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

DOI:
https://doi.org/10.1090/S0002-9947-1971-0283575-1

MathSciNet review:
0283575

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Abstract: A ring of continuous functions is a ring of the form , the ring of all continuous real-valued functions on a completely regular Hausdorff space . For an arbitrary ideal in , the author shows that the maximal ideals of are precisely the ideals of the form , for some maximal ideal in not containing .

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 onto a maximal idea in can be extended to an isomorphism of onto *.

Another of the main theorems characterizes those isomorphisms between and (the subrings of bounded functions in and respectively) which can be extended to isomorphisms between and .

The author proves that, given any ideal in , there exists a space so that the uniform closure of is isomorphic to a maximal ideal in .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0283575-1

Keywords:
Real-valued continuous functions,
structure spaces,
extending isomorphisms,

Article copyright:
© Copyright 1971
American Mathematical Society