Generalized Zhou inverses in rings

Abstract

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