Some remarks on generalized GCD domains

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10 Citations (Scopus)

Abstract

An integral domain R is a generalized GCD (GGCD) domain if the semigroup of invertible ideals of R is closed under intersection. In this article we extend the definition of PF-prime ideals to GGCD domains and develop a theory of these ideals which allows us to characterize Prfer and π-domains among GGCD domains. We also introduce the concept of generalized GCD modules as a natural generalization of GGCD domains to the module case. An R-module M is a GGCD module if the set of invertible submodules of M is closed under intersection. We show that an integral domain R is a GGCD domain if and only if a faithful multiplication R-module M is a GGCD module. Various properties and characterizations of faithful multiplication GGCD modules over integral domains are considered and consequently, necessary and sufficient conditions for a ring R(M), the idealization of M, to be a GGCD ring are given.

Original languageEnglish
Pages (from-to)142-164
Number of pages23
JournalCommunications in Algebra
Volume36
Issue number1
DOIs
Publication statusPublished - Jan 2008

Keywords

  • GGCD domain
  • GGCD module
  • Invertible submodule
  • Multiplication module
  • PF-prime ideal

ASJC Scopus subject areas

  • Algebra and Number Theory

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