Divisorial Prime Ideals in Prüfer Domains

For an integral domain R with quotient field K ≠ R, the inverse of a nonzero fractional ideal I of R is the set (R: I) = {t ∈ K ∈ tI ⊆ R}. The divisorial closure of I with respect to R is the fractional ideal (R: (R: I)). In addition I is divisorial as an ideal of R if I = (R: (R: I)). Of concern here are divisorial prime ideals in Prüfer domains. In some cases one can have a pair of comparable Prüfer domains T ⊊ R $$T \subsetneq R$$ with a common nonzero prime ideal P such that P is divisorial as an ideal of T but is not divisorial as an ideal of R. For example, if P = P 2 is a nonzero nonmaximal prime of a valuation domain V, then P is divisorial as an ideal of V but P = PV P is not divisorial as an ideal of V P . We review several relevant results on divisorial primes and present some new sufficient conditions on when P is divisorial as an ideal of R, and if not when a T ⊊ R $$T \subsetneq R$$ exists such that P = P ∩ T is divisorial as an ideal of T.