= Solution
Every <integer> is definable without parameters in the <ring> <first-order language>. For $a\ge0$, use $x=\bar a$; for $a=-m<0$, use $x+\bar m=0$. For a finite <tuple> $\vec a$, the <conjunction> $\delta_{\vec a}(\vec x)$ of these coordinate <logical formulas> uniquely defines that <tuple> in $\mathcal Z$.
If $\phi(\vec x)$ belongs to its <complete type>, the <first-order sentence>
$$
\forall\vec x\ (\delta_{\vec a}(\vec x)\to\phi(\vec x))
$$
is true in $\mathcal Z$, since there is only the specified <tuple> to check. It therefore belongs to $T=\operatorname{Th}(\mathcal Z)$. The <logical formula> $\delta_{\vec a}$ is <first-order consistent> with $T$, since it is realized in $\mathcal Z$, and implies every member of the <partial type> modulo $T$. Hence \b[every <complete type> realized in $\mathcal Z$ is a <principal complete type>], isolated by its tuple-defining <logical formula>.
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