Even Integers form Commutative Ring

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 $28$