Rational Numbers under Multiplication form Commutative Monoid

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

Proof
First we note that Rational Multiplication is Associative so $\left({\Q, \times}\right)$ is a semigroup.

Then we have:

Identity
Rational Multiplication Identity is $1$.

Commutativity
Rational Multiplication is Commutative.

Infinite
Rational Numbers are Countably Infinite.