# Integers form Commutative Ring

## Theorem

The set of integers $\Z$ forms a commutative ring under addition and multiplication.

## Proof

We have that:

From Integers under Addition form Abelian Group, the algebraic structure $\struct {\Z, +}$ is an abelian group.
From Integers under Multiplication form Countably Infinite Commutative Monoid, the algebraic structure $\struct {\Z, \times}$ is a commutative monoid and therefore a commutative semigroup.
Integer Multiplication Distributes over Addition.

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

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

$\blacksquare$