everything that was known except for Schreier domains and that seemed a little odd to me. So I have put together something on Riesz monoids to remedy that. (Still working on fine-tuning it.) – mzafrullah Nov 28 '19 at 00:27
  • @BillDubuque: Regarding the first bullet point, about UFDs: won't G(R), as an abstract group, be a sum of copies of $\mathbb{Z}$ for any Dedekind domain, for instance? Perhaps the statement holds once you introduce an appropriate partial order though? – ndkrempel May 08 '22 at 18:37
  • @ndkrempel I appended an excerpt which gives more detail (too long for a comment). – Bill Dubuque May 08 '22 at 19:34
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"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

mzafrullah
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  • In fact I do strive to spread the Schreier gospel here - having mentioned it and related divisibility properties many times here since the dawn of the site, including links to your work (I discovered your answer only just now after your comment on mine). – Bill Dubuque Nov 27 '19 at 11:49