Proposition 6.3.4.label Any proper regular ideal of an abelian Banach algebra $A$ is contained in a maximal regular ideal.
Proof. Let $e$ be an identity modulo a proper ideal $\frak{m}$ then any ideal containing $\frak{m}$ is regular. The family $\cali{F}$ of all ideals of $A$ containing $\frak{m}$ but not $e$ is an inductive set under inclusion ordering, i.e. every chain $\cali{C}\suf \cali{F}$, the union $\cups{}{A\in\cali{C}}A$ is an upper bound of $\cali{C}$ and belongs to $\cali{F}$. Hence $\cali{F}$ has a maximal element of Zorn’s lemma.$\square$