On some generalized inverses and partial orders in $\ast$-rings

Abstract

Let $\mathcal{R}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(\mathbb{C)}$, the set of all $m\times n $ matrices over $\mathbb{C}$, to the set $\mathcal{R}% ^{\dagger}$ of all Moore-Penrose invertible elements in $\mathcal{R}$. We study partial orders on $\mathcal{R}^{\dagger}$ that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart $\ast $-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.