Integer Multiples Closed under Multiplication

Theorem

Let $n \Z$ be the set of integer multiples of $n$.

Then the algebraic structure $\struct {n \Z, \times}$ is closed under multiplication.

Proof

Let $x, y \in n \Z$.

Then $\exists p, q \in \Z: x = n p, y = n q$.

So $x y = n p \cdot n q = n \paren {n p q}$ where $n p q \in \Z$.

Thus $x y \in n \Z$ and so $\struct {n \Z, \times}$ is closed.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.1$