  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 $b\in R$
such that $bab=b, b\in comm^2(a), a^n-ab\in \sqrt{J(R)}$ for some $n\in {\Bbb
N}$. We prove that $a\in R$ has generalized Zhou inverse if and only if there
exists $p=p^2\in comm^2(a)$ such that $a^n-p\in \sqrt{J(R)}$ for some $n\in
{\Bbb N}$. Cline's formula and Jacobson's Lemma for generalized Zhou inverses
are established. In particular, the Zhou inverse in a ring is characterized.
