General linear group over subring of field need not be conjugacy-closed
It is possible to have a field , a unital subring of (note, in particular, that is an integral domain), and a natural number such that there exist matrices such that and are conjugate in the General linear group (?) but not in the general linear group .
Example of the integers and the rationals
Let be the ring of integers and be the field of rational numbers . Consider the matrices:
These matrices are conjugate in , by the matrix:
On the other hand, the matrices are not conjugate in . (Not ethat although the above matrix has integer entries, it is not in because its inverse does not have integer entries).