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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On isomorphisms between ideals in rings of continuous functions


Author: David Rudd
Journal: Trans. Amer. Math. Soc. 159 (1971), 335-353
MSC: Primary 46.55
MathSciNet review: 0283575
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Abstract: A ring of continuous functions is a ring of the form $ C(X)$, the ring of all continuous real-valued functions on a completely regular Hausdorff space $ X$. For an arbitrary ideal $ I$ in $ C(X)$, the author shows that the maximal ideals of $ I$ are precisely the ideals of the form $ I \cap M$, for some maximal ideal $ M$ in $ C(X)$ not containing $ I$.

The author shows that any ring isomorphism between ideals in any two rings of continuous functions preserves order, boundedness, and lattice structure; and he uses these results to obtain one of the main theorems: An isomorphism of a maximal ideal in $ C(X)$ onto a maximal idea in $ C(Y)$ can be extended to an isomorphism of $ C(X)$ onto $ C(Y)$.

Another of the main theorems characterizes those isomorphisms between $ {C^ \ast }(X)$ and $ {C^ \ast }(Y)$ (the subrings of bounded functions in $ C(X)$ and $ C(Y)$ respectively) which can be extended to isomorphisms between $ C(X)$ and $ C(Y)$.

The author proves that, given any ideal $ I$ in $ C(X)$, there exists a space $ X(I)$ so that the uniform closure of $ I$ is isomorphic to a maximal ideal in $ C(X(I))$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0283575-1
PII: S 0002-9947(1971)0283575-1
Keywords: Real-valued continuous functions, structure spaces, extending isomorphisms, $ \beta X,\upsilon X$
Article copyright: © Copyright 1971 American Mathematical Society