Incomparable Prime Ideals in Commutative Radical Fréchet Algebras

Let R be a commutative radical Fréchet algebra having a nonnilpotent element a with a ∈ Ra. Then R contains a continuum of incomparable prime ideals. In [3], J. Esterle proved the following result; it is a main ingredient in his proof that epimorphisms from C0(Ω) onto Banach algebras are continuous. Theorem (Esterle). Let R be a commutative radical Banach algebra. Suppose that there exists a non-nilpotent element a ∈ R with a ∈ Ra. Then the set of prime ideals in R, ordered by inclusion, does not form a chain. Thus, each algebra R as in the theorem contains at least two prime ideals which are incomparable. In [1], Bouloussa extended the result to commutative radical Fréchet algebra. In this paper, we shall extend this by producing a continuum of pairwise incomparable prime ideals. This will be proved as a consequence of a result on ideals in (not necessarily commutative) radical Fréchet algebras and the existence of a continuum of ”almost disjoint” subsets of N due to Sierpinski. 1. Preliminaries More details of the following can be found, for example, in [2]. A Fréchet algebra is a topological algebra A whose topology is determined by a sequence of algebra seminorms (pn) such that