I'm reading Hungerford, Abstract Algebra: An Introduction (1E, 1990), and the last example in Ch. 6 on ideals and quotient rings (after defining prime ideals, maximal ideals, and giving the theorem that an ideal is maximal iff the associated quotient ring is a field) says this:
EXAMPLE Let $T$ be the ring of continuous functions from $\mathbb R$ to $\mathbb R$ and let $I$ be the ideal of all functions $g$ such
that $g(2) = 0$. On pages 140-141 we saw that $T/I$ is a field
isomorphic to $\mathbb R$. Therefore $I$ is a maximal ideal in $T$. It
can be shown that every maximal ideal in $T$ is of this type.
Emphasis mine. Note that the latter claim appears in the 1E (1990) Hungerford book, but has been removed as of the 3E (2014).
Now, in the highlighted claim, what is meant by "of this type"? Does it mean a function of the form $g(c) = 0$ for some real number $c$? Or something else? And where is a proof of that fact given (whatever it indicates)?
Best Answer
Yes, you got the type right, I think. The type here being ideals of the form $I_c=\{f\in C^0~|~f(c)=0\}$ for some fixed $c\in\mathbb R$, where $C^0$ is the ring of continuous functions $\mathbb R\to\mathbb R$.
But the statement is not true. Consider for instance the ideal $C^0_c$ of continuous functions with compact support (that is, they are $0$ outside of some compact set). This ideal is obviously not trivial ($\neq C^0$), and we can prove that it is not contained in any ideal of the form $I_c$, so there must be a maximal ideal not of the form $I_c$ which contains $C^0_c$.
The ideal is not contained in any of the ideals $I_c$ because $C^0_c$ contains functions which don't vanish on the interval $(-a,a)$. But if $a> c$, then $c$ is not a root of that function, so can't be a common root of all functions in $C^0_c$, which is a prerequisite to being contained in $I_c$.
As mentioned before, $C^0_c$ being neither trivial, nor contained in any of the ideals $I_c$, implies that there must be a maximal ideal not of the form $I_c$.
The statement is true, however, if we let the domain of our functions be compact instead of $\mathbb R$. See here.