In abstract algebra, an ordered ring is a commutative ring with a total order ≤ such that for all a, b, and c in R:
- if a ≤ b then a + c ≤ b + c.
- if 0 ≤ a and 0 ≤ b then 0 ≤ ab.
Ordered rings are familiar from arithmetic. Examples include the real numbers. (The rationals and reals in fact form ordered fields.) The complex numbers do not form an ordered ring (or ordered field).
In analogy with real numbers, we call an element c ≠ 0, of an ordered ring positive if 0 ≤ c and negative if c ≤ 0. The set of positive (or, in some cases, nonnegative) elements in the ring R is often denoted by R+.
If a is an element of an ordered ring R, then the absolute value of a, denoted |a|, is defined thus:
where -a is the additive inverse of a and 0 is the additive identity element.
A discrete ordered ring or discretely 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.
Read more about Ordered Ring: Basic Properties
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