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.
Categories: math.CT