| We study the relative properties of (b, c)-inverses with respect to a ring endomorphism. A new class of generalized inverses named α-(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 α-(b, c)-inverse. The condition that an α-(b, c)- invertible element is precisely a (b, c)-invertible element is investigated. We also study the strongly clean decompositions for α-(b; c)-inverses. Some well-known results on (b, c)-inverses are extended and unified |
