On partial orders in proper $*$-rings

Abstract

We study orders in proper $*$-rings that are derived from the core-nilpotent decomposition. The notion of the C-N-star partial order and the S-star partial order is extended from $M_ {n} ( \mathbb{C)}$, the set of all $n \times n$ complex matrices, to the set of all Drazin invertible elements in proper $*$-rings with identity. Properties of these orders are investigated and their characterizations are presented. For a proper $*$-ring $\mathcal{A}$ with identity, it is shown that on the set of all Drazin invertible elements $a \in \mathcal{A}$ where the core part of $a$ is an EP element, the C-N-star partial order implies the star partial order.