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 language | English |
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Pages (from-to) | 142-164 |
Number of pages | 23 |
Journal | Communications in Algebra |
Volume | 36 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jan 2008 |
Keywords
- GGCD domain
- GGCD module
- Invertible submodule
- Multiplication module
- PF-prime ideal
ASJC Scopus subject areas
- Algebra and Number Theory