Regular t-ideals of polynomial rings and semigroup rings with zero divisors

Abstract

For an integrally closed domain D with quotient field K, it is well known that every fractional t-ideal of D[X] has the form hI[X] with h∈K(X) and I a t-ideal of D. We generalize this useful result to certain polynomial rings and semigroup rings with zero divisors. As an application of our work, we characterize the semigroup rings with torsion-free cancellative monoids of exponents that are Prüfer v-multiplication rings or (generalized) greatest common divisor rings. By showing that Glaz's (generalized) greatest common divisor ring property ascends to polynomial rings, we affirmatively settle one of her conjectures.