Ordered ring

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Definitions

In abstract algebra, an ordered ring is a commutative ring <math>R</math> with a total order <math>\leq </math> such that

• if <math>a\leq b</math> and <math>c\in R</math>, then <math>a+c \leq b+c</math>

• if <math>0 \leq a</math> and <math>0\leq b</math>, then <math>0 \leq ab</math>.

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 <math>0\leq c, c\neq 0</math> and negative if <math>c\leq 0, c\neq0</math>. The set of positive (or, in some cases, nonnegative) elements in the ring <math>R</math> is often denoted by <math>R_+</math>.

If <math>a</math> is an element of an ordered ring <math>R</math>, then the absolute value of <math>a</math>, denoted <math>|a|</math>, is defined thus:

<math>|a| := \begin{cases} a, & \mbox{if } 0 \leq a \\ -a, & \mbox{otherwise} \end{cases} </math>,

where <math>-a</math> is the additive inverse of <math>a</math> and <math>0</math> is the additive identity element.

Basic properties

1. If <math>a\leq b</math> and <math>0\leq c</math>, then <math>ac\leq bc</math> Template:Ref. This property is sometimes used to define ordered rings instead of the second property in the definition above.
2. If <math>a,b \in R</math>, then <math>|ab|=|a||b|</math> Template:Ref.
3. If <math>a\in R</math>, then either <math>a\in R_+</math>, or <math>-a \in R_+</math>, or <math>a=0\,</math> Template:Ref. This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
4. An ordered ring <math>R</math> has no zero divisors if and only if <math>R_+</math> is closed under multiplication—that is, <math>ab</math> is positive if both <math>a</math> and <math>b</math> are positive Template:Ref.
5. In an ordered ring, no negative element is a square Template:Ref. This is because if <math>a\neq 0</math> and <math>a=b^2</math> then <math>b\neq 0</math> and <math>a=(-b)^2</math>; as either <math>b</math> or <math>-b</math> is positive, <math>a</math> must be positive.

The names below refer to theorems formally verified by the IsarMathLib project.

1. Template:Note OrdRing_ZF_2_L8
2. Template:Note OrdRing_ZF_2_L5
3. Template:Note OrdRing_ZF_3_L2, see also OrderedGroup_ZF_1_L13
4. Template:Note OrdRing_ZF_3_L3
5. Template:Note OrdRing_ZF_3_L12