Ring

**mathemanyak** · October 2nd 2008, 03:43 AM

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.

**ThePerfectHacker** · October 2nd 2008, 07:41 AM

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.

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