The inverse along a product and its applications

TL;DR

This paper investigates the inverse along a product in ring theory, providing criteria for its existence and applications to matrices, thereby advancing understanding of generalized inverses in algebraic structures.

Contribution

It introduces an existence criterion for the inverse along a product in rings and applies this to derive conditions and expressions for the inverse along matrices.

Findings

Established a criterion for the existence of the inverse along a product in rings.

Derived equivalent conditions for the inverse along a matrix.

Provided explicit expressions for the inverse along matrices.

Abstract

In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix.