Prove that the ring $\mathbb{Z}/n\mathbb{Z}$ does not have non trivial nilpotent elements if and only if $n$ is squarefree

Question

Prove that the ring $\mathbb{Z}/n\mathbb{Z}$ does not have non trivial nilpotent elements if and only if $n$ is squarefree.

Some idea?

What does it mean that $n$ is squarefree?

Each prime factor of $n$ appears only once.
https://en.wikipedia.org/wiki/Square-free_integer — user113102, Apr 03 '19 at 17:34
The actual version of this question (one not requesting an easily searched definition but rather the problem itself) https://math.stackexchange.com/q/749275/29335 — rschwieb, Apr 03 '19 at 18:09
Seriously, there is no excuse not to google this. When you google "squarefree meaning" you get Squarefree -- from Wolfram MathWorld, Square-free integer - Wikipedia, Square-free element - Wikipedia, Square-free number - Groupprops, Squarefree numbers - OeisWiki, Square Free Number - GeeksforGeeks, squarefree - Wiktionary — rschwieb, Apr 03 '19 at 18:11

score 2 · Answer 1 · answered Apr 03 '19 at 17:46

An integer $n$ is squarefree if (and only if), for all primes $p$, $p^2$ does not divide $n$.

Now try to see what a nilpotent element in $\mathbb{Z}/n\mathbb{Z}$ should look like: if $k=p_1^{r_1}p_2^{r_2}\dots p_s^{r_s}$ and $k^m\in n\mathbb{Z}$, then…

Prove that the ring $\mathbb{Z}/n\mathbb{Z}$ does not have non trivial nilpotent elements if and only if $n$ is squarefree

1 Answers1