I am working on the following:

An ideal $N$ is called nilpotent if $N^n$ is the zero ideal for some $n\geq1$. Prove that the ideal $p\mathbb{Z}/p^m\mathbb{Z}$ is a nilpotent ideal in the ring $\mathbb{Z}/p^m\mathbb{Z}$.

I think I've constructed a valid proof but I want to verify; is $N^n$ the zero ideal iff $m | n$?

  • 1
    $\begingroup$ It looks to me like it is true if $m|n$, but choosing $n = m$ seems like a fine way to proceed. $\endgroup$ May 7, 2013 at 6:21
  • $\begingroup$ Right, I just realized that. Thanks! $\endgroup$
    – Danny
    May 7, 2013 at 6:22

1 Answer 1


$N^n$ is the zero ideal if and only if $n \geq m$.


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