In the rings Z8 and Z6 find the units, zero divisors and nilpotent elements.
Recall that an element a in a ring is called nilpotent if an = 0R for some
positive integer n.
The units in $\displaystyle \mathbb{Z}_6$ are $\displaystyle [1],[5]$ since $\displaystyle \gcd(1.6)=\gcd(5.6)=1$. While $\displaystyle [2],[3],[4]$ are zero divisiors since $\displaystyle [2][3]=[0]$ and $\displaystyle [3][4]=[0]$. The nilpotent elements can only be among the zero divisors. Let us check one element and you try the other two. We will check if $\displaystyle [2]$ is nilpotent: $\displaystyle [2]^2 = [4],[2]^3 = [2], [2]^4=[4], ... $ and this pattern repeats. Therefore, $\displaystyle [2]$ is not nipotent. Now test the other two.