Are the integers definable in $\mathbb{Z}_{(p)}$?

Question

I am familiar with the statement (not the proof) of Robinson's definition of $\mathbb{Z}$ in $\mathbb{Q}$ in the language of rings. I would like to ask the same question for $\mathbb{Z}_{(p)}$ in place of $\mathbb{Q}$ (again in $L_r$, the language of rings), where by $\mathbb{Z}_{(p)}$ I mean the set of rationals with denominator coprime to $p$. As a second part, I would be interested to know whether a uniform definition exists. Let me also give some context: my initial question was whether $\mathbb{Z}_{(p)}$ is undecidable in $L_r$ and the most natural approach that came to my mind was to define the integers in there.

Answer 1

Yes. One way to do it is to interpret $\mathbb{Q}$ in $\mathbb{Z}_{(p)}$, use Robinson's definition of $\mathbb{Z}$ in $\mathbb{Q}$, and then pull this set back into $\mathbb{Z}_{(p)}$. It may be that there's a more direct approach, but this way really only uses generalities about interpretations.

So first, note that we can define the set $Q = \{(x,y)\mid y\neq 0\}\subseteq (\mathbb{Z}_{(p)})^2$. Think of an element $(a,b)\in Q$ as a formal fraction $\frac{a}{b}$. Now there is a definable equivalence relation $\sim$ on $Q$ by $(a,b)\sim (c,d)$ iff $ad = bc$ expressing equality of fractions. And we can define $\sim$-classes $0_Q,1_Q$ and operations $+_Q,-_Q,\times_Q$ on $Q$ which respect by $\sim$ by the usual definitions of arithmetic with fractions. Then you can check that $(Q/{\sim};0_Q,1_Q,+_Q,-_Q,\times_Q)\cong (\mathbb{Q};0,1,+,-,\times)$.

Now take a formula $\varphi(x)$ in the language of rings defining $\mathbb{Z}$ in $\mathbb{Q}$. Transform it into a formula $\varphi'(x,y)$ by replacing the symbols in the language of rings with their counterparts in $Q$, replacing $=$ by the definition of $\sim$, and replacing quantifiers by quantifiers over $Q$. Then $\varphi'(x,y)$ defines the isomorphic copy of $\mathbb{Z}$ inside $Q/{\sim}$.

Finally, the natural inclusion $\mathbb{Z}_{(p)}\hookrightarrow \mathbb{Q}$ is given by $a\mapsto \frac{a}{1}$, so the formula $\varphi'(a,1)$ expresses that $\frac{a}{1}\in \mathbb{Z}$ and defines $\mathbb{Z}$ in $\mathbb{Z}_{(p)}$.