then how do I explain this weird pattern when I multiplied the zero divisors to all elements in the ring:
e.g.
2*{0,1,2,3}={0,2,0,2}
2*{0,1,2,3,4,5} = {0,2,4,0,2,4}
3*{0,1,2,3,4,5} = {0,3,0,3,0,3}
2*{0,1,2,3,4,5,6,7}={0,2,4,6,0,2,4,6}
4*{0,1,2,3,4,5,6,7}={0,4,0,4,0,4,0,4}
6*{0,1,2,3,4,5,6,7}={0,6,4,2,0,6,4,2}
What are these repeating periods I am seeing here whenever the whole ring is being multiplied by the zero divisors?