1. ## Ring

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.

2. Originally Posted by mathemanyak
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 $\mathbb{Z}_6$ are $[1],[5]$ since $\gcd(1.6)=\gcd(5.6)=1$. While $[2],[3],[4]$ are zero divisiors since $[2][3]=[0]$ and $[3][4]=[0]$. 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 $[2]$ is nilpotent: $[2]^2 = [4],[2]^3 = [2], [2]^4=[4], ...$ and this pattern repeats. Therefore, $[2]$ is not nipotent. Now test the other two.