A note on weakly clean rings. (Q653857)

Let \(R\) be an associative ring. An element \(x\in R\) is called weakly clean if \(x\) is the sum of a unit element and an idempotent of \(R\). A ring is weakly clean if every element of it is weakly clean. As the main theorem the authors prove that a ring \(R\) is weakly clean if and only if for any \(x\in R\) there is an idempotent \(e\in R\) such that \(e\in xR\) and either \(1-e\in(1-x)R\) or \(1-e\in(1+x)R\). Two characterizations of unit regular rings are given in Theorems 2.2 and 2.3.