Inner-star and star-inner partial orders in ∗-rings

Abstract

The notions of inner-star and star-inner generalized inverses are introduced on the set of all regular elements in a ∗-ring [Formula: see text]. We thus extend the concept of inner-star and star-inner complex matrices. We study properties of these hybrid generalized inverses on the set [Formula: see text] of all Moore–Penrose invertible elements in [Formula: see text] and thus generalize some known results. Partial orders that are induced by inner-star and star-inner inverses are introduced on [Formula: see text], their properties are examined, and their characterizations are presented.