Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 46, No. 2, pp. 447466 (2005) 

Generalized GCD modulesMajid M. Ali and David J. SmithDepartment of Mathematics, Sultan Qaboos University and email: mali@squ.edu.om Department of Mathematics, University of Auckland email: smith@math.auckland.ac.nzAbstract: 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 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. Keywords: Multiplication module, projective module, flat module, invertible ideal, p.p. ring, greatest common divisor, least common multiple Classification (MSC2000): 13C13, 13A15 Full text of the article:
Electronic version published on: 18 Oct 2005. This page was last modified: 29 Dec 2008.
© 2005 Heldermann Verlag
