An element is nilpotent in $\mathbb{Z}/n\mathbb{Z}$ if and only if all prime divisors of $n$ divide the element.
14403
If $n = p_1^{a_1} ... p_r^{a_r}$ the set of nilpotents of $\mathbb{Z}_n$ is $(p_1 p_2 ... p_{r-1} p_r)\Bbb Z_n$
133046
Showing $[L,rad(L)]$ is nilpotent.
23394
If $G/Z(G)$ is cyclic, then $G$ is abelian
76813
The ring $ℤ/nℤ$ is a field if and only if $n$ is prime
64654
Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
103518
If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents
19202
$\mathbb{F}_{p^d}\subseteq\mathbb{F}_{p^n}$ if and only if $d$ divides $n$
34183
An equivalent condition with $\{0\}$ being the only nilpotent ideal
834
If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.
156443
if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
39788
$a+1,a-1$ invertible for nilpotent element
15191
If $a^2$ divides $b^2$, then $a$ divides $b$
67944
Nilpotent Elements and Intersection of Prime Ideals
69012
How to directly prove every element in a ring with exactly three ideals is either a unit or a zero divisor?
78455
Prove that a nonzero integer p is prime if and only if the ideal (p) is maximal in Z.
103830
Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal
102112
Idempotents in $\mathbb Z_n$