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

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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

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).