Nil-quasi-clean companion matrices

Abstract

Let \(R\) be a ring with identity. An element \(e\) in \(R\) is called a quasi-idempotent element if \(e^2=ke\) for some central unit \(k\) in \(R\).  For an element \(b\) in \(R\), if there is a positive integer \(m\) such that \(b^m=0\), then \(b\) is called a nilpotent element of \(R\). An element \(r\) in \(R\) is called a nil-quasi-clean element if \(r\) is a sum of a quasi-idempotent and a nilpotent. If every element of \(R\) is nil-quasi-clean, then \(R\) is called a nil-quasi-clean ring. This paper completely determines nil-quasi-clean companion matrices over a field.