On Medium *-Clean Rings

Abstract

A *-ring R is called a medium *-clean ring if every element in R is the sum or difference of an element in its Jacobson radical and a projection that commute. We prove that a ring R is medium *-clean if and only if R is strongly *-clean and R / J(R) is a Boolean ring, $${\mathbb {Z}}_3$$ or the product of such rings, if and only if R weakly J-*-clean and $$a^2\in R$$ is uniquely *-clean for all $$a\in R$$ , if and only if every idempotent lifts modulo J(R), R is abelian and R / J(R) weakly *-Boolean. A subclass of medium *-clean rings with many nilpotents is thereby characterized.