Topologically Noetherian Banach algebras

In this work, we introduce and study the properties of topologically Noetherian Banach algebras. In particular, we prove, if every prime closed ideal of a commutative Banach algebra A is maximal, then A is finite dimensional. Finally, we show that if every maximal ideal of a Banach algebra is generated by idempotent element then the Banach algebra is finite dimensional.