The inverse along an element in rings

TL;DR

This paper investigates the properties and characterizations of the inverse along an element in unitary rings, including special cases like the group, Drazin, and Moore-Penrose inverses, providing a comprehensive theoretical framework.

Contribution

It introduces new characterizations and descriptions of the inverse along an element in rings, including commuting cases and special inverse types, expanding existing algebraic theory.

Findings

Characterization of the existence of the inverse along an element

Full description of invertible elements along a fixed element

Characterization of commuting inverse along an element

Abstract

In this article several properties of the inverse along an element will be studied in the context of unitary rings. New characterizations of the existence of this inverse will be proved. Moreover, the set of all invertible elements along a fixed element will be fully described. Futhermore, commuting inverse along an element will be characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) will be also considered.