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In abstract algebra, an ordered ring is a commutative ring with a total order such that
if and , then
if and , then .
Ordered rings are familiar from arithmetic. Examples include the integers, the rational numbers, and the real numbers. (The rationals and reals in fact form ordered fields.) The complex numbers do not form an ordered ring (or field).
In analogy with ordinary numbers, we call an element c of an ordered ring positive if and negative if . The set of positive (or, in some cases, nonnegative) elements in the ring is often denoted by .
If is an element of an ordered ring , then the absolute value of , denoted , is defined thus:
,
where is the additive inverse of and is the additive identity element.
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Basic properties
If and , then This property is sometimes used to define ordered rings instead of the second property in the definition above.
If , then
An ordered ring that is not trivial is infinite.
If , then either , or , or This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
An ordered ring has no zero divisors if and only if is closed under multiplication—that is, is positive if both and are positive.
In an ordered ring, no negative element is a square. This is because if
eq 0 and then and ; as either or is positive, must be positive.
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Notes
The names below refer to theorems formally verified by the IsarMathLib project.
OrdRing_ZF_3_L2, see also OrdGroup_decomp
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