Principal ideals in -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 is locally compact; let x be a nonisolated point of Spec , and let 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 is finitely generated, then there exists an open set U containing x such that is generated by 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, can be generated by an element f such that generates no other even when the are principal.
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Keywords: Principal maximal ideal, strong topological divisor of zero, Shilov boundary
Article copyright: © Copyright 1976 American Mathematical Society