On G-Drazin partial order in rings

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