Some Remarks on Generalized GCD Domains |
| |
Authors: | Majid M Ali |
| |
Institution: | 1. Department of Mathematics and Statistics , College of Science, Sultan Qaboos University , Muscat, Oman mali@squ.edu.om |
| |
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. |
| |
Keywords: | GGCD domain GGCD module Invertible submodule Multiplication module PF-prime ideal |
|
|