Integers under Multiplication form Countably Infinite Commutative Monoid

Theorem
The set of integers under multiplication $$\left({\Z, \times}\right)$$ is a countably infinite commutative monoid.

Proof
First we note that Integer Multiplication forms Semigroup.

Then we have:

Identity
Integer Multiplication Identity is $1$.

Commutativity
Integer Multiplication is Commutative.

Infinite
Integers are Countably Infinite.