Definition:Ordered Ring

Definition

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.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields