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