C-infinity ring

Summary

A $C^\infty$-ring is a set on which every smooth function $f \colon \R^n \to \R$ acts as an $n$-ary operation, compatibly with composition. The basic example is the ring $C^\infty(M)$ of smooth functions on a smooth manifold, where the operations are composition with $f$. Every $C^\infty$-ring is in particular a commutative $\R$-algebra, but it also knows how to apply $\exp$, $\sin$ and all other smooth functions. $C^\infty$-rings play for smooth manifolds the role that commutative rings play in algebraic geometry, and they give models of synthetic differential geometry.