Corollary 2

Let A be a commutative regular Banach algebra without radical. Let $x, y\in A$. If the support of $\mathscr{G}(x)$ is compact and contained in the set of characters $\chi$ such that $\mathscr{G}(y)(\chi )\not= 0$, then $x$ is a multiple of $y$ in A.

Let I be the ideal $Ay$ of A. Then V(I) is the set of zeros of $\mathscr{G}(y)$. Since the support F of $\mathscr{G}(x)$ is compact and disjoint from V(I), we have $x\in I$ (Prop. 4 applied to F).