R is a commutative ring with an identity element. Prove or disprove that if
$a \in R ,\; a \neq 0 \;$and is not invertible so $a$ is a zero divisor.
I tried to pick $\;x \in R,\; x\neq 0\;$ s.t $\;ax = 0$
My first try led me to pick $\;x = ax-x$
$ax = a(ax-x) = a^2x - ax\;$ Then I thought that if i could prove that $\;a\;$ is idempotence I could prove that $\;ax = 0$
Take $R=\mathbb{Z}$ an element distinct of $-1$ and $1$ and $0$ is not invertible and is not a zero divisor