Multiplicative lattices: maximal implies prime and related questions

TL;DR

This paper explores the properties of multiplicative lattices, establishing conditions under which maximal elements are prime, and investigates related structures like m-systems from both algebraic and topological perspectives.

Contribution

It introduces a Prime Ideal Principle for multiplicative lattices, showing maximal elements are prime in various contexts, including commutative rings with identity.

Findings

Maximal elements are prime in several classes of multiplicative lattices.

The Prime Ideal Principle is established for these lattices.

Topological properties of m-systems are analyzed.

Abstract

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of m-systems. The notion of m-system is also studied from the topological point of view.