Integer Multiples Closed under Addition

Theorem

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

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


Proof

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

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

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

Thus $x + y \in n \Z$ and so $\struct {n \Z, +}$ 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 $8.1$
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