ordered ring

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ordered ring

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Noun edit

ordered ring (plural ordered rings)

(algebra, order theory, ring theory) A ring, R, equipped with a partial order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.
- 1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
  If $\leq$ is an ordering on $A$ compatible with its ring structure, we shall say that $(A,\ +,\ \cdot ,\leq )$ is an ordered ring. An element $x$ of an ordered ring $A$ is positive if $x\geq 0$ , and $x$ is strictly positive if $x>0$ .
  
  The set of all positive elements of an ordered ring $A$ is denoted by $A_{+}$ , and the set of all strictly positive elements of $A$ is denoted by $A_{+}^{*}$ .
  
  If $(A,\ +,\ \cdot ,\leq )$ is an ordered ring and if $\leq$ is a total ordering, we shall, of course, call $(A,\ +,\ \cdot ,\leq )$ a totally ordered ring; if $(A,\ +,\ \cdot )$ is a field, we shall call $(A,\ +,\ \cdot ,\leq )$ an ordered field, and if, moreover, $\leq$ is a total ordering, we shal call $(A,\ +\ \cdot ,\leq )$ a totally ordered field.
- 1990, P. M. Cohn, J. Howie (translators), Nicolas Bourbaki, Algebra II: Chapters 4-7, [1981, N. Bourbaki, Algèbre, Chapitres 4 à 7, Masson], Springer, 2003, Softcover reprint, page 19,
  DEFINITION 1. — Given a commutative ring $A$ , we say that an ordering on $A$ is compatible with the ring structure on $A$ if it is compatible with the additive group structure of $A$ , and if it satisfies the following axiom:
  
  (OR) The relations $x\geq 0$ and $y\geq 0$ imply $xy\geq 0$ .
  
  The ring $A$ , together with such an ordering, is called an ordered ring.
  
  Examples. — 1) The rings $\mathbb {Q}$ and $\mathbb {Z}$ , with the usual orderings, are ordered rings.
  
  2) A product of ordered rings, equipped with the product ordering, is an ordered ring. In particular, the ring $A^{E}$ of mappings from a set $E$ to an ordered ring $A$ is an ordered ring.
  
  3) A subring of an ordered ring, with the induced ordering, is an ordered ring.
(algebra, order theory, ring theory) A ring, R, equipped with a total order, ≤, such that for arbitrary a, b, c ∈ R, if a ≤ b then a + c ≤ b + c, and if, additionally, 0 ≤ c, then both ca ≤ cb and ac ≤ bc.
- 2013, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, Abstract Algebra: An Inquiry Based Approach, CRC Press, page 253:
  The positive elements in an ordered ring allow us to compare elements to 0, but we know in the integers that we can compare any two elements to each other. For example, we know that $4>2$ because $4-2>0$ . We can extend this idea to any ordered ring. If $R$ is an ordered ring and $a,b\in R$ , then we know by trichotomy that exactly one of the following must be true: $a-b>0$ , $a-b=0$ , or $-(a-b)>0$ .
- 2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra, Johns Hopkins University Press, page 77,
  Definition 3.5.4. A ring $R$ is an ordered ring if there exists a distinguished set $R^{+}$ , $R^{+}\subset R$ , called the set of positive elements, with the properties that:
  (1) The set $R^{+}$ is closed under addition and multiplication.
  
  (2) If $x\in R$ then exactly one of the following is true: (trichotomy law)
  (a) $x=0$ ,
  
  (b) $x\in R^{+}$ ,
  
  (c) $-x\in R^{+}$ .
  
  If further $R$ is an integral domain we call $R$ an ordered integral domain.
  
  […]
  
  Lemma 3.5.9. If $R$ is an ordered ring and $a\in R$ is a positive element, then the set $\{na:n\in \mathbb {N} \}\subset R^{+}$ .
  
  […]
  
  Theorem 3.5.2. An ordered ring must be infinite.

Usage notes edit

While the ring is, strictly speaking, not necessarily associative or commutative, it may be defined as either or both by authors working within an overarching theory.
The property ${\textsf {if}}\ a\leq b\ {\textsf {and}}\ 0\leq c\ {\textsf {then}}\ ca\leq cb\ {\textsf {and}}\ ac\leq bc$ in the definition is sometimes replaced by the equivalent ${\textsf {if}}\ 0\leq a\ {\textsf {and}}\ 0\leq b\ {\textsf {then}}\ 0\leq ab$ .
The order is said to be compatible with the ring structure of $R$ (in the sense that order is preserved by addition and, to an extent, multiplication).
A partial order $\leq$ is a total order if and only if the trichotomy condition holds: in other words, $P\cup -P=R$ , where $P=\left\{x:x\in R,0\leq x\right\}$ is the positive cone of $R$ and $-P=\left\{-x:x\in P\right\}$ .
- Consequently, in the total order case, it makes sense to define an absolute value applicable to every element of $R$ : $|x|=\left\{{\begin{array}{rl}x,&{\textsf {if}}\ 0\leq x\\-x,&{\textsf {if}}\ 0>x.\end{array}}\right.$