Rational Numbers under Multiplication form Monoid

Theorem
The set of rational numbers under multiplication $\struct {\Q, \times}$ forms a monoid.

Proof
Taking the monoid axioms in turn:

$\text S 0$: Closure
Rational Multiplication is Closed.

$\text S 1$: Associativity
Rational Multiplication is Associative.

$\text S 2$: Identity
Rational Multiplication Identity is $1$.

Hence the result.