Centralizer’s applications to the (b, c)-inverses in rings

Abstract

We give several conditions in order that the absorption law for one sided (b, c)-inverses in rings holds. Also, by using centralizers, we obtain the absorption law for the (b, c)-inverse and the reverse order law of the (b, c)-inverse in rings. As applications, we obtain the related results for the inverse along an element, Moore-Penrose inverse, Drazin inverse, group inverse and core inverse.

Characterizations of annihilator $(b,c)$-inverses in arbitrary rings

“…For more applications of centralizers to generalized inverses, we refer readers to [16][17][18]. Likewise, our results in this section seem to propose a new approach to study m-EP elements in * -rings.…”

“…xF(y) = F(xy)) for all x, y ∈ S. Corollary 3.7 (cf. [16,Lemma 3.6]). Let a i , b i , c i ∈ S, and suppose that a i is (b i , c i )invertible with the (b i , c i )-inverse x i , i = 1, 2.…”