I am studying Hilbert's tenth problem over rings of algebraic integers, which for a fixed such ring $R$ can be viewed as deciding the theory $H10(R)$ defined by $$ϕ ∈ H10(R) ⇔ ϕ = ∃ x_1, …, x_k : ψ(x_1, …, x_k), ψ \text{ is atomic, and } R \models ϕ.$$ If $R$ is computable then $H10(R)$ is easily seen to be computably enumerable.
There are well known results like the DPRM-theorem stating that a subset $S$ of $ℕ$ is computably enumerable if and only if it is Diophantine over $ℤ$ and all witnesses are contained in $ℕ$, i.e. if there exists an atomic formula $ψ(x, y_1, …, y_n)$ such that $$α ∈ S ⇔ ℤ \models ∃ y_1, …, y_n : ψ(α, y_1, …, y_n) ∧ \bigwedge_{i = 1}^n y_i ∈ ℕ.$$
This is easily seen to be a Diophantine definition of the set $S$ over $ℤ$ by Lagrange's four-square theorem. In particular, the halting set $\mathcal{K}$ is Diophantine over $ℤ$, and the computable function $$x ↦ \mathrm{gn}(∃ y_1, …, y_n : ψ_{\mathcal{K}}(x, y_1, …, y_n)),$$ where $∃ y_1, …, y_n : ψ_{\mathcal{K}}(x, y_1, …, y_n)$ is a Diophantine definition of the halting set and $\mathrm{gn}$ denotes the Gödel number, shows that the halting set is many-one reducible to $H10(ℤ)$.
Several other results over other rings of algebraic integers $R$ have been proven by Denef, Lipshitz, Pheidas, Shlapentokh, and Shapiro to name just a few and they always proved, that $ℤ$ has a Diophantine definition over $R$. In fact, Denef and Lipshitz [2] conjectured that $ℤ$ is Diophantine over all rings of algebraic integers. But by a result of Davis, Matijasevič, and Robinson [1] this implies that all computably enumerable subsets of $R$, are Diophantine over $R$. In particular, we can embed the halting set $\mathcal{K}$ into $R$ in a Diophantine way and deduce that $H10(R)$ is many-one equivalent to $\mathcal{K}$.
So to finally come to my question: Is there a known example of a ring $R$ (commutative, with unity) such that $H10(R)$ is a creative set? If not, are there other (algebraic) structures known, where the primitive positive theory is creative?
Note that I was intentionally vague on wether I allow only the constant symbols $\mathtt{0, 1}$ in the atomic formula $ψ$, or constant symbols $c_α$ for all elements $α ∈ R$, as whenever $ℤ$ is Diophantine over such a ring of algebraic integers $R$ both theories are equally hard to decide from the view of computability theory. This follows again from a result of Davis, Matijasevič, and Robinson [1].
[1]: Davis, M.; Matijasevič, Y. V. & Robinson, J. Hilbert's tenth problem: Diophantine equations: positive aspects of a negative solution Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), Amer. Math. Soc., Providence, R. I., 1976 , 323-378
[2]: Denef, J. & Lipshitz, L. Diophantine sets over some rings of algebraic integers J. London Math. Soc. (2), 1978 , 18 , 385-391