For orders in number fields, a subring is associated if and only if it is both ideal-preserving and locally associated, yielding a quadratic-order classification of half-factorial domains.
Overrings of half-factorial orders
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abstract
The behavior of factorization properties in various ring extensions is a central theme in commutative algebra. Classically, the UFDs are (completely) integrally closed and tend to behave well in standard ring extensions, with the notable exception of power series extension. The half-factorial property is not as robust; HFDs need not be integrally closed and the half-factorial property is not necessarily preserved in integral extensions or even localizations. Here we exhibit classes of HFDs that behave well in (almost) integral extensions, resolve an open question on the behavior of the boundary map, and give a squeeze theorem for elasticity in certain domains.
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Multiplicative Relationships of Subrings and their Applications to Factorization
For orders in number fields, a subring is associated if and only if it is both ideal-preserving and locally associated, yielding a quadratic-order classification of half-factorial domains.