Definition:Ordering Compatible with Ring Structure

Definition
Let $$\left({R, +, \circ}\right)$$ be a ring whose zero is $$0_R$$.

An ordering $$\preceq$$ on $$R$$ is compatible with the ring structure $$R$$ iff:


 * $$(1) \quad \preceq$$ is compatible with $$+$$


 * $$(2) \quad \forall x, y \in R: 0_R \preceq x, 0_R \preceq y \implies 0_R \preceq x \circ y$$