Generalized GCD modules

Majid M. Ali, David J. Smith

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 languageEnglish
Pages (from-to)447-466
Number of pages20
JournalBeitrage zur Algebra und Geometrie
Volume46
Issue number2
Publication statusPublished - 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

Ali, M. M., & Smith, D. J. (2005). Generalized GCD modules. Beitrage zur Algebra und Geometrie, 46(2), 447-466.