Principal ideals in $F$-algebras

Author:
James M. Briggs

Journal:
Proc. Amer. Math. Soc. **60** (1976), 231-234

MSC:
Primary 46J20

DOI:
https://doi.org/10.1090/S0002-9939-1976-0423085-1

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.

- J. M. Briggs,
*Finitely generated ideals in regular $F$-algebras*, Pacific J. Math.**61**(1975), no. 2, 339–350. MR**470683**
R. M. Brooks, - 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**

*Finite modules over F-algebras*(preprint).

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Keywords:
Principal maximal ideal,
strong topological divisor of zero,
Shilov boundary

Article copyright:
© Copyright 1976
American Mathematical Society