Some Remarks on Generalized GCD Domains

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 Prüfer 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.

Key Words:

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

2000 Mathematics Subject Classification:

• 13C13
• 13A15

ACKNOWLEDGMENT

Dedicated to David J. Smith on the occasion of his retirement.

Notes

Communicated by I. Swanson.