Here is a somewhat different way to think about it, although the result is equivalent to the theories in the other answers.

Begin with the observation that the structures $\langle\mathbb{N},+,\cdot,0,1,<\rangle$ and $\langle\mathbb{Z},+,\cdot,0,1,<\rangle$ are [bi-interpretable](https://en.wikipedia.org/wiki/Interpretation_(model_theory)) as first-order structures. What this means, to say it simply, is that each can define a copy of the other in such a way that when one composes the interpretations, each model can see how it is copied into the interpretation-of-the-interpretation. 

To define a copy of the integers in the natural numbers, one can think of every integer as the difference of two natural numbers (as in Matt's [answer](https://mathoverflow.net/a/430600)), and then use pairs of numbers under the *same-difference* relation 
 $$(a,b)\sim(c,d)\quad\text{ if and only if }\quad a+d=b+c.$$
The integers are the resulting quotient structure (but in fact one can avoid the quotient by definably picking least representatives). The converse interpretation is simply to take the non-negative integers. 

The importance of the bi-interpretation perspective is that the constructions mentioned in the other answers can in fact be undertaken *inside* the models, rather than in some larger metatheoretic context. Being bi-interpretable, these two structures are thus revealed to have exactly the same semantic content, merely presented in a different form.

Now, the main point is that this bi-interpretation works in any model of PA, not just the standard model, and PA proves that it is a bi-interpretation. Every model of PA is bi-interpretable with the ring arising via the interpretation with the same-difference quotient. 

Because of this, it is natural to take the desired theory of the integer ring to be exactly the theory that PA proves about $\mathbb{Z}$ through this interpretation. The resulting theory will be bi-interpretable with PA, and the bi-interpretation of the structures $\mathbb{N}$ and $\mathbb{Z}$ are an instance of that bi-interpretation.

In fact, this theory will be equivalent to the theories offered in the other answers.