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:

Monoid Axiom $\text S 0$: Closure

Rational Multiplication is Closed.

$\Box$

Monoid Axiom $\text S 1$: Associativity

Rational Multiplication is Associative.

$\Box$

Monoid Axiom $\text S 2$: Identity

Rational Multiplication Identity is $1$.

$\Box$

Hence the result.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids