2. Ideals of functions

Proposition. Let ${\cal R}$ be the ring of all functions from R$\rightarrow$   R [reals to reals], or the ring of all continuous functions, or all differentiable functions, or all polynomials. For a particular number $x$, let $M _ x = \{f \in {\cal R} \vert f(x) = 0\}$, the ideal of all functions that vanish at $x$. Then $M _ x$ is a maximal ideal of ${\cal R}$.

Proof. Consider the evaluation map $e _ x: {\cal R} \rightarrow$   R given by $e _ x (f) = f(x)$. Since $e _ x$ is a homomorphism of ${\cal R}$ onto R with $M _ x$ as its kernel, $M _ x$ is maximal.