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, Z3 or the product of such rings, if and only if R weakly J-*-clean and a2R is uniquely *-clean for all aR, 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.