## On isomorphisms between ideals in rings of continuous functions

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- by David Rudd PDF
- Trans. Amer. Math. Soc.
**159**(1971), 335-353 Request permission

## 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

- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**159**(1971), 335-353 - MSC: Primary 46.55
- DOI: https://doi.org/10.1090/S0002-9947-1971-0283575-1
- MathSciNet review: 0283575