Derivations On a C∞-ring

Abstract

The main purpose of this paper is to provide several results on objects lying between differential geometry and algebraic geometry such as C∞-rings and derivations on a C∞-ring. A C∞-ring is defined as a set with operations by C∞-functions on Euclidean spaces. A derivation on a C∞-ring is defined in two method, an ℝ-derivation and a C∞-derivation. The main result of this paper is to show that any ℝ-derivation is a C∞-derivation for some classes of C∞-rings (Theorems 3.1, 3.2).