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

In this paper we present the notion of a von Neumann regular ${\mathcal{C}}^{\mathrm{\infty }}-$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 ${\mathcal{C}}^{\mathrm{\infty }}-$rings is a reflective subcategory of ${\mathcal{C}}^{\mathrm{\infty }}\mathbf{R}\mathbf{n}\mathbf{g}$. We prove that every homomorphism between Boolean algebras can be represented by a ${\mathcal{C}}^{\mathrm{\infty }}-$ring homomorphism between von Neumann regular ${\mathcal{C}}^{\mathrm{\infty }}-$rings.