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Math Help - Ring

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    Junior Member mathemanyak's Avatar
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    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.
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    Quote Originally Posted by mathemanyak View Post
    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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