Integers form Commutative Ring

Theorem
The integers $\Z$ 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 Zero, the zero is $0$.


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