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 \mathcal{V}(_{R}M) of varieties of submodules of M is a distributive lattice. If M is a semisimple R-module and the unary operation ^{\prime} on \mathcal{V}(_{R}M) is defined by (V(N))^{\prime}=V(\tilde{N}), where M=N\oplus \tilde{N}, then the lattice \mathcal{V}(_{R}M) with ^{\prime} forms a Boolean algebra. In this paper, we examine the properties of certain mappings between \mathcal{V}(_{R}R) and \mathcal{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 \mathcal{V}(_{R}R) and \mathcal{V}(_{R}M) are isomorphic lattices, and therefore \mathcal{V}(_{R}M) and the lattice \mathcal{R}(R) of radical ideals of R are anti-isomorphic lattices. Moreover, if R is a semisimple ring, then \mathcal{V}(_{R}R) and \mathcal{V}(_{R}M) are isomorphic Boolean algebras, and therefore \mathcal{V}(_{R}M) and \mathcal{L}(R) are anti-isomorphic Boolean algebras.

کلیدواژه ها:

Lattice homomorphism‎ ، ‎‎ ‎omega-module‎ ، ‎Primeful module‎ ، ‎Semisimple ring