Ideals, quotients, and continuity of the Cuntz semigroup for rings
Ramon Antoine, Pere Ara, Joan Bosa, Francesc Perera, Eduard Vilalta
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 $Λ(R)$. 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 $Λ(R)$. 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 $Λ(R)$ whenever $R$ is a dense ring.