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Given an integral domain $D$, there are several ways how to construct a new integral domain related to D. For example, one can consider a ring of polynomials/formal power series/formal Laurent series in one/two/infinite indeterminates, a localization $S^{-1}D$, where $S \subseteq D$ is some multiplicatively closed subset, etc.

Note that all these constructions "depend" only on one domain, i.e. $D$.

In the case of general rings, there is in a way much more possibilities: Since category of rings is both complete and co-complete, one can consider limits/colimits of arbitrary diagram in $\mathcal{Ring}$. In particular, it is possible to have products of arbitrary nonempty family of rings. This is, however, almost never an integral domain (i.e. if the family contains at least two rings).

So the question is the following:

Is there some universal construction, which, given two or more abstract integral domains, produces a new integral domain, related to the given ones?

By abstract domains I mean that the construction does not assume some relation between the domains (i.e. one doesn't need to be, for example, subring of the other, so that the construction would produce some ring in between).

By universal I mean that the construction does not assume much about the structure of the given domains except the fact that they are domains (for example, does not need them to be of the form $F[x], K[x]$ for some fields $F,K$). (In fact, if the "initial" rings were not domains and the resulting ring would be, such construction would interest me as well).

Edit: An example I encountered is: Ultraproduct of collection of integral domains is again an integral domain. This construction satisfies the conditions universal and abstract as I described them, however, it can give a "new" domain only if we consider infinite collections of domains. Another interesting example is mentioned by Zhen Lin in comments.

Pavel Čoupek
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    The tensor product of two integral domains over an algebraically closed field is again an integral domain: see here. – Zhen Lin Jun 22 '13 at 12:51
  • Thank you. Although it is quite specific, it is the best example I have found so far. – Pavel Čoupek Jun 22 '13 at 17:54
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    I quite like this question. More generally, one might ask how to construct an algebraic variety from two abstract algebraic varieties with no relation to each other. Over an algebraically closed field we can take their fiber product. – Martin Brandenburg Oct 02 '13 at 22:26

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Inductive limits over filtering preordered sets of integral rings are integral. A proof can be found in EGA 0.6.1.6 (2nd edition).

(I am not sure though if this fulfills your criterion of "abstract"; there are of course some "relations" between the given rings.)

Fred Rohrer
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  • PavelC writes: "By abstract domains I mean that the construction does not assume some relation between the domains (i.e. one doesn't need to be, for example, subring of the other, so that the construction would produce some ring in between)." Therefore directed colimits don't really fit to the question. Besides, that they preserve "integral" is a trivial exercise, no EGA reference is necessary, right? – Martin Brandenburg Oct 02 '13 at 22:21
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    @Martin: I already mention this; it is not clear to me what PavelC means with "abstract". Further, "trivial exercise" does not make sense in my opinion. If you want to say that it is easy then I agree. Finally, references are never necessary, but sometimes appreciated. – Fred Rohrer Oct 03 '13 at 04:39