  We generalise Gabidulin codes to the case of infinite fields, eventually with
characteristic zero. For this purpose, we consider an abstract field extension
and any automorphism in the Galois group. We derive some conditions on the
automorphism to be able to have a proper notion of rank metric which is in
coherence with linearized polynomials. Under these conditions, we generalize
Gabidulin codes and provide a decoding algorithm which decode both errors and
erasures. Then, we focus on codes over integer rings and how to decode them. We
are then faced with the problem of the exponential growth of intermediate
values, and to circumvent the problem, it is natural to propose to do
computations modulo a prime ideal. For this, we study the reduction of
generalized Gabidulin codes over number ideals codes modulo a prime ideal, and
show they are classical Gabidulin codes. As a consequence, knowing side
information on the size of the errors or the message, we can reduce the
decoding problem over the integer ring to a decoding problem over a finite
field. We also give examples and timings.
