  A dimension group is a partially ordered countable group such that (1) every
finite subset is contained in an ordered subgroup which is a finite direct
power of Z and (2) the group has an order unit i.e. a positive element u such
that every group element is smaller than a multiple of u. For every subgroup H
of the multiplicatice groups of the positive rational numbers a dimension group
is constructed whose order-preserving automorphism group is H and H acts on an
order unit by multiplication as rational numbers. This implies that for every
equivalence relation on the positive integers for which n and m are equivalent
if and only if nk and mk are equivalent, there is a ring over which the ring of
n times n matrices and the ring of m times m matrices are equivalent if and
only if n and m are equivalent.
