ordered ring

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