Core partial order in rings with involution

TL;DR

This paper explores the properties and characterizations of the core partial order in rings with involution, including the reverse order law and its relations to other partial orders.

Contribution

It provides new characterizations of core partial order and investigates the reverse order law for core invertible elements in rings with involution.

Findings

Characterizations of core partial order in rings with involution

Conditions for the reverse order law for core invertible elements

Relationships between core partial order and other partial orders

Abstract

Let $R$ be a unital ring with involution. We give several characterizations and properties of core partial order in $R$. In particular, we investigate the reverse order law $(ab)^{\tiny\textcircled{\tiny\#}} = b^{\tiny\textcircled{\tiny\#}} a^{\tiny\textcircled{\tiny\#}}$ for two core invertible elements $a,b\in R$. Some relationships between core partial order and other partial orders are obtained.