About locally m-convex algebras with dense finitely generated ideals

Abstract

It is well known, as a consequence of a theorem of Richard Arens, that a commutative Fréchet locally $m$-convex algebra $E$ with unit does not have dense finitely generated ideals. We shall see that this result can no longer be true if $E$ is not complete and metrizable. We observe that the same is true for the theorem of Arens; that is, this theorem can no longer be true if $E$ is not complete and metrizable. Moreover, several conditions for a unital commutative (not necessarily complete) locally $m$-convex algebra are given, for which all maximal ideals have codimension one.