# Even Integers form Commutative Ring

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## Theorem

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

Then $\struct {2 \Z, +, \times}$ is a commutative ring.

However, $\struct {2 \Z, +, \times}$ 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.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 18$. Definition of a Ring: Example $28$