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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.2011-09-22
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    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.2011-09-22

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