On rings without a certain divisibility property (Q1106275)

An integral domain R with field of fractions F is called D-ring if whenever \(f,g\in R[X]\) are nonzero and such that if f(a) divides g(a) for all but finitely many \(a\in R\), then f divides g in F[X]. The author studies Dedekind non-D-rings. If R is a non-D-ring and \(f\in R[X]\), then f is called unit-valued if f(R)\(\subset U(R)\) (units of R). If R is a Dedekind non-D-ring with ideal class group Cl(R) and \(f\in R[X]\) is a unit-valued polynomial, \(\deg (f)=n\geq 2\), then Cl(R) is a torsion group with exponent d for some divisor d of n. Applications to algebraic number fields are given.