On Uniquely Clean Rings

Abstract

A ring R is uniquely (nil) clean in case for any a ∈ R there exists a unique idempotent e ∈ R such that a − e ∈ R is invertible (nilpotent). We prove in this article that a ring R is uniquely clean if and only if R is an exchange ring with all idempotents central, and that for all maximal ideals M of R, R/M ≅ ℤ₂. It is shown that every uniquely clean ring of which every prime ideal of R is maximal is uniquely nil clean. Further, we prove that a ring R is uniquely nil clean if and only if R/J(R) is Boolean and R is a π-regular ring with all idempotents central.

Key Words:

• Maximal ideal
• Uniquely clean ring
• Uniquely nil clean ring

2000 Mathematics Subject Classification:

• 16E50
• 16U10

ACKNOWLEDGMENTS

The research of the author is supported by the Natural Science Foundation of Zhejiang Province (Y6090404) and the Fund of Hangzhou Normal University (200901).

Notes

Communicated by V. A. Artamonov.