I was given the following question. Let A be a commutative ring with unity, and J an ideal of A. Prove J is a prime ideal if and only if A/J is an integral domain.

This is how I proved it but I dont know if it is complete enough. If anyone has any suggestions please let me know

Suppose that there exists a,b $in$ A/J, so that ab = 0. This also means that ab $in$ J. We need to prove that there are no zero divisors in A/J. However, since J is a prime ideal, it means that a $in$ J or b $in$ J. Since a $in$ J or b $in$ J, this means that a = 0 or b = 0. Thus there means that there are no zero divisors. Hence A/J is an integral domain.

Suppose A/J is an integral domain. Let ab $in$ A. Since A/J is an integral domain it has no zero divisors, this means that ab = 0 and either a = 0 or b = 0. Which means that a $in$ J or b $in$ J and ab $in$ J. Thus J is a prime ideal.