Ring without zero divisor (Proof)

Question

$A$ is commutative ring and $I$ is ideal in $A .\ $ Then prove

A is a ring without zero divisor iff $A/I$ is a ring without divisor.

Please guide me how can I do this. Thank you.

Answer 1

Counter-example:

$A=\Bbb Z$ is an integral domain, but $\Bbb Z/I$, with $I=(4)$ is not.

In general, $A/I$ is an integral domain (no zero-divisors) $\iff I\subset A$ is a prime ideal. This is because if $(a+I)(b+I)=ab+I=0+I\Rightarrow ab\in I$, then since $I$ is a prime ideal, $a\in I$ or $b\in I$. That is, $a+I=0+I$ or $b+I=0+I$.