Revista de la
Unión Matemática Argentina
On partial orders in proper $*$-rings
Janko Marovt
Volume 59, no. 1 (2018), pp. 193–204


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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.