Can you describe all the ways to "un-localize" a ring?

Question

Start with a commutative unital ring $X$, and consider the collection $\mathcal{L}$ of all rings $R$ such that there exists a multiplicative subset $S \subset R$ where $R\left[S^{-1}\right] \cong X$. Is there necessarily an $R \in \mathcal{L}$ such that the only non-units of $R$ are $\pm 1$? I.e. is there a "best" un-localization of $X$? And is this $R$ going to be unique in any sense? Is every other ring $R' \in \mathcal{L}$ going to be some localization of $R$ too? Generally what is the structure of $\mathcal{L}$ for a given ring $X$? Is this question more interesting if we reformulate it for not-necessarily-unital rings instead?

Note that I'm aware this might be a poor question; I'm very much spit-balling ideas I've never considered before, this sort of "inverse problem" for localization, to see if it's meaningful and if folks have thought about it before.

Answer 1

I'm not sure about your more general question about the structure of $\mathcal L$ but any finite field not equal to $\Bbb F_2$ or $\Bbb F_3$ (or more generally any integral domain containing a finite field $F\neq\Bbb F_2,\Bbb F_3$) will give an example of a ring which does not have an "un-localization"

Answer 2

I will only talk about unital commutative rings (and I mostly think about integral domains).

From your description, I think what you have in mind of a "best un-localization" is a universal property:

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Definition: An un-localization ("unloc" for short) of a ring $X$ is a triple $(R, S, \iota)$, where:

• $R$ is a ring;
• $S \subseteq R$ is a multiplicative subset;
• $\iota$ is a homomorphism from $R$ to $X$, such that it induces isomorphism between $S^{-1}R$ and $X$.

A best unloc of $X$ is an unloc $(R, S, \iota)$ which satisfies the following universal property:

For any unloc $(R', S', \iota')$ of $X$, there exists a unique homomorphism $\tau$ from $R$ to $R'$, such that $\tau(S) \subseteq S'$ and $\iota' \circ \tau = \iota$.

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The best unloc, if exists, is then unique up to unique isomorphism.

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Example:

1. $\mathbb{Z}$ is the best unloc of $\mathbb{Q}$.

2. $\mathbb{F}_p$ is the best unloc of itself.

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However a best unloc may not exist in general.

For example, consider $X = k(t)$, the field of rational fields over a field $k$.

The inclusion $\tau:A = k[t] \hookrightarrow X$ identifies $X$ with the fraction field of $A$, hence gives an unloc of $X$.

Similarly, $\tau':A' = k[t^{-1}] \hookrightarrow X$ also gives an unloc of $X$.

Now if $\iota: R\rightarrow X$ is a best unloc, then the image $\iota(R)$ should be contained in both $\tau(A)$ and $\tau'(A')$, whose intersection is $k$. This of course is not possible.