Exercise 2, § 5

Let A be a commutative regular Banach algebra without radical, I an ideal of A$,\chi$ an element of V(I) and J the set of the $x\in A$ such that $\mathscr{G}x$ has compact support disjoint from $\{\chi \}$. Then I + J is the smallest of the ideals R of A such that $R\supset I$ and $V(R) =\{\chi \}$.