The dual notion of strong irreducibility

TL;DR

This paper unifies the concepts of strong irreducibility and its duals in lattice theory, linking them to ring theory and topology, and explores their properties and applications in module theory and torsion theories.

Contribution

It provides a unifying lattice-theoretic framework for strong irreducibility and its duals, extending their analysis to localization and torsion theories.

Findings

Unified characterization of strong irreducibility and duals in lattices

Analysis of behavior under central localization

Application to hereditary torsion theories

Abstract

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.