Lifting Central Idempotents Modulo Jacobson Radicals and Ranks of K₀ Groups

Abstract

Let R be a ring and let J(R) be the Jacobson radical of R. We discuss the problem of determining when the central idempotents in R/J(R) can be lifted to R. If R is a noetherian (artinian) ring, we give some conditions relative to the ranks of K ₀ groups under which the central idempotents in R/J(R) can be lifted. In particular, when R is semilocal, these conditions are necessary and sufficient. Moreover, we consider ranks of K ₀ groups of pullbacks of rings and obtain the upper and lower bounds on them under some suitable conditions.

Key Words:

• Jacobson radical
• Lifting central idempotents
• K ₀ groups
• Pullbacks

2000 Mathematics Subject Classification:

• 16E20
• 18F25
• 18F30
• 19A49

ACKNOWLEDGMENTS

The author would like to give many thanks to the referee for many useful comments and suggestions. This work was partially supported by the Natural Science Foundation of Jiangsu province of China (BK 2007133).

Notes

Communicated by M. Ferrero.