# 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 $1$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 21$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 1.3$