Maximal ideals of the ring of all continuous functions

Question

Let $R$ be the ring of all continuous functions from the interval $[0,1]$ to $\Bbb R$. For each $a\in[0,1]$ let $$A_a=\{f\in R \mid f(a)=0\}$$

Now firstly, this is part of an assignment problem, please give me hints/insight into the problem, and not a solution.

I want to show that $A_a$ is a maximal ideal - but I am not sure how to think of my set. I have dealt with polynomial rings, perhaps some analogy?

Now, how to know this is a subring? Well if $f_1(a)=0$ and $f_2(a)=0$ clearly this is closed under subtraction and multiplication. Showing it is maximal is harder and I am not sure where to start.

Answer 1

You're supposed to see that $A_a$ is an ideal. That's stronger than being a subring. Anyway...

but I am not sure how to think of my set.

Hint: it's a kernel.

Answer 2

Consider the application $$\phi_a:R\to\Bbb R$$ defined by $\phi_a(f)=f(a)$.

Is $\phi_a$ a morphism? Is it surjective? What is its kernel? Remember that given a ring $S$ and an ideal $\mathfrak m$, then $\mathfrak m$ is maximal if and only if the quotient ring $S/\mathfrak m$ is a field.