A Very Brief Note on Some Commutative Algebraic Properties of a Dirac-Fueter Modified Equation

TL;DR

This paper investigates the algebraic properties of solutions to a modified Fueter-Dirac equation in four dimensions, revealing a structured relationship with a special commutative ring related to quaternions, and exploring their local algebraic behavior.

Contribution

It introduces a novel algebraic framework for solutions of a modified Fueter-Dirac equation, highlighting their structure within a specific commutative ring with non-invertibles and localization.

Findings

Solutions exhibit a structured algebraic behavior.

Solutions relate to a commutative ring with non-invertibles.

Localization of the ring reveals additional properties.

Abstract

We call attention to the unusual properties that the 4 dimensional solutions for a modified Fueter-Dirac equations satisfy: In a coordinate-free, constant-free and strictly mathematical way it is possible to show that all the solutions for a modified Fueter-Dirac Equation, which are radial symmetric when restricted to the 3-space, have a nice algebraic structure. Locally, these solutions behave naturally in a algebraic way determined by a certain commutative ring related to the quaternions with non-invertibles. This ring has a nontrivial localization. By using left and right versions of the operator we obtain quirality.