Inabstract algebra, anordered ring is a (usuallycommutative)ringR with atotal order ≤ such that for alla,b, andc inR:^[1]

• ifa ≤b thena +c ≤b +c.
• if 0 ≤a and 0 ≤b then 0 ≤ab.

Ordered rings are familiar fromarithmetic. Examples include theintegers, therationals and thereal numbers.^[2] (The rationals and reals in fact formordered fields.) Thecomplex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 andi.

In analogy with the real numbers, we call an elementc of an ordered ringRpositive if 0 <c, andnegative ifc < 0. 0 is considered to be neither positive nor negative.

The set of positive elements of an ordered ringR is often denoted byR₊. An alternative notation, favored in some disciplines, is to useR₊ for the set of nonnegative elements, andR₊₊ for the set of positive elements.

If${\displaystyle a}$ is an element of an ordered ringR, then theabsolute value of${\displaystyle a}$, denoted${\displaystyle |a|}$, is defined thus:

where${\displaystyle -a}$ is theadditive inverse of${\displaystyle a}$ and 0 is the additiveidentity element.

Adiscrete ordered ring ordiscretely 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.

For alla,b andc inR:

• Ifa ≤b and 0 ≤c, thenac ≤bc.^[3] This property is sometimes used to define ordered rings instead of the second property in the definition above.
• |ab| = |a| |b|.^[4]
• An ordered ring that is nottrivial is infinite.^[5]
• Exactly one of the following is true:a is positive, −a is positive, ora = 0.^[6] This property follows from the fact that ordered rings areabelian,linearly ordered groups with respect to addition.
• In an ordered ring, no negative element is a square:^[7] Firstly, 0 is nonnegative. Now ifa ≠ 0 anda =b² thenb ≠ 0 anda = (−b)²; as eitherb or −b is positive,a must be nonnegative.

• Ordered field – Algebraic object with an ordered structure
• Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Ordered topological vector space
• Ordered vector space – Vector space with a partial order
• Partially ordered ring – Ring with a compatible partial order
• Partially ordered space – Partially ordered topological space
• Riesz space – Partially ordered vector space, ordered as a lattice, also called vector lattice
• Ordered semirings

The list below includes references to theorems formally verified by theIsarMathLib project.

• Ordered groups
• Real algebraic geometry

• Pages displaying short descriptions of redirect targets via Module:Annotated link