Definition:Ordering Compatible with Ring Structure

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) $$\preceq$$ is compatible with $$+$$;
 * 2) $$\forall x, y \in R: 0_R \preceq x, 0_R \preceq y \implies 0_R \preceq x \circ y$$.