On the Maximal Spectrum of Semiprimitive Multiplication Modules

AnR-moduleMis called a multiplication module if for each submoduleNofM,N=IMfor some idealIofR. As defined for a commutative ringR, anR-moduleMis said to be semiprimitive if the intersection of maximal submodules ofMis zero. The maximal spectra of a semiprimitive multiplication moduleMare studied. The isolated points of Max(M) are characterized algebraically. The relationships among the maximal spectra ofM, Soc(M) and Ass(M) are studied. It is shown that Soc(M) is exactly the set of all elements ofMwhich belongs to every maximal submodule ofMexcept for a finite number. If Max(M) is infinite, Max(M) is a one-point compactification of a discrete space if and only ifMis Gelfand and for some maximal submoduleK, Soc(M) is the intersection of all prime submodules ofMcontained inK. WhenMis a semiprimitive Gelfand module, we prove that every intersection of essential submodules ofMis an essential submodule if and only if Max(M) is an almost discrete space. The set of uniform submodules ofMand the set of minimal submodules ofMcoincide. Ann(Soc(M))Mis a summand submodule ofMif and only if Max(M) is the union of two disjoint open subspacesAandN, whereAis almost discrete andNis dense in itself. In particular, Ann(Soc(M)) = Ann(M) if and only if Max(M) is almost discrete.