Generalized GCD modules

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.