### 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 |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Communications in Algebra*,

*36*(1), 142-164. https://doi.org/10.1080/00927870701665271