Definition:Ordering Compatible with Ring

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

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 23$