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 $+$


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

Also see

 * Ordered Ring