C-infinity ring

Remarks

Equivalently, a $C^\infty$-ring is a model of the Lawvere theory whose $n$-ary operations are the smooth maps $\R^n \to \R$. In other words, it is a product-preserving functor to sets from the category of the spaces $\R^n$ with smooth maps. Using $f(x, y) = x + y$, $f(x, y) = xy$ and the constant functions, every $C^\infty$-ring is a commutative ring and an $\R$-algebra.

Example: for a smooth manifold $M$, let $A = C^\infty(M)$ and $\Phi_f(a_1, \dots, a_n) = f \circ (a_1, \dots, a_n)$. Clause 1 holds since $\pi_i \circ (a_1, \dots, a_n) = a_i$, and clause 2 is associativity of composition. The ring of germs of smooth functions at a point is a $C^\infty$-ring in the same way. The ring $\R[\varepsilon]/(\varepsilon^2)$ is a $C^\infty$-ring with $\Phi_f(a + b\varepsilon) = f(a) + f'(a) b \varepsilon$ for $n = 1$, and the analogous formula with the gradient for larger $n$; clause 2 is the chain rule.

Non-example: on $A = \R$, the rule $\Phi_f(a_1, \dots, a_n) = f(0, \dots, 0)$ fails clause 1, since $\Phi_{\pi_1}(a) = 0 \ne a$ for $a \ne 0$.