Von Neumann Regular $\mathcal{C}^{\infty}-$Rings and Applications

TL;DR

This paper introduces and studies von Neumann regular $ ext{C}^ty$-rings, establishing their properties, categorical relationships, and representation of Boolean algebra homomorphisms within this framework.

Contribution

It defines von Neumann regular $ ext{C}^ty$-rings, proves their categorical properties, and shows how Boolean algebra homomorphisms can be represented by $ ext{C}^ty$-ring homomorphisms.

Findings

The category of von Neumann regular $ ext{C}^ty$-rings is reflective in $ ext{C}^ty$-rings.

Homomorphisms between Boolean algebras can be represented by $ ext{C}^ty$-ring homomorphisms.

Properties of von Neumann regular $ ext{C}^ty$-rings are characterized and analyzed.

Abstract

In this paper we present the notion of a von Neumann regular $\mathcal{C}^{\infty}-$ring, we prove some results about them and we describe some of their properties. We prove, using two different methods, that the category of von Neumann regular $\mathcal{C}^{\infty}-$rings is a reflective subcategory of $\mathcal{C}^{\infty}{\rm \bf Rng}$. We prove that every homomorphism between Boolean algebras can be represented by a $\mathcal{C}^{\infty}-$ring homomorphism between von Neumann regular $\mathcal{C}^{\infty}-$rings.