ordered ring

Noun

ordered ring (plural ordered rings)

1. (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.
   • 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.1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217

   Synonyms:

   partially ordered ring

2. (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.
   • 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...2013, Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, Abstract Algebra: An Inquiry Based Approach, CRC Press, page 253
   • (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...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...

   Synonyms:

   totally ordered ring