Meaning of ordered ring | Babel Free

Definitions

1. 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.
2. 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.

Examples

“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.”

“(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.”

“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.”

“(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.”

CEFR level