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 $a \in \Z_{\ne 0}$.
By Integer Divides its Absolute Value:
- $a \divides \size a$ and $\size a \divides a$
Hence $\size a$ is an associate of $a$.
Now we prove its uniqueness.
Let $b, c \in \Z_{\ne 0}$ 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 $\mathbf Z$