Example 2

Let T be a locally compact space, and let $\mathscr{C}_0(T)$ be the commutative $\mathbf{C}$-algebra of continuous mappings from T to $\mathbf{C}$ that tend to 0 at infinity (Gen. Top., X, §4, No. 4, p. 316); it is unital if and only if T is compact. Let F be a closed subset of T, and let $\mathfrak{a}_F$ be the set of elements of $\mathscr{C}_0(T)$ whose restriction to F is the zero function; it is an ideal of $\mathscr{C}_0(T)$. An element $u$ of $\mathscr{C}_0(T)$ is a unit modulo $\mathfrak{a}_F$ if and only if we have $u(t) = 1$ for every $t\in F$. Such an element $u$ exists, that is, $\mathfrak{a}_F$ is regular, if and only if F is compact. The mapping $t\mapsto \mathfrak{a}_{\{t\}}$ is a bijection from T to the set of regular maximal ideals of $\mathscr{C}_0(T)$ (TS, I, §3, n$^o2$, p. 32, corollaire 1). Suppose that T is not compact, and denote by $\mathfrak{a}$ the subset of $\mathscr{C}_0(T)$ consisting of the functions with compact support; then $\mathfrak{a}$ is an ideal of $\mathscr{C}_0(T)$ that is not contained in any regular ideal of $\mathscr{C}_0(T)$.