Rings in which the power of every element is the sum of an idempotent and a unit

Abstract

A ring R is uniquely ?-clean if the power of every element can be uniquely written as the sum of an idempotent and a unit. We prove that a ring R is uniquely ?-clean if and only if for any a ? R, there exists an integer m and a central idempotent e ? R such that am ? e ? J(R), if and only if R is Abelian; idempotents lift modulo J(R); and R/P is torsion for all prime ideals P ? J(R). Finally, we completely determine when a uniquely ?-clean ring has nil Jacobson radical.