On strongly \(\pi\)-regular group rings (Q1396727)

An element \(x\) in a ring \(R\) is said to be left or right \(\pi\)-regular if there exists \(y\in R\) and a positive integer \(n\) such that \(x^n=yx^{n+1}\) or \(x^n=x^{n+1}y\), respectively. If \(x\) is both left and right \(\pi\)-regular, then it is strongly \(\pi\)-regular, and \(R\) is said to be a strongly \(\pi\)-regular ring if all its elements have this property. This paper determines some necessary and some sufficient conditions for a group ring to be strongly \(\pi\)-regular.