$K_0$ of purely infinite simple regular rings

TL;DR

This paper extends the concept of purely infinite simple C*-algebras to unital rings, exploring their K-Theory properties, constructing new examples, and showing that every countable abelian group can be realized as their K_0 group.

Contribution

It introduces a ring-theoretic framework for purely infinite simple rings, develops construction techniques, and characterizes their K-theoretic invariants.

Findings

K_0(R)^+ = K_0(R) for purely infinite simple rings

The monoid of finitely generated projective modules is isomorphic to K_0(R) with a zero element added

Every countable abelian group is isomorphic to K_0 of some purely infinite simple regular ring

Abstract

We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if $R$ is a purely infinite simple ring, then $K_0(R)^+= K_0(R)$, the monoid of isomorphism classes of finitely generated projective $R$-modules is isomorphic to the monoid obtained from $K_0(R)$ by adjoining a new zero element, and $K_1(R)$ is the abelianization of the group of units of $R$. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we compute the Grothendieck groups of these examples. In particular, we prove that every countable abelian group is isomorphic to $K_0$ of some purely infinite simple regular ring. Finally, some known examples are analyzed within this framework.