additive order of any zero divisor in $Z_{p^2}$ is p, is it true?

Question

This result was used in a proof of a theorem, i am not sure if it's true. can someone tell the proof idea.

Can it be generalized to additive order of any zero divisor in $Z_{p^k}$, is there any formula to calculate the additive order of zero divisors of $Z_{p^k}$ in general ?

Answer 1

Well, the zero divisors of ${\Bbb Z}_{p^k}$ are exactly the nonzero elements $a$ with $\gcd(a,p^k)\ne 1$, i.e., the zero divisors are exactly the nonzero multiples of $p$. Then the $p^{k-1}$-multiple of a zero divisor $a$ written as $a=pb$ is $p^{k-1}\cdot a = p^{k-1}pb = p^kb=0$.