Weak clean index of a ring

摘要

Motivated by the concept of clean index of rings (Lee and Zhou in Commun Algebra 40:807-822, 2012), we introduce the concept of weak clean index of rings. For any element a of a ring R with unity, we define χ(a) = {e ∈ R | e~2 = e and a -e or a + e is a unit}. The weak clean index of R is defined as sup{|χ(a)| : a ∈ R} and it is denoted by Win(R), where |χ(a)| denotes the cardinality of the set χ(a). In this article, we characterize rings of weak clean indices 1, 2 and 3.