# Definition:Ordering Compatible with Ring Structure

## Definition

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

An ordering $\preccurlyeq$ on $R$ is compatible with the ring structure of $R$ if and only if:

 $(\text {OR} 1)$ $:$ $\preccurlyeq$ is compatible with $+$: $\ds \forall a, b, c \in R:$ $\ds a \preccurlyeq b$ $\ds \implies$ $\ds \paren {a + c} \preccurlyeq \paren {b + c}$ $(\text {OR} 2)$ $:$ Product of Positive Elements is Positive $\ds \forall a, b \in R:$ $\ds 0_R \preccurlyeq x, 0_R \preccurlyeq y$ $\ds \implies$ $\ds 0_R \preccurlyeq x \circ y$