Definition:Ordered Ring

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Let $\struct {R, +, \circ}$ be a ring.

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

Then $\struct {R, +, \circ, \preceq}$ is an ordered ring.

Totally Ordered Ring

Let $\struct {R, +, \circ, \preceq}$ be an ordered ring.

Let the ordering $\preceq$ be a total ordering.

Then $\struct {R, +, \circ, \preceq}$ is a totally ordered ring.

Also see

  • Results about ordered rings can be found here.