Boolean Algebras, Generalized Abelian Rings, and Grothendieck Groups

Abstract

ABSTRACT A ring R is called generalized Abelian if for each idempotent e in R, eR and (1 − e)R have no isomorphic nonzero summands. The class of generalized Abelian rings properly contains the class of Abelian rings. We denote by GAERS − 1 the class of generalized Abelian exchange rings with stable range 1. In this article we prove, by introducing Boolean algebras, that for any R ∈ GAERS − 1, the Grothendieck group K 0(R) is always an Archimedean lattice-ordered group, and hence is torsion free and unperforated, which generalizes the corresponding results of Abelian exchange rings. Our main technical tool is the use of the ordered structure of K 0(R)+, which provides a new method in the study of Grothendieck groups.