On some generalized inverses and partial orders in ∗-rings
Janko Marovt, Dijana Mosić, Insa Cremer · Journal of Algebra and Its Applications · 2022
Let [Formula: see text] be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all Moore–Penrose invertible elements in [Formula: see text]. We study partial orders on [Formula: see text] that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart ∗-rings the concept of another partial order, called the plus order, which has been recently introduced on the set of all bounded linear operators between Hilbert spaces. Properties of these relations are investigated and some known results are thus generalized.