# Even Integers do not form Integral Domain

## Theorem

Let $2 \Z$ be the set of even integers.

Then $\struct {2 \Z, +, \times}$ is not an integral domain.

## Proof

From Integer Multiples form Commutative Ring, $\struct {2 \Z, +, \times}$ is a commutative ring.

As $2 \ne 1$, we also have from Integer Multiples form Commutative Ring that $\struct {2 \Z, +, \times}$ has no unity.

Hence by definition it is not an integral domain.

$\blacksquare$