Resumen de Generalizations of Cline's formula for three generalized inverses

Yong Jiang, Yongxian Wen, Qingping Zeng

• It is shown that an element a in a ring is Drazin invertible if and only if so is an; the Drazin inverse of a is given by that of an, and vice versa. Using this result, we prove that, in the presence of aba=aca, for any natural numbers n and m, (ac)n is Drazin invertible in a ring if and only if so is (ba)m; the Drazin inverse of (ac)n is expressed by that of (ba)m, and vice versa. Also, analogous results for the pseudo Drazin inverse and the generalized Drazin inverse are established on Banach algebras.