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.
Elasticity in polynomial-type extensions
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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.