  Messages in communication networks often are considered as "discrete" taking
values in some finite alphabet (e.g. a finite field). However, if we want to
consider for example communication based on analogue signals, we will have to
consider messages that might be functions selected from an infinite function
space. In this paper, we extend linear network coding over finite/discrete
alphabets/message space to the infinite/continuous case. The key to our
approach is to view the space of operators that acts linearly on a space of
signals as a module over a ring.
  It turns out that modules over many rings $R$ leads to unrealistic network
models where communication channels have unlimited capacity. We show that a
natural condition to avoid this is equivalent to the ring $R$ being Dedekind
finite (or Neumann finite) i.e. each element in $R$ has a left inverse if and
only if it has a right inverse. We then consider a strengthened capacity
condition and show that this requirement precisely corresponds to the class of
(faithful) modules over stably finite rings (or weakly finite).
  The introduced framework makes it possible to compare the performance of
digital and analogue techniques. It turns out that within our model, digital
and analogue communication outperforms each other in different situations. More
specifically we construct: 1) A communications network where digital
communication outperforms analogue communication. 2) A communication network
where analogue communication outperforms digital communication.
  The performance of a communication network is in the finite case usually
measured in terms band width (or capacity). We show this notion also remains
valid for finite dimensional matrix rings which make it possible (in principle)
to establish gain of digital versus analogue (analogue versus digital)
communications.
