Integers form Integral Domain

Theorem
The integers $\Z$ form an integral domain under addition and multiplication.

Proof
First we note that the integers form a commutative ring with unity whose zero is $0$ and whose unity is $1$.

Next we see that the $\left({\Z, +, \times}\right)$ has no divisors of zero.

So, by definition, the algebraic structure $\left({\Z, +, \times}\right)$ is an integral domain whose zero is $0$ and whose unity is $1$.