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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Principal ideals in $ F$-algebras


Author: James M. Briggs
Journal: Proc. Amer. Math. Soc. 60 (1976), 231-234
MSC: Primary 46J20
MathSciNet review: 0423085
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0423085-1
PII: S 0002-9939(1976)0423085-1
Keywords: Principal maximal ideal, strong topological divisor of zero, Shilov boundary
Article copyright: © Copyright 1976 American Mathematical Society