Generalized Zhou inverses in rings

We introduce and study a new class of generalized inverses in rings. An element a in a ring R has generalized Zhou inverse if there exists b∈R such that bab=b,b∈comm2(a),an−ab∈J(R) for some n∈N. We prove that a∈R has generalized Zhou inverse if and only if there exists p=p2∈comm2(a) such that an−p∈J(R) for some n∈N. Cline’s formula and Jacobson’s lemma for generalized Zhou inverses are established. In particular, the Zhou inverse in a ring is characterized.