Generalized Core Inverse in a proper $*$-ring

TL;DR

This paper introduces and characterizes weak core and central weak core inverses in proper *-rings, providing representations, properties, and examples, including applications to quaternion algebra.

Contribution

It defines new classes of generalized inverses in proper *-rings and explores their properties, representations, and connections with other inverses and algebraic structures.

Findings

Characterizations and representations of weak core and central weak core inverses.

Additive properties and explicit expressions for these inverses.

Examples including quaternion algebra illustrating the concepts.

Abstract

In this paper, we introduce the notion of weak core and central weak core inverse in a {\it proper $*$-ring}. We further elaborate on these two classes by producing a few representations and characterizations of the weak core and central weak core invertible elements. We investigated additive properties and a few explicit expressions for these two classes of inverses through other generalized inverses. In addition, numerical examples are provided to validate claims on weak core inverses. Following {\it proper $*$-ring} and their interconnections with Clifford algebra, we also present examples of the group inverse and the weak core inverse of a non-zero non-invertible quaternion $\mathbb{H}_s$.