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 $A,B\in GL_{n}(R)$ such that $A$ and $B$ are conjugate in the General linear group (?) $GL_{n}(K)$ but not in the general linear group $GL_{n}(R)$.

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:

$A={\begin{pmatrix}1&1\\0&1\end{pmatrix}},B={\begin{pmatrix}1&2\\0&1\end{pmatrix}}$

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

${\begin{pmatrix}2&0\\0&1\end{pmatrix}}$.

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).