The Cuntz semigroup of a ring

Abstract

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