  An element of a ring is unique clean if it can be uniquely written as the sum
of an idempotent and a unit. A ring $R$ is uniquely $\pi$-clean if some power
of every element in $R$ is uniquely clean. In this article, we prove that a
ring $R$ is uniquely $\pi$-clean if and only if for any $a\in R$, there exists
an $m\in {\Bbb N}$ and a central idempotent $e\in R$ such that $a^m-e\in J(R)$,
if and only if $R$ is abelian; every idempotent lifts modulo $J(R)$; and $R/P$
is torsion for all prime ideals $P$ containing the Jacobson radical $J(R)$.
Further, we prove that a ring $R$ is uniquely $\pi$-clean and $J(R)$ is nil if
and only if $R$ is an abelian periodic ring, if and only if for any $a\in R$,
there exists some $m\in {\Bbb N}$ and a unique idempotent $e\in R$ such that
$a^m-e\in P(R)$, where $P(R)$ is the prime radical of $R$.
