ordered ring

In abstract algebra, an ordered ring is a ring that is a totally ordered set. Ordered rings are familiar from arithmetic. Examples include the integers, the rational numbers and the real numbers. The complex numbers do not form an ordered field. In analogy with ordinary numbers, we call an element c of an ordered ring positive if c > 0 and negative if c < 0. The order on a ring is required to satisfy certain conditions that are familiar properties of ordinary numbers:

If a < b then for any c, a + c < b + c.
If 0 < a and 0 < b then 0 < ab.

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