Mappings between the lattices of varieties of submodules

Let R be a commutative ring with identity and M be an R-module. It is shown that the usual lattice V(RM) of varieties of submodules of M is a distributive lattice. If M is a semisimple R-module and the unary operation ′ on V(RM) is defined by (V(N))′=V(N~), where M=N⊕N~, then the lattice V(RM) with ′ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between V(RR) and V(RM), in particular considering when these mappings are lattice homomorphisms. It is shown that if M is a faithful primeful R-module, then V(RR) and V(RM) are isomorphic lattices, and therefore V(RM) and the lattice R(R) of radical ideals of R are anti-isomorphic lattices. Moreover, if R is a semisimple ring, then V(RR) and V(RM) are isomorphic Boolean algebras, and therefore V(RM) and L(R) are anti-isomorphic Boolean algebras.