Definition:Ordered Ring

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 ordering $$\preceq$$ is a total ordering, then $$\left({R, +, \circ; \preceq}\right)$$ is a totally ordered ring.