Core partial order and core orthogonality comparing with star partial order and star orthogonality

In this paper, we find that the core partial order and the star partial order not only have the same properties, but also have the difference in an arbitrary ring or a \(*\) $\ast$ -reducing ring. A number of new characterizations for the core, star partial order and the strong core, star orthogonality are presented, the proofs of which are only based on the ring theory. In particular, several examples are given to illustrate our results.