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
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