Classifying topos of integral domains

On the opposite of finitely presented commutative rings, cover the zero ring by the empty family and cover $A$ by $A/(a),A/(b)$ whenever $ab=0$. The generated topology classifies nontrivial integral domains, with generic model the sheafified tautological ring. It is not subcanonical: at $\mathbb{Z}/4$, the covering quotient to $\mathbb{Z}/2$ identifies the distinct sections $0$ and $2$ of the polynomial-ring representable.

In the generic domain, negation of simultaneous invertibility of finitely many elements implies that one is zero. At a ring stage, localize at their product $t$. All become units, so a tuple satisfying the negation makes the localized stage empty. The empty-cover criterion for the domain-classifying site then says $A[1/t]=0$, hence $t$ is nilpotent. Repeated zero-product covers force one factor to vanish locally. The two-variable property conversely forces the integral-domain axiom in any nontrivial internal ring.

For a finitely presented ring $A$, its sheafified representable is initial exactly when $A$ is the zero ring. The zero ring has a generating empty cover. Every nonzero ring maps to a field by quotienting by a maximal ideal, and that set-based domain gives a point at which the representable has a section, precluding initiality.