A reduced, cancellative, torsion-free, commutative monoid M can
be embedded in an integral domain R, where the atoms (irreducible elements)
of M correspond to a subset of the atoms of R. This fact was used by J.
Coykendall and B. Mammenga to show that for any reduced, cancellative,
torsion-free, commutative, atomic monoid M, there exists an integral domain
R with atomic factorization structure isomorphic to M. More generally, we
show that any “nice” subset of atoms of R can be realized as the set of atoms
of an integral domain T that contains R. We will also give several applications
of this result.
Cancellative commutative monoid, reduced monoid, atomic monoid, integral domain