Generalized GCD modules

Majid M. Ali and David J. Smith

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 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

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