Quasi-clean rings

Exchange rings have been characterized by the property that, for all a∈R, there exists an idempotent e∈aR such that 1-e∈(1-a)R. In this thesis, we introduce the notion of quasi-exchange rings via quasi-idempotent elements. We prove that a ring R is quasi-exchange if and only if, given a right ideal I of a ring R and x∈R with kx-x^2∈I where k is a central unit of R, there exist a quasi-idempotent q^2=kq such that q-x∈I. We also show that R is a quasi-exchange ring if and only if R⁄(J(R)) a quasi-exchange ring and R satisfies the previons equivalent condition for the Jacobson radical J(R). Some relations between quasi-exchange rings and (strongly) quasi-clean rings are also obtained.