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

In this paper, we explore which part of the ideal lattice of a general ring is parametrized by its Cuntz semigroup [Formula: see text] and its ambient semigroup [Formula: see text]. We identify these classes of ideals as the quasipure ideals (a generalization of pure ideals) in the case of [Formula: see text], and what we term decomposable ideals in the case of [Formula: see text]. For an ([Formula: see text]-)unital ring [Formula: see text], 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 [Formula: see text] is an abstract [Formula: see text]-semigroup whenever [Formula: see text] is left normal and, for such rings, the assignment [Formula: see text] is continuous. We prove a parallel result for [Formula: see text] whenever [Formula: see text] is a dense ring.