  Let $R$ be a ring, $e$ an idempotent of $R$ and $\delta(R)$ denote the
intersection of all essential maximal right ideals of $R$ which is called Zhou
radical. In this paper, the Zhou radical of a ring is applied to the
$e$-reduced property of rings. We call the ring $R$ {\it Zhou right} (resp.
{\it left}) {\it $e$-reduced} if for any nilpotent $a$ in $R$, we have $ae\in
\delta(R)$ (resp. $ea\in \delta(R))$. Obviously, every ring is Zhou $0$-reduced
and a ring $R$ is Zhou right (resp., left) $1$-reduced if and only if
$N(R)\subseteq \delta(R)$. So we assume that the idempotent $e$ is nonzero. We
investigate basic properties of Zhou right $e$-reduced rings. Furthermore, we
supply some sources of examples for Zhou right $e$-reduced rings. In this
direction, we show that right $e$-semicommutative rings (and so right
$e$-reduced rings and $e$-symmetric rings), central semicommutative rings and
weak symmetric rings are Zhou right $e$-reduced. As an application, we deal
with some extensions of Zhou right $e$-reduced rings. Full matrix rings need
not be Zhou right $e$-reduced, but we present some Zhou right $e$-reduced
subrings of full matrix rings over Zhou right $e$-reduced rings.
