# Non-Zero Integer has Unique Positive Integer Associate

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## Theorem

Let $a \in \Z$ be an integer such that $a \ne 0$.

Then $a$ has a unique associate $b \in \Z_{>0}$.

## Proof

Let $a, b, c \in \Z_{\ne 0}$ such that $b > 0$ and $c > 0$.

Let $a \sim b$ and $a \sim c$ where $\sim$ denotes the relation of associatehood.

By definition of associatehood:

- $a \divides b$ and $b \divides a$

and:

- $a \divides c$ and $c \divides a$

From Divisor Relation is Antisymmetric/Corollary/Proof 2:

- $a = \pm b$

and

- $a = \pm c$

That is:

- $\pm b = \pm c$

which means:

- $b = c$ or $b = -c$

But as both $b > 0$ and $c > 0$:

- $b = c$

Hence the result.

$\blacksquare$

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$