Central Drazin inverses

We introduce and study a subclass of the Drazin invertible elements in a ring [Formula: see text], which are called central Drazin invertible. An element [Formula: see text] is said to be central Drazin invertible if there exists [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for some integer [Formula: see text]. Some basic properties of the central Drazin inverse are obtained. Of particular interest are the central Drazin invertible elements that are simultaneously group invertible, which we show have a property generalizing strong cleanness. Some well-known results related to the cleanness of rings and the reverse order law are generalized.