Core partial order in rings with involution

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 $(ab{)}^{\stackrel{◯}{\#}}=b^{\stackrel{◯}{\#}}{a}^{\stackrel{◯}{\#}}$ for two core invertible elements $a,b\in R$. Some relationships between core partial order and other partial orders are obtained.