General linear group over subring of field need not be conjugacy-closed

Statement

It is possible to have a field $K$, a unital subring $R$ of $K$ (note, in particular, that $R$ is an integral domain), and a natural number $n$ such that there exist matrices $A,B\in G{L}_n(R)$ such that $A$ and $B$ are conjugate in the General linear group (?) $G{L}_n(K)$ but not in the general linear group $G{L}_n(R)$.

Related facts

• General linear group over subfield is conjugacy-closed

Proof

Example of the integers and the rationals

Let $R$ be the ring of integers $\mathbb{Z}$ and $K$ be the field of rational numbers $\mathbb{Q}$. Consider the matrices:

$A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),B=\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$

These matrices are conjugate in $GL(2,\mathbb{Q})$, by the matrix:

$\left(\begin{array}{cc}2& 0\\ 0& 1\end{array}\right)$.

On the other hand, the matrices are not conjugate in $GL(2,\mathbb{Z})$. (Not ethat although the above matrix has integer entries, it is not in $GL(2,\mathbb{Z})$ because its inverse does not have integer entries).