Ordered ring

English

Noun

1. Template:Lb 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 $\le$ is an ordering on $A$ compatible with its ring structure, we shall say that $(A,+,\cdot ,\le )$ is an ordered ring. An element $x$ of an ordered ring $A$ is positive if $x\ge 0$, 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 $(A,+,\cdot ,\le )$ is an ordered ring and if $\le$ is a total ordering, we shall, of course, call $(A,+,\cdot ,\le )$ a totally ordered ring; if $(A,+,\cdot )$ is a field, we shall call $(A,+,\cdot ,\le )$ an ordered field, and if, moreover, $\le$ is a total ordering, we shal call $(A,+\cdot ,\le )$ a totally ordered field.

   • 1990, P. M. Cohn, J. Howie (translators), Template:W, 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 $x\ge 0$ and $y\ge 0$ imply $xy\ge 0$.

     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.

2. Template:Lb 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.
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   • 2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra, Template:W, 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.

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     Lemma 3.5.9. If $R$ is an ordered ring and $a\in R$ is a positive element, then the set $\{na:n\in \mathbb{N}\}\subset R^{+}$.

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     Theorem 3.5.2. An ordered ring must be infinite.

Usage notes

• 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 $\mathsf{\text{if}}a\le b\mathsf{\text{and}}0\le c\mathsf{\text{then}}ca\le cb\mathsf{\text{and}}ac\le bc$ in the definition is sometimes replaced by the equivalent $\mathsf{\text{if}}0\le a\mathsf{\text{and}}0\le b\mathsf{\text{then}}0\le ab$.
• 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 $\le$ is a total order if and only if the trichotomy condition holds: in other words, $P\cup -P=R$, where $P=\{x:x\in R,0\le x\}$ is the Template:M of $R$ and $-P=\{-x:x\in P\}$.
  • Consequently, in the total order case, it makes sense to define an Template:M applicable to every element of $R$: $|x|=\{\begin{array}{cc}x,& \mathsf{\text{if}}0\le x\\ -x,& \mathsf{\text{if}}0>x.\end{array}$

Synonyms

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Hyponyms

• Template:Sense Template:L (= Template:L), Template:L

Derived terms

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See also

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Further reading

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• Ordered ring on Template:W