Generalization of sharp and core partial order using annihilators

TLDR: In this article, the sharp order is extended to the set of elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent.

Abstract: The sharp order is a well known partial order defined on the set of complex matrices with index less or equal one. Following Semrl's approach, Efimov extended this order to the set of those bounded Banach space operators $A$ for which the closure of the image and kernel are topologically complementary subspaces. In order to extend the sharp order to arbitrary ring $R$ (particulary to Rickart and Rickart $*$-rings) we use the notions of annihilators. The concept of the sharp order is extended to the set $\mathcal{I}_R$ of those elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent. It is proved that the sharp order is a partial order relation on $\mathcal{I}_R$. Following the idea we also extend and discuss the recently introduced concept of core partial order.