ordered ring
短语词形变化
释义与例句
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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.
数学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.
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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.
数学(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.