SUBLATTICES OF R-TORS INDUCED BY A-MODULES

Year 2012, Volume: 12 Issue: 12, 17 - 36, 01.12.2012

Jaime Castro Pérez Gerardo Aguilar Sánchez

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).

Keywords

torsion theories, lattices, A-modules, atomic dimension, Gabriel dimension, left τ-semiartinian rings, left τ-artinian rings

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• Departamento de Matem´aticas Instituto Tecnol´ogico y de Estudios Superiores de Monterrey Calle del Puente 222, Tlalpan M´exico, D.F. M´exico e-mail: [e-mail] Gerardo Aguilar S´anchez Departamento de F´ısica y Matem´aticas Divisi´on de Ingenier´ıa y Arquitectura Tecnol´ogico de Monterrey Campus Ciudad de M´exico M´exico e-mail: [e-mail]