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

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