K0 of purely infinite simple regular rings

Abstract

We extend the notion of a purely in nite 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 in nite simple ring, then K0(R)+ = K0(R), the monoid of isomorphism classes of nitely generated projective R-modules is isomorphic to the monoid obtained from K0(R) by adjoining a new zero element, and K1(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 K0 of some purely in nite simple regular ring. Finally, some known examples are analyzed within this framework.