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 $\preccurlyeq$ satisies the ring compatible ordering axioms:

$(\text {OR} 1)$	$:$		$\preccurlyeq$ 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 a, 0_R \preccurlyeq b $	$\ds \implies $	$\ds 0_R \preccurlyeq a \circ b $

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers

\((\text {OR} 1)\)	$:$		$\preccurlyeq$ 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	\(\ds \forall a, b \in R:\)		\(\ds 0_R \preccurlyeq a, 0_R \preccurlyeq b \)	\(\ds \implies \)	\(\ds 0_R \preccurlyeq a \circ b \)