Let A be a regular,
semisimple, commutative F-algebra with identity. For each point in the spectrum of
A, let 𝒜p denote the local algebra of germs at p of elements of A and let ℳp denote
its maximal ideal. When ℳp is finitely generated we show to what extent
representatives of its generators are generators of the maximal ideals in the algebras
of functions locally belonging to A on some neighborhood of p. We show
that if ℳp is finitely generated, then all point derivations of A at p are
continuous. Using this last fact, we describe the generators of maximal ideals when
the polynomials in finitely many elements of the algebra are dense in the
algebra.
