Integers form Integral Domain

Theorem
The integers 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.


 * $$(1)$$: $$\left({\Z, +, \times}\right)$$ has a unity, and the unity is $1$.
 * $$(2)$$: $\left({\Z, +, \times}\right)$ 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 $$1$$.