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I tried a bit of thinking, but I haven't worked all the details. I have a hint though that may lead to the answer of your question. You may want to regard the continuous functions over an open set as a ring. This ring is reduced and commutative (thus there is a so-called rational completion) and we could then look at rational completion of them and this may lead to an answer.

A good and downloadable reference of this is found here. A classical reference (and also the best one) is the book of Lambek "Lectures on Rings and Modules" by Lambek (please don't confuse it with the book of Lam, who happens to have the same first 3 letters in his last name, entitled "Lectures on Modules and Rings"), see for instance sections 2.3 and 4.4 of the book.

A few years ago, I had written a small entry in Planetmath that characterized rational extensions of commutative reduced rings. And you can use that as an easy definition.
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I tried to a bit of thinking, but I haven't worked all the details. I have a hint though that may lead to the answer of your question. You may want to regard the continuous functions over an open set as a ring. This ring is reduced and commutative (thus there is a so-called rational completion) and we could then look at rational completion of them and this may lead to an answer. A good and downloadable reference of this is found here. A classical reference (and also the best oneI think) is the book of Lambek "Lectures on Rings and Modules" by Lambek (please don't confuse it with the book of Lam, who happens to have the same first 3 letters in his last name, entitled "Lectures on Modules and Rings").

The problem with my idea that I , see here is that even if for instance sections 2.3 and 4.4 of the condition you write holdsbook.

A few years ago, the restriction C(X) --> C(U) (where, C(U) is the ring of continuous functions over U having values in $\mathbb R$) is not necessarily I had written a monomorphism and small entry in Planetmath that characterized rational extensions are extensions of commutative reduced rings. And you can use that as an easy definition.
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I tried to a bit of thinking, but I haven't worked all the details. I have a hint though that may lead to the answer of your question. You may want to regard the continuous functions over an open set as a ring. This ring is reduced and commutative (thus there is a so-called rational completion) and we could then look at rational completion of them and this may lead to an answer. A good reference of this is found here. A classical reference (and also the best one I think) is the book of Lambek "Lectures on Rings and Modules" by Lambek (please don't confuse it with the book of Lam, who happens to have the same first 3 letters in his name, entitled "Lectures on Modules and Rings").

The problem with my idea that I see here is that even if the condition you write holds, the restriction C(X) --> C(U) (where, C(U) is the ring of continuous functions over U having values in $\mathbb R$) is not necessarily a monomorphism and rational extensions are extensions of rings.