An ordered ring is a commutative ring with a total ordering such that, for every :

1. If , then
2. If and , then

An ordered field is an ordered ring where is also a field.

Examples of ordered rings include:

• The integers , under the standard ordering .
• The real numbers under the standard ordering.
• The polynomial ring in one variable over , under the relation if and only if has nonnegative leading coefficient.

Examples of rings which do not admit any ordering relation making them into an ordered ring include:

• The complex numbers .
• The finite field , where is any prime.