Ordered ring

The real numbers are an ordered ring which is also an ordered field. The integers, a subset of the real numbers, are an ordered ring that is not an ordered field.

In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R:[1]

  • if a ≤ b then a + c ≤ b + c.
  • if 0 ≤ a and 0 ≤ b then 0 ≤ ab.
  1. ^ Lam, T. Y. (1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, American Mathematical Society, ISBN 0-8218-0702-1, Zbl 0516.12001