Integers form Integral Domain

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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 $\struct {\Z, +, \times}$ has no divisors of zero.


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

$\blacksquare$


Sources

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