Extensions of Jacobson's lemma for generalized inverses in a ring

Let $R$ be an associative ring with unit $1$, and $a,b,c\in R$ satisfy $a(ba{)}^2=abaca=acaba=(ac{)}^{2}a$, this paper proves that $1-ac$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively) if and only if $1-ba$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively). In particular, we obtain new common spectral properties for $ac$ and $ba$ in Banach algebras. As applications, new extension of Jacobson’s lemma for B-Fredholm elements and generalized Fredholm elements in rings is established.