Principal ideals in $F$-algebras

Abstract:

This paper is concerned with generalizations to F-algebras of theorems which Gleason has proved for finitely generated maximal ideals in Banach algebras. Let A be a uniform commutative F-algebra with identity such that Spec $(A)$ is locally compact; let x be a nonisolated point of Spec $(A)$, and let $\ker (x)$ denote the maximal ideal of all elements of A which vanish at x. In this paper it is shown that: If f is an element of A vanishing only at x, then the principal ideal Af generated by f is closed in A. If the polynomials in the element f are dense in A and if $\ker (x)$ is finitely generated, then there exists an open set U containing x such that $\ker (y)$ is generated by $f - f(y)$ for all y in U. An example is given which shows that if A is not uniform, the conclusion of the last result may not be true. In fact, the example shows that it is possible to have a nonisolated finitely generated maximal ideal in the algebra. A second example shows that in a uniform F-algebra with locally compact spectrum, $\ker (x)$ can be generated by an element f such that $f - f(y)$ generates no other $\ker (y)$ even when the $\ker (y)$ are principal.