ordered ring
Language: English
Part of speech: noun
Forms: form: ordered rings
Senses: examples: text: 1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
If < is an ordering on A compatible with its ring structure, we shall say that (A,+,·,<) is an ordered ring. An element x of an ordered ring A is positive if x>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,+,·,<) is an ordered ring and if < is a total ordering, we shall, of course, call (A,+,·,<) a totally ordered ring; if (A,+,·) is a field, we shall call (A,+,·,<) an ordered field, and if, moreover, < is a total ordering, we shal call (A,+·,<) a totally ordered field.
text: (OR) The relations x>0 and y>0 imply xy>0.
The ring A, together with such an ordering, is called an ordered ring.
Examples. — 1) The rings Q and 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 Aᴱ 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.
ref: 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
topics: algebra
mathematics
order-theory
sciences
qualifier: ring theory
glosses: 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.
examples: text: 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 4gt;2 because 4-2gt;0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b#92;inR, then we know by trichotomy that exactly one of the following must be true: a-bgt;0, a-b#61;0, or -(a-b)gt;0.
ref: 2013, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, Abstract Algebra: An Inquiry Based Approach, CRC Press, page 253:
text: (1) The set R⁺ is closed under addition and multiplication.
(2) If x∈R then exactly one of the following is true: (trichotomy law)
(a) x=0,
(b) x∈R⁺,
(c) -x∈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∈R is a positive element, then the set na:n∈ N⊂R⁺.
[…]
Theorem 3.5.2. An ordered ring must be infinite.
ref: 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⁺⊂R, called the set of positive elements, with the properties that
topics: algebra
mathematics
order-theory
sciences
qualifier: ring theory
glosses: 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.
Synonyms: sense: ring equipped with a partial order
word: partially ordered ring
sense: ring equipped with a total order
word: totally ordered ring
Hyponyms: sense: both senses
alt: = discretely ordered ring
word: discrete ordered ring
sense: both senses
word: ordered field
Derived terms: lattice-ordered ring
Related terms: ordered field