If $n>1$ is not a prime number, $\mathbb Z/n \mathbb Z$ is commutative ring, where $n>1$ is a fixed natural number, then $\mathbb Z/n\mathbb Z$ is not an integral domain.
natural number is either a prime number or a product of two numbers... integral domain is $ac=ab$, then $b=c$ ($a$ is not $0$) its mean, zero divisor does not work in The law concellation. If $\mathbb Z/n\mathbb Z$ has zero divisor, then this proof will be easy. But how to prove $\mathbb Z/n\mathbb Z$ has zero divisor?
If a number $n$ is the product of two numbers $a$ and $b$ where both are not equal to 1, which is possible since $n$ is not prime, then the residue classes of $a$ and $b$ are both nonzero but their product is zero mod $n$.