Totally Ordered Ring Zero Precedes Element or its Inverse

Theorem
Let $\left({R, +, \circ, \preceq}\right)$ be an ordered ring.

From the definition of ordered ring, $\preceq$ is compatible with $+$.

Let $0_R$ be the zero element of the ring.

Let $x \ne 0_R$ be a non-zero element of the ring, and let $-x$ be its ring negative.

Then $0_R \prec x \lor 0_R \prec -x$, but not both.

Proof
By the definition of total ordering, $\preceq$ is connected.

As $x \ne 0_R$, one of the following is true, but not both:


 * $(1): 0_R \prec x$


 * $(2): x \prec 0_R$

If $(2)$, because $\prec$ is compatible with $+$: