\begin{definition}[Definition:Ordering Compatible with Ring Structure]
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
An ordering $\preccurlyeq$ on $R$ is '''compatible with the ring structure of $R$''' {{iff}}:
{{begin-axiom}}
{{axiom | n = \text {OR} 1
        | lc= $\preccurlyeq$ is compatible with $+$:
        | q = \forall a, b, c \in  R
        | ml= a \preccurlyeq b
        | mo= \implies
        | mr= \paren {a + c} \preccurlyeq \paren {b + c}
}}
{{axiom | n = \text {OR} 2
        | lc= Product of Positive Elements is Positive
        | q = \forall a, b \in R
        | ml= 0_R \preccurlyeq x, 0_R \preccurlyeq y
        | mo= \implies
        | mr= 0_R \preccurlyeq x \circ y
}}
{{end-axiom}}
\end{definition}