Integers form Commutative Ring

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

Proof
We have that:


 * From Integers under Addition form Abelian Group, the algebraic structure $\left({\Z, +}\right)$ is an abelian group.


 * From Integers under Multiplication form Countably Infinite Commutative Monoid, the algebraic structure $\left({\Z, \times}\right)$ is a commutative monoid and therefore a commutative semigroup.


 * Integer Multiplication Distributes over Addition.

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

By Integer Multiplication has Zero, the zero is $0$.