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An example of such an ring is the Prufer group.

My question is this - is there a way to use some sort of "compactness" theorem, to show that such a ring exists? Here by compactness I mean something along the lines of: "If a model exists for all finite subsets of some set of sentences, then a model exists for the whole set."

I don't think it's possible to state the needed sentences in predicate logic, because we're making statements about specific substructures of the ring, not just elements of the ring.

Is there some sort of analogue of the first-order logic completeness, that could show the existence of such a ring?

I might be reaching a bit too far given my knowledge, but I'd still like to get an answer to this question if it's possible - ideally, an answer that doesn't assume the knowledge of things like higher-order logic. I'd appreciate an answer I can understand without having to study up on general higher-order logic too much.

Hopefully, I'm not asking to get spoonfed information too much, and I'm sorry if that's the case.

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    A few comments: (1) why would you want a non-constructive way of showing the existence of something like the Prüfer groups? (2) the problem with doing this in first-order logic is axiomatising the Artinian property: if you could do that, you could use a compactness argument to conjure up the modules you are looking for. (3) higher-order logic isn't that hard (but doesn't lead to an answer to your question as far as I know). – 2017-02-26
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    Thanks for the comment. I simply found the idea of showing the existence of such a ring in this way interesting. It's a concrete problem, which makes it seem easier to think about (at least for me), and it doesn't seem completely out of my grasp. I did consider phrasing the question more generally: "If I have an infinite amount of statements concerning a lattice of some ring (lattice of the substructures), and for every finite subset of these statements I have a ring that satisfies them, can I somehow show that there exists a ring that satisfies all of them? Perhaps using compactness?" – 2017-02-26

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