Equivalence of three different definitions of irreducible element

TL;DR

This paper compares three different definitions of irreducibility in rings, showing their equivalence within a specific class of rings called 'rings with only harmless zero divisors'.

Contribution

It clarifies the relationship among three concepts of irreducibility and identifies conditions under which they are equivalent.

Findings

The three definitions are equivalent in rings with only harmless zero divisors.

The paper delineates the specific class of rings where these concepts coincide.

It enhances understanding of irreducibility in algebraic structures.

Abstract

In this work, we try to draw a comparison among the three different concepts of irreducibility, given by Fletcher, Galovich, Bouvier; these three concepts are equivalent in a particular class of rings, called 'rings with only harmless zero divisors'.