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Find the zero divisors and the units of ##\mathbb Z[X]/<X^3>##

  1. Sep 13, 2014 #1
    The problem statement, all variables and given/known data

    Find the zero divisors and the units of the quotient ring ##\mathbb Z[X]/<X^3>##


    The attempt at a solution

    If ##a \in \mathbb Z[X]/<X^3>## is a zero divisor, then there is ##b \neq 0_I## such that ##ab=0_I##. I think that the elements ##a=X+<X^3>## and ##b=X^2+<X^3>## are zero divisors because we have

    ##ab=XX^2+<X^3>=X^3+<X^3>=<X^3>##.

    I couldn't think of any other divisors so I suspect these two are the only ones. Am I correct? If that is the case, how could I show these are the only zero divisors?

    As for the units I don't know what to do. Any suggestions would be appreciated.
     
  2. jcsd
  3. Sep 18, 2014 #2
    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
     
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