Core partial order in rings with involution

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.