Principal ideals in -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 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.

**[1]**J. M. Briggs,*Finitely generated ideals in regular 𝐹-algebras*, Pacific J. Math.**61**(1975), no. 2, 339–350. MR**0470683****[2]**R. M. Brooks,*Finite modules over F-algebras*(preprint).**[3]**Ronn L. Carpenter,*Principal ideals in 𝐹-algebras*, Pacific J. Math.**35**(1970), 559–563. MR**0282213****[4]**Andrew M. Gleason,*Finitely generated ideals in Banach algebras*, J. Math. Mech.**13**(1964), 125–132. MR**0159241****[5]**Ernest A. Michael,*Locally multiplicatively-convex topological algebras*, Mem. Amer. Math. Soc.,**No. 11**(1952), 79. MR**0051444**

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

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

Article copyright:
© Copyright 1976
American Mathematical Society