Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / B / 7 / ii / Solution

Apply downward Löwenheim-Skolem to the extension from part (i), keeping all standard integers and the new prime $c$. The first-order language and this prescribed subset are countable, so there is a countable elementary substructure $N$ containing them. In particular $N\equiv \mathcal{Z}$ and contains an element outside the standard copy of $\mathbb{Z}$.

No ring isomorphism $N\to \mathcal{Z}$ can exist. Such an isomorphism fixes $0$ and $1$, hence every nonnegative integer numeral. It also fixes each negative integer, which is the unique solution of $x+\bar{n}=0$ for the corresponding positive $n$. Consequently it fixes the whole standard copy. The image of $c$ would be some standard integer $a$, already the image of that same named standard element, violating injectivity.

Thus $N$ and $\mathcal{Z}$ are two nonisomorphic countably infinite first-order models of $T$, so