The surreal numbers have a subring, the ring of "omnific integers" or $\mathbf{Oz}$, which have the property that every surreal number is a quotient of two omnific integers. That is, the field of fractions of the omnific integers is the entire surreal number field, which in particular includes all the reals.

A ring whose field of fractions includes all the reals seems like a useful thing. The omnific integers would seem to be much larger than necessary if that is what we want. So we can ask for simpler examples.

Of course, $\Bbb R$ is a trivial example of a ring whose field of fractions includes all of $\Bbb R$. So, to be precise, I am interested in rings which do not already have all the reals, but whose field of fractions does have all the reals.

In particular, I have the following questions:

- Does there exist some ring $R$, which is not a superset of $\Bbb R$ but whose field of fractions is a superset of $\Bbb R$, that is "smallest" in the sense that $R$ is isomorphic to a subring of any other ring with this property?
- If we add the requirement that $R$ be an ordered ring, does that have any effect on #1?
- Does the ring of omnific integers have any smallest subring with the above property?
- Would any ring with the above property embed into the omnific integers anyway, making these criteria all equivalent?