Integers form Commutative Ring

Theorem
The integers form a commutative ring under addition and multiplication.

Proof
We have that:


 * The algebraic structure $$\left({\Z, +}\right)$$ is an abelian group.


 * The algebraic structure $$\left({\Z, \times}\right)$$ is a monoid and therefore a semigroup.


 * Integer Multiplication Distributes over Addition.

Thus all the ring axioms are fulfilled, and $$\left({\Z, +, \times}\right)$$ is a ring.

Also, by Integer Multiplication has a Zero, the zero is $$0$$.


 * Finally: $$\left({\Z, +, \times}\right)$$ is a commutative ring as Integer Multiplication is Commutative.