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

Abstract

Let $\mathfrak{S}_p(_RM)$ be the lattice of all saturated submodules of an $R$-module $M$ with respect to a prime ideal $p$ of a commutative ring $R$. We examine the properties of the mappings $\eta:\mathfrak{S}_p(_RR)\rightarrow \mathfrak{S}_p(_RM)$ defined by $\eta(I)=S_p(IM)$ and $\theta:\mathfrak{S}_p(_RM)\rightarrow \mathfrak{S}_p(_RR)$ defined by $\theta(N)=(N:M)$, in particular considering when these mappings are lattice homomorphisms. It is proved that if $M$ is a semisimple module or a projective module, then $\eta$ is a lattice homomorphism. Also, if $M$ is a faithful multiplication $R$-module, then $\eta$ is a lattice epimorphism. In particular, if $M$ is a finitely generated faithful multiplication $R$-module, then $\eta$ is a lattice isomorphism and its inverse is $\theta$. It is shown that if $M$ is a distributive module over a semisimple ring $R$, then the lattice $\mathfrak{S}_p(_RM)$ forms a Boolean algebra and $\eta$ is a Boolean algebra homomorphism.