Colombeau's Algebra of full Generalized Numbers

TL;DR

This paper explores the topological and algebraic properties of Colombeau's full generalized numbers, establishing an ultra-metric structure and initiating their study within commutative algebra and topology.

Contribution

It introduces an ultra-metric on Colombeau's full generalized numbers and begins the investigation of their algebraic and topological properties.

Findings

The ring can be endowed with an ultra-metric making it a topological ring.

Initial questions about its algebraic and topological structure are addressed.

The work opens pathways for further research in algebra and topology of generalized number rings.

Abstract

Let $\bar{\Kset}_f$ denote the commutative unital ring of Colombeau's full generalized numbers. This ring can be endowed with an ultra-metric in such a way that it becomes a topological ring. There are many interesting question about $\bar{\Kset}_f$ in the framework of Commutative Algebra and General Topology as well as of the superposition of these two subjects. The purpose of this paper aims to give an initial step toward the study of this ring.