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

Abstract

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. % from 𝓡. 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.