Definition:Ordered Field

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

If $\left({R, +, \circ}\right)$ is a field, then $\left({R, +, \circ, \preceq}\right)$ is an ordered field.

Totally Ordered Field
If $\le$ is a total ordering, then $\left({R, +, \circ, \le}\right)$ is a totally ordered field.