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