  This paper establishes the Gleason result for finitely generated ideals in
the context of Frechet algebras, and, in particular, provides an affirmative
answer to a question about the Gleason result in commutative Frechet algebras
(Carpenter posed this question for uniform Frechet algebras in 1970). As a
welcome bonus of our method, locally Stein algebras are completely
characterized, and, as an application of this characterization, an affirmative
answer to the Gleason problem (1964) for such algebras is provided recapturing
all the classical results on the Gleason problem in the theory of several
complex variables.
