# The Rings Between Z and Q

We start with a definition of the rings in question whose first appearance on the scene in [49, 50] dates back to 1994.

Definition 3.6 *(A Ring in between)* Let *S* be a set of primes of Q. Let *R _{S}* be the following subring of Q.

If S = 0, then *R _{S} =* Z. If S contains all the primes of Q, then

*R*Q. If S is finite, we call the ring

_{S}=*small.*If S is infinite, we call the ring

*big.*

Some of these rings have other (canonical) names: the small rings are also called rings of S-integers, and when S contains all but finitely many primes, the rings are called semi-local subrings of Q. To measure the “size” of big rings we use the natural density of prime sets defined below.

Definition 3.7 *(Natural Density)* If A is a set of primes, then the natural density of A is equal to the limit below (if it exists):

The big and small rings are not hard to construct.

*Example 3.1 (A Small Ring not Equal* to Z)

*Example 3.2 (A Big Ring not Equal to* Q)

Given a big or a small ring *R* we can now ask the following questions which were raised above with respect to Q:

- • Is HTP solvable over R?
- • Do integers have a Diophantine definition over R?
- • Is there a Diophantine model of integers over R?

Here one could hope that understanding what happens to HTP over a big ring can help to understand HTP over Q.