### 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 |
---|---|

Pages (from-to) | 142-164 |

Number of pages | 23 |

Journal | Communications in Algebra |

Volume | 36 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2008 |

### Fingerprint

### 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

**Some remarks on generalized GCD domains.** / Ali, Majid M.

Research output: Contribution to journal › Article

*Communications in Algebra*, vol. 36, no. 1, pp. 142-164. https://doi.org/10.1080/00927870701665271

}

TY - JOUR

T1 - Some remarks on generalized GCD domains

AU - Ali, Majid M.

PY - 2008/1

Y1 - 2008/1

N2 - 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.

AB - 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.

KW - GGCD domain

KW - GGCD module

KW - Invertible submodule

KW - Multiplication module

KW - PF-prime ideal

UR - http://www.scopus.com/inward/record.url?scp=38649119172&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38649119172&partnerID=8YFLogxK

U2 - 10.1080/00927870701665271

DO - 10.1080/00927870701665271

M3 - Article

VL - 36

SP - 142

EP - 164

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 1

ER -