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

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