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 RS be the following subring of Q.
If S = 0, then RS = Z. If S contains all the primes of Q, then RS = Q. If S is finite, we call the ring 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.