Ideals, quotients, and continuity of the Cuntz semigroup for rings

TL;DR

This paper investigates how the Cuntz semigroup relates to the ideal structure of rings, identifying classes of ideals parametrized by these semigroups and establishing their behavior under quotients and limits.

Contribution

It introduces the concepts of quasipure and decomposable ideals, and proves the continuity of the Cuntz semigroup functor for certain classes of rings.

Findings

$ ext{S}(R)$ is an abstract Cu-semigroup for left normal rings.

The assignment $R o ext{S}(R)$ is continuous for left normal rings.

$ ext{S}(R)$ and $ ext{Λ}(R)$ behave well with respect to quotients.

Abstract

In this paper we explore which part of the ideal lattice of a general ring is parametrized by its Cuntz semigroup $\mathrm{S}(R)$ and its ambient semigroup $\Lambda(R)$. We identify these classes of ideals as the quasipure ideals (a generalization of pure ideals) in the case of $\mathrm{S}(R)$, and what we term decomposable ideals in the case of $\Lambda(R)$. For an ($s$-)unital ring $R$, the latter class exhausts all ideals of the ring. We prove that these constructions behave well with respect to quotients. In order to study the passage to inductive limits, we introduce the classes of dense and left normal rings. We show that $\mathrm{S}(R)$ is an abstract Cu-semigroup whenever $R$ is left normal and, for such rings, the assignment $R\mapsto \mathrm{S}(R)$ is continuous. We prove a parallel result for $\Lambda(R)$ whenever $R$ is a dense ring.