Clean Index of Rings

Abstract

A ring is called clean (resp., uniquely clean) if each of its elements can be (resp., uniquely) expressed as the sum of an idempotent and a unit. Motivated by recent work on uniquely clean rings in [6], we introduce the clean index of a ring R. For a ∈ R, let ℰ(a) = {e ∈ R: e ² = e, a − e ∈ U(R)} where U(R) is the group of units of R and the clean index of R, denoted in(R), is defined by in(R) = sup{|ℰ(a)|: a ∈ R}. Thus, R is uniquely clean if and only if R is clean with in(R) = 1. So far, uniquely clean rings are the only clean rings whose structure is fully understood (see [6]). In this article, we characterize the (arbitrary) rings of clean indices 1, 2, 3 and determine the abelian rings of finite clean index. Applications to semipotent rings, semiprime rings, and clean rings are discussed.

Key Words:

• Clean index
• Clean ring
• Formal triangular matrix ring
• Local ring
• Semipotent ring
• Uniquely clean ring

2000 Mathematics Subject Classification:

• Primary 16U99

ACKNOWLEDGMENTS

Part of the work was carried out when the second author was visiting the National Taiwan University sponsored by NCTS of Taipei. He gratefully acknowledges the financial support from NCTS and kind hospitality from the host university. The research of the first author was supported by NSC of Taiwan, and that of the second author by NSERC of Canada.

Notes

Communicated by T. Albu.