Exercise 4, § 5

Let A be a commutative regular Banach algebra without radical, satisfying the Ditkin condition, I a closed ideal of A$,x\in \Upsilon (V(I))$ and F the boundary of $V(Ax)$. If $F\cap V(I)$ contains no nonempty perfect set, then $x\in I$. (Imitate the reasoning of Prop. 5 of I, p. 94.)