On some partial orders on a certain subset of the power set of rings

Let be a ring with identity and let be a collection of subsets of such that their left and right annihilators are generated by the same idempotent. % from . We extend the notion of the sharp, the left-sharp, and the right-sharp partial orders to , present equivalent definitions of these orders, and study their properties. We also extend the concept of the core and the dual core orders to , show that they are indeed partial orders when is a Baer *-ring, and connect them with one-sided sharp and star partial orders.