Show that Zn has a nonzero nilpotent element if and only

Step 1 of 2

If n is divisible by the square of some prime, then \(p^{2} \mid n\)        

Then we can write \(n=p^{2} q\) where p is prime.

Let

\(\begin{aligned} p q \in Z_{n} & \Rightarrow(p q)^{2} \in Z_{n} \\ & \Rightarrow p^{2} q^{2} \in Z_{n} \end{aligned}\)

Here, \(p^{2} q^{2}=0\) (since \(\left(p^{2} q^{2}\right) \bmod p q=0\))

Therefore, \(n q=0\).

That is, pq is a non-zero nilpotent element in \(Z_{n}\).