SUBLATTICES OF R-TORS INDUCED BY A-MODULES

Abstract

This article is concerned with the study of the sublattice gen (ρ) of R-tors, where ρ is some arbitrary but fixed member of R-tors. We use the concept of the ρ-A-module (M is a ρ-A-module, if M is ρ-torsion free and ρ ∨ξ ({M}) is an atom in gen (ρ)) and we define an equivalence relation in the sublattice gen (ρ). The partition associated to this equivalence relation allows us to get interesting information about this sublattice. As an application, we obtain new characterizations of ρ-artinian rings, ρ-semiartinian rings (a ring R is ρ-semiartinian if every non-zero ρ-torsion free R-module contains a τ- cocritical submodule) and rings with ρ-atomic dimension (rings such that for all σ ∈ gen (ρ) with σ 6= χ, there exists a σ-A-module).