Nil-Clean Rings with Involution

摘要

A*-ring R is called a nil *-clean ring if every element of R is a sum of a projection and a nilpotent.Nil*-clean rings are the version of nil-clean rings introduced by Diesl.This paper is about the nil*-clean property of rings with emphasis on matrix rings.We show that a*-ring R is nil*-clean if and only if J(R)is nil and R/J(R)is nil*-clean.For a 2-primal*-ring R,with the induced involution given by(aij)*=(a*ij)^(T),the nil*-clean property of Mn(R)is completely reduced to that of Mn(Zn).Consequently,Mn(R)is not a nil*-clean ring for n=3,4,and M2(R)is a nil*-clean ring if and only if J(R)is nil,R/J(R)is a Boolean ring and a*-a∈J(R)for all a∈R.