Principal ideals in -algebras
James M. Briggs
Proc. Amer. Math. Soc. 60 (1976), 231-234
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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.
M. Briggs, Finitely generated ideals in regular
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- J. M. Briggs, Finitely generated ideals in regular F-algebras, Pacific J. Math. 61 (1975), 339-350. MR 0470683 (57:10429)
- R. M. Brooks, Finite modules over F-algebras (preprint).
- R. L. Carpenter, Principal ideals in F-algebras, Pacific J. Math. 35 (1970), 559-563. MR 43 #7926. MR 0282213 (43:7926)
- A. M. Gleason, Finitely generated ideals in Banach algebras, J. Math. Mech. 13 (1964), 125-132. MR 28 #2458. MR 0159241 (28:2458)
- E. A. Michael, Locally multiplicatively-complex topological algebras, Mem. Amer. Math. Soc. No. 11 (1952). MR 14, 482. MR 0051444 (14:482a)
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