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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Principal ideals in $F$-algebras
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by James M. Briggs PDF
Proc. Amer. Math. Soc. 60 (1976), 231-234 Request permission


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.
  • J. M. Briggs, Finitely generated ideals in regular $F$-algebras, Pacific J. Math. 61 (1975), no. 2, 339–350. MR 470683
  • R. M. Brooks, Finite modules over F-algebras (preprint).
  • Ronn L. Carpenter, Principal ideals in $F$-algebras, Pacific J. Math. 35 (1970), 559–563. MR 282213
  • Andrew M. Gleason, Finitely generated ideals in Banach algebras, J. Math. Mech. 13 (1964), 125–132. MR 0159241
  • Ernest A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 79. MR 51444
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 231-234
  • MSC: Primary 46J20
  • DOI:
  • MathSciNet review: 0423085