TITLE: A question about open induction
QUESTION [10 upvotes]: An old theorem of A. J. Wilkie (Some results and problems on weak systems of arithmetic, Logic Colloquium '77) asserts that  a discretely ordered ring $R$ can be extended to a model of open induction if and only if  for all $n>1$, there is a homomorphism from $R$ onto $\mathbb{Z}/n\mathbb{Z}$. 
Wilkie's proof proceeds by adjoining transcendental elements to $R$, but it is not clear that this is ever necessary:
Does every  ring that extends to a model of open induction have an algebraic extension to a model of open induction?
Does anyone know anything about this? I know of no place in the literature where the question is even mentioned, although it has come up more than once in conversation.
Here is simple test-case: Let $R$ be the ring $\mathbb{Z}[t,\sqrt{2}t-r]$, where $r$ is a real transcendental and $t$ is an indeterminate. Order $R$ by declaring $t$ positive infinite. It is not hard  to show that $R$ extends to a  model of true arithmetic. I don't know if $R$ has an algebraic extension to a model of open induction.

REPLY [6 votes]: I think that the answer in general is “no”, and that this follows from results by D’Aquino, Knight, and Lange [1].
Namely, by Theorem 3.5 and Remark 3, there exists a real-closed field $R$ with an element $r\in R$ and a $\mathbb Z$-ring $I\subseteq R$ such that no discretely ordered subring $I\subseteq J\subseteq R$ intersects $[r,r+1)$, but $b/i\in[r,r+1)$ for some $b,i\in I$. Then $I$ (being a $\mathbb Z$-ring) can be extended to a model of $\mathit{IOpen}$, but no such extension can be found in its real closure $\operatorname{rcl}(I)\subseteq R$ (since no discretely ordered $I\subseteq J\subseteq \operatorname{rcl}(I)$ can contain an integer part of $b/i$).
Reference:
[1] Paola D’Aquino, Julia Knight, Karen Lange: Limit computable integer parts, Archive for Mathematical Logic 50 (2011), no. 7, pp. 681–695, doi:10.1007/s00153-011-0241-z. Erratum in: Archive for Mathematical Logic 54 (2015), no. 3, pp. 487–489, doi:10.1007/s00153-015-0418-y