Mappings between the lattices of saturated submodules with respect to a prime ideal

Abstract

to a prime ideal p of a commutative ring R. We examine the properties of the mappings : Sp(RR) ! Sp(RM) defined by (I) = Sp(IM) and : Sp(RM) ! Sp(RR) defined by (N) = (N : M), in particular considering when these mappings are lattice homomor- phisms. It is proved that if M is a semisimple module or a projective module, then is a lattice homomorphism. Also, if M is a faithful multiplication R-module, then is a lattice epimorphism. In particular, if M is a finitely generated faithful multiplication R-module, then is a lattice isomorphism and its inverse is . It is shown that if M is a distributive module over a semisimple ring R, then the lattice Sp(RM) forms a Boolean algebra and is a Boolean algebra homomorphism.