ordered ring
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.
Noun
- 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 < is an ordering on A compatible with its ring structure, we shall say that (A,+,·,<) is an ordered ring. An element x of an...
- (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...
- 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...
- (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...