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.