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

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