On G-Drazin partial order in rings

Abstract

We extend the concept of a G-Drazin inverse from the set \(M_n\) of all \(n\times n\) complex matrices to the set \(\mathcal{R}^{D}\) of all Drazin invertible elements in a ring \(\mathcal{R}\) with identity. We also generalize a partial order induced by G-Drazin inverses from \(M_n\) to the set of all regular elements in \(\mathcal{R}^{D}\), study its properties, compare it to known partial orders, and generalize some known results.