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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.

References [Enhancements On Off] (What's this?)

  • 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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Keywords: Principal maximal ideal, strong topological divisor of zero, Shilov boundary
Article copyright: © Copyright 1976 American Mathematical Society