Let $L$ be a the first order language of rings with unity. Let $T$ be the first order theory of Valuation rings, which along with the first order axioms for integral domain, contains the following axiom : $\forall a \forall b ((\exists r (ra=b)) \lor (\exists s (sb=a)))$.
Is this first order theory of Valuation rings complete ? model complete ?
Note that models for this theory are both fields and non-fields.
In particular, the sentence $(\forall x)[(x \neq 0) \rightarrow (\exists y)(xy=1)]$ is not decidable from the axioms.
Similarly, embedding of a non-field valuation ring $D$ to its field of fraction $F$ will not be an elementary embedding. The formula $(\exists y)(xy=1)$ will be true about a non-zero element $a \in \mathfrak{m}$ in the maximal ideal of $D$ when taken in $F$, but not true in $D$.
Or do you consider something like $(\exists x)[(x\neq 0)\wedge(\forall y)(xy\neq 1)]$ an axiom of integral domains (excluding fields)?