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In abstract algebra (ring theory specifically), when we are dealing with factorization, UFD's, and so on, we are often only interested in the multiplicative structure of the ring, not the additive structure. So here is the basic situation we face: we a start with an integral domain (R,+,*) (i.e. a commutative ring with no zero divisors). Then from that we take the multiplicative monoid (M,*) where M = R - {0}. What structure does M have? It's a commutative monoid with the cancellation property: ab=ac implies b=c. (A monoid is a set endowed with an operation that is associative and has an identity.)

But we're still not done! In the context of factorization and associated topics, we also don't care about multiplication by units (invertible elements). We call two elements a and b of M "associates" if a=bu for some unit u. So my question is, can we usefully construct out of M yet another structure which consists of equivalence classes under the associate equivalence relation?

You can take a quotient of a group with a subset if the subset is a normal subgroup. You can take a quotient of a ring with a subset if the subset is an ideal. So what is the condition that a subset of a monoid has to satisfy in order to be able to construct a quotient monoid? And whatever the condition is, does the group G of units of M satisfy this condition, so that M/G would be our quotient monoid, and would be the quotient of M under the associate equivalence relation? And assuming that we are allowed to form the quotient monoid M/G, what is the structure of this monoid?

Any help would be greatly appreciated.

Thank You in Advance.

But we're still not done! In the context of factorization and associated topics, we also don't care about multiplication by units (invertible elements). We call two elements a and b of M "associates" if a=bu for some unit u. So my question is, can we usefully construct out of M yet another structure which consists of equivalence classes under the associate equivalence relation?

You can take a quotient of a group with a subset if the subset is a normal subgroup. You can take a quotient of a ring with a subset if the subset is an ideal. So what is the condition that a subset of a monoid has to satisfy in order to be able to construct a quotient monoid? And whatever the condition is, does the group G of units of M satisfy this condition, so that M/G would be our quotient monoid, and would be the quotient of M under the associate equivalence relation? And assuming that we are allowed to form the quotient monoid M/G, what is the structure of this monoid?

Any help would be greatly appreciated.

Thank You in Advance.

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