e-Reduced rings in terms of the Zhou radical

Let R be a ring, e an idempotent of R and δ(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 { Zhou right} (resp. { left}) { e-reduced} if for any nilpotent a in R, we have ae∈ δ(R) (resp. ea∈ δ(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)⊆ δ(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.