Reduced multiplication modules

Abstract

An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of $\mbox{\rm Spec}(M)$ . Also a finitely generated module M is a Baer module if and only if $\mbox{\rm Spec}(M)$ is an extremally disconnected space; if and only if it is a CS-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N, $\mbox{\rm Ann}(x) \not \subseteq (N:M).$ When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (VNM); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a VNM.