If an ideal $I$ is not prime in a ring $R$ then $R/I$ is not an integral domain
Let $R$ be a ring and $I$ an ideal such that $\exists\ a,b\notin I$ and $ab\in I\ \ \ $ ($I$ is not a prime ideal)
Then $a+I,\ b+I\in R/I$ and $(a+I)(b+I)=ab+I=0+I$ so $R/I$ is not an integral domain.
The contraposition is:
$R/I$ is an integral domain $\implies$ $I$ is a prime ideal
Proof:
Since $R/I$ is an integral domain $\nexists (a+I),(b+I)\ne 0+I$ s.t. we have that $ab+I=0$
In other words $\nexists a,b\notin I$ s.t. $ab\in I$ so if $ab\in I$ then $a\in I$ or $b\in I$
Is everything ok in these short proofs?