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