Show that a commutative ring with unit A is an integral ring if, and only if, $\langle 0 \rangle$ is a prime.

Question

$\quad$ I'm not convinced of my demonstration below, could someone give me a hint?

$(\Leftarrow)$ Is it correct?

$\quad$ Suppose $\{0\}$ be prime. Then for $x,y \in R\setminus\{0\}$ we have $xy\in \{0\}$ implies $x=0$ or $y=0$ and thus $A$ is an integral domain.

$(\Rightarrow)$ Is it correct?

$\quad$ If $A$ is an integral domain, then there are no zero divisors. So if $xy=0$ we have $x=0$ or $y=0$ meaning $\{0\}$ is a prime ideal.

$\square$