The quaternion over the ring of Colombeau's full generalized numbers

TL;DR

This paper extends the algebraic and topological analysis of quaternions over Colombeau's full generalized numbers, building on previous work on simplified versions, and explores their ring properties and ideal structures.

Contribution

It introduces and investigates the topological algebra of quaternions over Colombeau's full generalized numbers, extending prior results and analyzing their ring-theoretic properties.

Findings

Classified dense ideals of the algebraic structure

Established the existence of a maximal ring of quotients that is Von Neumann regular

Extended previous results from simplified to full Colombeau generalized numbers

Abstract

In this paper, we extend the results obtained by Cortes-Ferrero-Juriaans (2009) for the quaternion over the ring Colombeau's simplified generalized numbers, denoted by $\overline{\mathbb{H}}_s$, to the quaternion over the ring of Colombeau's full generalized numbers, denoted by $\overline{\mathbb{H}}$. In this paper, we introduce and investigate the topological algebra of the quaternion over the ring of Colombeau's full generalized numbers. This is an important object to study if one wants to build the algebraic theory of Colombeau's full generalized numbers $\overline{\mathbb{K}}$ studied by Aragona-Garcia-Juriaans (2013). We study some ring theoretical properties of $\overline{\mathbb{H}}$, we classify the dense ideals of $\overline{\mathbb{K}}$ in the algebraic sense, and as a consequence, it has a maximal ring of quotients which is Von Neumann regular.