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 whose zero is $0$.

Next we see that the additional properties are fulfilled for $\left({\Z, +, \times}\right)$ to be an integral domain.


 * $(1): \quad$ $\left({\Z, +, \times}\right)$ has a unity, and the unity is $1$.
 * $(2):\quad$ $\left({\Z, +, \times}\right)$ has no divisors of zero.

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