An element $a$ in a commutative ring $R$ is called nilpotent if $a^r = 0$ for some $r\in\Bbb N$.
$1.$ List all nilpotent elements in the ring $R = \Bbb Z/12\Bbb Z$.
$2.$ Let $a\in R\setminus\{0\}$ be nilpotent. Show that $a$ is a zero divisor.
My attempt:
$1.$ Because the prime factorization of $12=2^2\cdot3$, the prime factorization of $a$ contains $2,3$, so the nilpotent elements are $[0]_{12}$ and $[6]_{12}$. Is this correct?
$2.$ I'm having trouble on this one. :(