Does k Divide R's Order When R's Additive Group is Cyclic?

[tjkubo]

If R is a finite ring and its additive group is cyclic, then
R = <r> = {nr : n an integer} for some r in R.
r^2 is in R so r^2 = kr for some integer k.
Does k have to divide the order of R?

 

[Deveno]

the finite field of 5 elements is a ring with a cyclic additive group.

3 generates this ring, and 4= 3²= (3)(3), is it the case that 3 divides 5?