On EP-nilpotent decomposition in $\ast$-rings

Let $\mathcal{R}$ be a unital ring with involution. We generalize the notion of the EP-nilpotent decomposition from the set of all $n\times n$ complex matrices to the set of all core-EP invertible elements in $\mathcal{R}$, and study certain relations in $\mathcal{R}$ that are based on the EP-nilpotent decomposition. The notions of the ES and EN partial orders are extended to $\mathcal{R}$. Properties of these relations are investigated and new characterizations are presented.