Let $ X $ be a commutative Banach algebra, $ x \in X $, and let $ \lambda $ be a nonzero complex number. Let:
$$ J = \left \{ \lambda^{-1}xy - y : y \in X \right \} $$
I am trying to show that $ J $ is a maximal modular ideal of $ X $. It is easy to show that $ J $ is an ideal of $ X $ and that $ J $ is modular with modular unit $ \lambda^{-1}x $. I am having difficulties showing that $ J $ is maximal though. I am trying to prove this by contradiction.
Suppose that $ J $ is not a maximal ideal of $ X $. Let $ M $ be a maximal ideal of $ X $ containing $ J $ and let $ m \in M \setminus J $. Then $ m \neq \lambda^{-1}xy - y $ for any $ y \in X $. At this point I am stuck in finding a contradiction.
Any help would be greatly appreciated!