Comaximal Factorization Lattices

TL;DR

This paper extends the study of comaximal ideal factorizations from integral domains to the broader context of multiplicative lattices, providing a unified lattice-theoretic framework for these factorizations.

Contribution

It generalizes key results about comaximal factorizations from domains to multiplicative lattices, broadening the theoretical understanding.

Findings

Most results on comaximal factorizations in domains are obtainable in multiplicative lattices.

The prime spectrum structure influences factorization properties in lattices.

The paper establishes conditions under which lattice elements have prime radical or primary factorizations.

Abstract

Brewer and Heinzer studied the (integral) domains D having the property that each proper ideal A of D has a comaximal ideal factorization with some additional property. They proved that for a domain D, the following are equivalent: (1) Each proper ideal A of D has a comaximal factorization where the factors have prime radical (resp. are primary, resp. are prime powers). (2) The prime spectrum of D is a tree under inclusion and each ideal of D has only finitely many minimal primes (resp. D is one dimensional and each ideal of D has only finitely many minimal primes, resp. D is a Dedekind domain). The aim of this paper is to show that most of the results can be obtained in the setup of multiplicative lattices.