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.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 58$. Ideals