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