Collection of smooth real valued functions on smooth manifold has ring structure.

Question

For any smooth manifold $M$, how do I see that the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and that every point $x \in M$ determines a ring homomorphism $F \to \mathbb{R}$ and hence a maximal ideal in $F$?

Answer 1

For $f,g \in F$, define $f+g$ and $fg$ by $(f+g)(x)=f(x)+g(x)$ and $fg(x)=f(x)g(x)$. This makes $F$ into a ring (even more, it is a commutative $\Bbb{R}$-algebra).

Now, for every point $x\in M$, there is a homomorphism $F\to \Bbb{R}:f\mapsto f(x)$. This map is clearly surjective (because of constant functions for example) and its kernel is $\mathfrak{m}_x=\{f\in F\mid f(x)=0\}$. Since $F/\mathfrak{m}_x \cong \Bbb{R}$ is a field, $\mathfrak{m}_x$ is a maximal ideal of $F$.