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 as 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), 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. 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.
