SOME ASPECTS OF τ-FULL MODULES

Abstract

Let τ be a hereditary torsion theory on Mod-R. For a right τ-full R-module M, we establish that [τ,τ ∨ξ (M)] is a boolean lattice; we find necessary and sufficient conditions for the interval [τ,τ ∨ξ (M)] be atomic, and we give conditions for the atoms be of some specific type in terms of the internal structure of M. We also prove that there are lattice isomorphisms between the lattice [τ,τ ∨ξ (M)] and the lattice of τ-pure fully invariant submodules of M, under the additional assumption that M is absolutely τ-pure. With the aid of these results, we get a decomposition of a τ-full and absolutely τ-pure R-module M as a direct sum of τ-pure fully invariant submodules N and N with different atomic characteristics on the intervals [τ,τ ∨ξ (N)] and [τ,τ ∨ξ (N)], respectively.