Ordered ring

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Definitions

In abstract algebra, an ordered ring is a commutative ring [R] with a total order [\leq ] such that

• if [a\leq b] and [c\in R], then [a+c \leq b+c]

• if [0 \leq a] and [0\leq b], then [0 \leq ab].

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 [0\leq c, c\neq 0] and negative if [c\leq 0, c\neq0]. 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:

[|a| := \begin a, & \mbox 0 \leq a \\ -a, & \mbox \end ],

where [-a] is the additive inverse of [a] and [0] is the additive identity element.

Basic properties

• If [a\leq b] and [0\leq c], then [ac\leq bc.]^{[#endnote_ineq]} This property is sometimes used to define ordered rings instead of the second property in the definition above.
• If [a,b \in R], then [|ab|=|a||b|.]^{[#endnote_abs]}
• An ordered ring that is not trivial is infinite. ^{[#endnote_inf]}
• If [a\in R], then either [a\in R_+], or [-a \in R_+], or [a=0\,.]^{[#endnote_partition]} This property follows from the fact that ordered rings are abelian, linearly ordered groups with respect to addition.
• An ordered ring [R] has no zero divisors if and only if [R_+] is closed under multiplication—that is, [ab] is positive if both [a] and [b] are positive.^{[#endnote_equiv]}
• In an ordered ring, no negative element is a square.^{[#endnote_square]} This is because if [a\neq 0] and [a=b^2] then [b\neq 0] and [a=(-b)^2]; as either [b] or [-b] is positive, [a] must be positive.

Notes

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

1. ↑  OrdRing_ZF_1_L9
2. ↑  OrdRing_ZF_2_L5
3. ↑  ord_ring_infinite
4. ↑  OrdRing_ZF_3_L2, see also OrdGroup_decomp
5. ↑  OrdRing_ZF_3_L3
6. ↑  OrdRing_ZF_1_L12

 

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