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 (ordered) 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 (ordered) subring of any other (ordered) ring with this property?
- 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?

To add some detail to the above:

In the omnific integers, for any real number $r$, we have that $r \omega$ is an omnific integer. So, to start, we can look at the following fragment of the omnific integers, with all elements of the form

$$z + r_1\omega + r_2\omega^2 + ... + r_n\omega^n$$

where $z$ is an integer, and the $r_n$ are all real numbers. I will notate this ring as

$$\Bbb Z \: \tilde \oplus \: \Bbb R \: \tilde \oplus \: \Bbb R \: \tilde \oplus \: ...$$

where the $\tilde \oplus$ is a kind of modified direct sum in which the polynomial coefficient multiplication is used rather than pointwise multiplication, which is always possible as long as each ring is a subring of the ring after it.

It is easy to see that the above ring has $\Bbb R$ in its field of fractions. It is also easy to see that this is true for any ring of the form

$$\Bbb Z \: \tilde \oplus \: \Bbb Z \: \tilde \oplus \: ... \tilde \oplus \: \Bbb Z \: \tilde \oplus \: \Bbb R \: \tilde \oplus \: \Bbb R \: \tilde \oplus \: ...$$

which the first ring embeds into, and which also embeds into the first ring.

So, one thing we can do is ask if there exists a countable sequence of rings $R_n \neq \Bbb R$ such that

$$ R_1 \subset R_2 \subset R_3 \subset ...$$

and

$$\Bbb R \subset \text{Quot}(R_1 \: \tilde \oplus \: R_2 \: \tilde \oplus \: R_3 \: \tilde \oplus \: ...)$$

One way this could be is if there is a countable sequence of $R_n$ such that the union of all the $R_n$ is $\Bbb R$, so that each real number appears at some point in the $R_n$.