TITLE: Multiplicative monoid of a commutative ring
QUESTION [8 upvotes]: 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. :(

REPLY [3 votes]: "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