Centralizer's applications to the inverse along an element

TL;DR

This paper explores properties of the inverse along an element in ring theory, establishing absorption and reverse order laws, and characterizes the Moore-Penrose inverse using centralizers and one-sided invertibilities.

Contribution

It introduces new absorption and reverse order laws for the inverse along an element and characterizes the Moore-Penrose inverse via centralizers in rings.

Findings

Absorption law for the inverse along an element proven

Reverse order law established using centralizers

Moore-Penrose inverse characterized by one-sided invertibilities

Abstract

In this paper, we first prove that the absorption law for one-sided inverses along an element holds, deriving the absorption law for the inverse along an element. We then apply this result to obtain the absorption law for the inverse along different elements. Also, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring. Finally, we characterize the Moore-Penrose inverse by one-sided invertibilities in a ring with involution.