Absolute Value on Ordered Integral Domain is Strictly Positive except when Zero

Theorem
Let $\struct {D, +, \times}$ be an ordered integral domain.

For all $a \in D$, let $\size a$ denote the absolute value of $a$.

Then $\size a$ is always (strictly) positive except when $a = 0$.

Proof
Let $P$ be the positivity property on $D$, and let $<$ be the ordering induced by it.

From the trichotomy law, exactly one of three possibilities holds for any $ a \in D$:

$(1): \quad \map P a$:

In this case $0 < a$ and so $\size a = a$.

So:
 * $\map P a \implies \map P {\size a}$

$(2): \quad \map P {-a}$:

In this case $a < 0$ and so $\size a = -a$.

So:
 * $\map P {-a} \implies \map P {\size a}$

$(3): \quad a = 0$:

In this case $\size a = a$.

So:
 * $a = 0 \implies \size a = 0$.

Hence the result.

Also see

 * Absolute Value is Bounded Below by Zero