Module-Theoretic Characterizations of Generalized GCD Domains

Abstract

We give several module-theoretic characterizations of generalized GCD domains. For example, we show that an integral domain R is a generalized GCD domain if and only if semi-divisoriality and flatness are equivalent for torsion-free R-modules if and only if every w-finite w-module is projective if and only if R is w-Prüfer (in the sense of Zafrullah). We also characterize when a pullback R of a certain type is a generalized GCD domain. As an application, we characterize when R = D + XE[X] (here, D ⊆ E is an extension of domains and X is an indeterminate) is a generalized GCD domain.

Key Words:

• Flat
• GGCD domain
• Projective
• Prüfer domain
• Pullback
• w-Module
• w-Prüfer

2000 Mathematics Subject Classification:

• 13A15
• 13C10
• 13C11
• 13G05

ACKNOWLEDGMENTS

I would like to thank Hoseo University and the University of North Carolina at Charlotte (where the work was done) for allowing me to spend a sabbatical year and wish to thank the people for their hospitality. In particular, I am grateful to E. Houston for introducing the article [22] to me and helpful comments concerning this article. I would also like to express my sincere thanks to the referee for many useful comments and suggestions.

Notes

Communicated by A. Singh.

Dedicated to Evan G. Houston on the occasion of his sixtieth birthday.