I have been looking at periodic linear recurrences and have some conditions involving roots of unity. Now I can get the same results out in the context of a Ring if I assume that I have two elements r and s with the following properties:-
1) r and s commute 2) r^n = s^m = 1_R (i.e. they are 'roots' of the multiplicative identity) 3) r - s is not a zero divisor
Unfortunately my ring theory is not strong enough to identify any examples other than the complex numbers which were what I was trying to abstract in the first place. Can anyone help, or point out why I shouldn't expect more exotic examples to exist?