An Extension of the Reduced Property in Rings

Let [Formula: see text] be a ring, [Formula: see text] and [Formula: see text] stand for the intersection of all essential maximal right ideals of [Formula: see text] which is called the Zhou radical. The aim of this paper is to apply the Zhou radical to the [Formula: see text]-reducedness of rings. In this direction, a ring [Formula: see text] is said to be Zhou right (resp., left) [Formula: see text]-reduced if for any nilpotent [Formula: see text] in [Formula: see text], we have [Formula: see text] (resp., [Formula: see text]. For nonzero idempotents, we investigate properties and supply examples of Zhou right [Formula: see text]-reduced rings. Along these lines, we show that right [Formula: see text]-semicommutative rings (and so right [Formula: see text]-reduced rings and [Formula: see text]-symmetric rings), central semicommutative rings and weak symmetric rings are Zhou right [Formula: see text]-reduced. As an application, we deal with some extensions of Zhou right [Formula: see text]-reduced rings.