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In 1971, Grauert and Remmert proved that a commutative, complex, NoetherianBanach algebra is necessarily finite-dimensional. More precisely, they provedthat a commutative, complex Banach algebra has finite dimension over $\C$whenever all the closed ideals in the algebra are (algebraically) finitelygenerated. In 1974, Sinclair and Tullo obtained a non-commutative version ofthis result. In 1978, Ferreira and Tomassini improved the result of Grauert andRemmert by showing that the statement is also true if one replaces `closedideals' by `maximal ideals in the \v{S}ilov boundary of $A$'. We shall give ashorter proof of this latter result, together with some extensions and relatedexamples.

Generators of maximal left ideals in Banach algebras