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.
Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers.[2] (The rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and i.
In analogy with the real numbers, we call an element c of an ordered ring Rpositive if 0 < c, and negativeifc < 0. 0 is considered to be neither positive nor negative.
The set of positive elements of an ordered ring R is often denoted by R+. An alternative notation, favored in some disciplines, is to use R+ for the set of nonnegative elements, and R++ for the set of positive elements.
Adiscrete ordered ringordiscretely ordered ring is an ordered ring in which there is no element between 0 and 1. The integers are a discrete ordered ring, but the rational numbers are not.
An ordered ring that is not trivial is infinite.[5]
Exactly one of the following is true: a is positive, −a is positive, or a = 0.[6] This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
In an ordered ring, no negative element is a square:[7] Firstly, 0 is square. Now if a ≠ 0 and a = b2 then b ≠ 0 and a = (−b)2; as either b or −b is positive, a must be nonnegative.