Rings in which elements are sum of a central element and an element in the Jacobson radical

An element in a ring R is called CJ if it is of the form c+j, where c belongs to the center and j is an element from the Jacobson radical. A ring R is called CJ if each element of R is CJ. We establish the basic properties of CJ rings, give several characterizations of these rings, and connect this notion with many standard elementwise properties such as clean, uniquely clean, nil clean, CN, and CU. We study the behavior of this notion under various ring extensions. In particular, we show that the subring C + J is always a CJ ring and that if R[x] is a CJ ring then R satisfies the Köthe conjecture.