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Math Help - Primes and units etc

  1. #1
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    Primes and units etc

    Let p be prime, m be a positive integer.

    1-show that each element c of Z(p^m) is either a unit or is nilpotent.

    I know nilpotent is when there is a pos. int. k so that c^k=0, and that a unit is such that c*k=1, but I don't know how to show this.

    2- How many units are in Z(p^m)? nonunits? Explain.
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  2. #2
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    Quote Originally Posted by twittytwitter View Post
    Let p be prime, m be a positive integer.

    1-show that each element c of Z(p^m) is either a unit or is nilpotent.

    I know nilpotent is when there is a pos. int. k so that c^k=0, and that a unit is such that c*k=1, but I don't know how to show this.

    2- How many units are in Z(p^m)? nonunits? Explain.

    Hint for (1) and, after some thinking, for (2) as well: an element c\in\mathbb{Z}_{p^m} is a unit iff (c,p^m)=1\Longleftrightarrow p\nmid c \Longrightarrow the non-units are exactly the elements

    of the form c=ap^m\,,\,\,0\leq a<p, and now check what happens with a non-unit when you raise it to an appropiate power...

    Tonio
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