Stone Commutator Lattices and Baer Rings

TL;DR

This paper extends Davey's characterization of Stone lattices to commutator lattices and applies these results to congruence lattices of rings, establishing new links between lattice theory and ring theory.

Contribution

It generalizes Davey's theorem to commutator lattices and applies it to rings, bridging lattice theory and algebraic structures.

Findings

Characterization of prime, radical, and complemented elements in commutator lattices

Transfer of Davey's theorem to congruence lattices of rings

New insights into the structure of semiprime members in congruence-modular varieties

Abstract

In this paper, we transfer Davey`s characterization for $\kappa $--Stone bounded distributive lattices to lattices with certain kinds of quotients, in particular to commutator lattices with certain properties, and obtain related results on prime, radical, complemented and compact elements, annihilators and congruences of these lattices. We then apply these results to certain congruence lattices, in particular to those of semiprime members of semi--degenerate congruence--modular varieties, and use this particular case to transfer Davey`s Theorem to commutative unitary rings.