Boolean Algebras, Generalized Abelian Rings, and Grothendieck Groups

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.

Key Words:

• Archimedean property
• Boolean algebra
• Exchange ring
• Generalized Abelian ring
• Grothendieck group
• Lattice-ordered group (ℓ-group)
• Stable range 1

AMS Subject Classification (2000):

• 19A49
• 16E20
• 06F15
• 06E20

ACKNOWLEDGMENTS

The authors would like to thank the referees for their many helpful comments and valuable suggestions, which have greatly improved the presentation of this article. Also, the first author was partially supported by the NSFC Grant, and the second author was partially supported by the National Distinguished Youth Science Foundation of China Grant.

Notes

Communicated by A. Facchini.