Generalized Drazin inverses in a ring

Abstract

An element $a$ in a ring $R$ has generalized Drazin inverse if and only if there exists $b\in R$ such that $b=b^2a, b\in comm^2(a), a-a^2b
\in R^{qnil}.$ We prove that $a\in R$ has generalized Drazin inverse if and only if
there exists $p^3=p\in comm^2(a)$ such that $a+p\in U(R)$ and $ap\in R^{qnil}$. An element $a$ in a ring $R$ has pseudo Drazin inverse if and only if there exists $b\in comm^2(a)$ such that $b=b^2a, a^k-a^{k+1}b
\in J(R)$ for some $k\in {\Bbb N}$. We also characterize pseudo inverses by means of tripotents in a ring. Moreover, we prove that $a\in R$ has pseudo Drazin inverse
if and only if there exists $b\in comm^2(a)$ and $m,k\in {\Bbb N}$ such that $b^m=b^{m+1}a, a^k-a^{k+1}b\in J(R).$