### Abstract

In recent work we called a ring R a GGCD ring if the semigroup of finitely generated faithful multiplication ideals of R is closed under intersection. In this paper we introduce the concept of generalized GCD modules. An R-module M is a GGCD module if M is multiplication and the set of finitely generated faithful multiplication submodules of M is closed under intersection. We show that a ring R is a GGCD ring if and only if some R-module M is a GGCD module. Glaz defined a p.p. ring to be a GGCD ring if the semigroup of finitely generated projective (flat) ideals of R is closed under intersection. As a generalization of a Glaz GGCD ring we say that an R-module M is a Glaz GGCD module if M is finitely generated faithful multiplication, every cyclic submodule of M is projective, and the set of finitely generated projective (flat) submodules of M is closed under intersection. Various properties and characterizations of GGCD modules and Glaz GGCD modules are considered.

Original language | English |
---|---|

Pages (from-to) | 447-466 |

Number of pages | 20 |

Journal | Beitrage zur Algebra und Geometrie |

Volume | 46 |

Issue number | 2 |

Publication status | Published - 2005 |

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### Keywords

- Flat module
- Greatest common divisor
- Invertible ideal
- Least common multiple
- Multiplication module
- p.p. Ring
- Projective module

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Beitrage zur Algebra und Geometrie*,

*46*(2), 447-466.

**Generalized GCD modules.** / Ali, Majid M.; Smith, David J.

Research output: Contribution to journal › Article

*Beitrage zur Algebra und Geometrie*, vol. 46, no. 2, pp. 447-466.

}

TY - JOUR

T1 - Generalized GCD modules

AU - Ali, Majid M.

AU - Smith, David J.

PY - 2005

Y1 - 2005

N2 - In recent work we called a ring R a GGCD ring if the semigroup of finitely generated faithful multiplication ideals of R is closed under intersection. In this paper we introduce the concept of generalized GCD modules. An R-module M is a GGCD module if M is multiplication and the set of finitely generated faithful multiplication submodules of M is closed under intersection. We show that a ring R is a GGCD ring if and only if some R-module M is a GGCD module. Glaz defined a p.p. ring to be a GGCD ring if the semigroup of finitely generated projective (flat) ideals of R is closed under intersection. As a generalization of a Glaz GGCD ring we say that an R-module M is a Glaz GGCD module if M is finitely generated faithful multiplication, every cyclic submodule of M is projective, and the set of finitely generated projective (flat) submodules of M is closed under intersection. Various properties and characterizations of GGCD modules and Glaz GGCD modules are considered.

AB - In recent work we called a ring R a GGCD ring if the semigroup of finitely generated faithful multiplication ideals of R is closed under intersection. In this paper we introduce the concept of generalized GCD modules. An R-module M is a GGCD module if M is multiplication and the set of finitely generated faithful multiplication submodules of M is closed under intersection. We show that a ring R is a GGCD ring if and only if some R-module M is a GGCD module. Glaz defined a p.p. ring to be a GGCD ring if the semigroup of finitely generated projective (flat) ideals of R is closed under intersection. As a generalization of a Glaz GGCD ring we say that an R-module M is a Glaz GGCD module if M is finitely generated faithful multiplication, every cyclic submodule of M is projective, and the set of finitely generated projective (flat) submodules of M is closed under intersection. Various properties and characterizations of GGCD modules and Glaz GGCD modules are considered.

KW - Flat module

KW - Greatest common divisor

KW - Invertible ideal

KW - Least common multiple

KW - Multiplication module

KW - p.p. Ring

KW - Projective module

UR - http://www.scopus.com/inward/record.url?scp=30644462566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30644462566&partnerID=8YFLogxK

M3 - Article

VL - 46

SP - 447

EP - 466

JO - Beitrage zur Algebra und Geometrie

JF - Beitrage zur Algebra und Geometrie

SN - 0138-4821

IS - 2

ER -