  The rate of a network code is the ratio of the block size of the network's
messages to that of its edge codewords. We compare the linear capacities and
achievable rate regions of networks using finite field alphabets to the more
general cases of arbitrary ring and module alphabets. For non-commutative
rings, two-sided linearity is allowed. Specifically, we prove the following for
directed acyclic networks:
  (i) The linear rate region and the linear capacity of any network over a
finite field depend only on the characteristic of the field. Furthermore, any
two fields with different characteristics yield different linear capacities for
at least one network.
  (ii) Whenever the characteristic of a given finite field divides the size of
a given finite ring, each network's linear rate region over the ring is
contained in its linear rate region over the field. Thus, any network's linear
capacity over a field is at least its linear capacity over any other ring of
the same size. An analogous result also holds for linear network codes over
module alphabets.
  (iii) Whenever the characteristic of a given finite field does not divide the
size of a given finite ring, there is some network whose linear capacity over
the ring is strictly greater than its linear capacity over the field. Thus, for
any finite field, there always exist rings over which some networks have higher
linear capacities than over the field.
