Exercise 5, § 5

Let $A$ be a regular commutative Banach algebra without radical. Suppose that every closed ideal of $A$ is intersection of regular maximal ideals. Let $x\in A$. Let F be the set of zeros of $\mathscr{G}'(x)$. Then $x$ is the limit of elements $u_nx$, where $u_n\in A$ and $\mathscr{G}(u_n)$ vanishes in a neighbourhood of F. (Let J be the set of $y\in A$ such that $\mathscr{G}(y)$ has compact support disjoint from F. We have $x\in \overline{J}$. Use prop. 4 of I, p. 91.) In particular, A satisfies the Ditkin condition.