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

A ring R is uniquely π-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 π-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 π-clean ring has nil Jacobson radical.