Divisorial Prime Ideals in Prüfer Domains

Thomas G. Lucas ()
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Thomas G. Lucas: University of North Carolina Charlotte, Department of Mathematics and Statistics

A chapter in Rings, Polynomials, and Modules, 2017, pp 281-298 from Springer

Abstract: Abstract 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.

Keywords: Prüfer domain; Divisorial ideal; [MSC 2010] Primary 13F05; 13A15 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-65874-2_14

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DOI: 10.1007/978-3-319-65874-2_14

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