For me x modulo n is a kind of abuse anyway, as it means to
choose something from a set, the equivalence class. 

But since one does it for rationals as well and there seems
to be no good way to work with sets - why not ...

However it leads to the confusing discussion what to do with
rational functions as the thread about 'eval' shows.

For the ring or group definition Z ---> Z / nZ my view of x mod n is 
to choose *the* representant in the integer interval 0 ... abs(n)-1,
while for n=0 there is nothing to choose.

Why is it a problem to define it that way? It can only be some,
if one insists in viewing at it as 'division with reminder', but
in Algebra you work beyond that (just use it, if it applies).


PS: especially I dislike negative representants.