The Cuntz semigroup of a ring

TL;DR

This paper introduces a new invariant called the Cuntz semigroup for rings, extending concepts from C*-algebra theory to algebraic structures, and explores its properties and computations in various ring classes.

Contribution

It defines the Cuntz semigroup for rings, provides two equivalent descriptions, and computes it for several classes of rings, connecting it to existing invariants and C*-algebra theory.

Findings

Defined the Cuntz semigroup for rings.

Provided computations for specific ring classes.

Connected the invariant to C*-algebra Cuntz semigroup.

Abstract

For any ring $R$, we introduce an invariant in the form of a partially ordered abelian semigroup $\mathrm{S}(R)$ built from an equivalence relation on the class of countably generated projective modules. We call $\mathrm{S}(R)$ the Cuntz semigroup of the ring $R$. This construction is akin to the manufacture of the Cuntz semigroup of a C*-algebra using countably generated Hilbert modules. To circumvent the lack of a topology in a general ring $R$, we deepen our understanding of countably projective modules over $R$, thus uncovering new features in their direct limit decompositions, which in turn yields two equivalent descriptions of $\mathrm{S}(R)$. The Cuntz semigroup of $R$ is part of a new invariant $\mathrm{SCu}(R)$ which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both $\mathrm{S}(R)$ and $\mathrm{SCu}(R)$ in a number of interesting situations, such as unit-regular rings, semilocal rings, and in the context of nearly simple domains. We also relate our construcion to the Cuntz semigroup of a C*-algebra.