Proposition 6.3.6 (Maximal Ideal).label Let $x\in A$ be a non-invertible element of a unital abelian Banach algebra then $x\in\frak{m}$ for some maximal ideal $\frak{m}$ of $A$.
Proof. By assumption, $1\nin Ax$ hence it is a proper ideal. Since $A$ is unital, any ideal is regular so by Proposition 6.3.3 is contained in some maximal ideal $\frak{m}$.$\square$