Ideal of Ring/Examples/Set of Even Integers

Example of Ideal of Ring
The set $2 \Z$ of even integers forms an ideal of the ring of integers.

Proof
Let $x \in 2 \Z$.

Then:
 * $\forall y \in \Z: x y \in 2 \Z$

and:
 * $\forall y \in \Z: y x \in 2 \Z$

Hence the result by definition of ideal.