Relative $(b, c)$-Inverses with Respect to a Ring Endomorphism

Abstract:

      We study the relative properties of $(b, c)$-inverses with respect to a ring endomorphism. A new class of generalized inverses named $\alpha$-$(b, c)$-inverse is introduced and studied in a more general setting. We show by giving an example that $(b, c)$-inverses behave quite differently from $\alpha$-$(b, c)$-inverses. The condition that an $\alpha$-$(b, c)$-invertible element is precisely a $(b, c)$-invertible element is investigated. We also study the strongly clean decompositions for $\alpha$-$(b, c)$-inverses. Some well-known results on ($b, c$)-inverses are extended and unified.