Definition:Ordered Ring

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\preceq$ be an ordering compatible with the ring structure of $\left({R, +, \circ}\right)$.

Then $\left({R, +, \circ, \preceq}\right)$ is an ordered ring.

Totally Ordered Ring
If the ordering $\preceq$ is a total ordering, then $\left({R, +, \circ, \preceq}\right)$ is a totally ordered ring.