ordered ring

Noun

 * 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) * 1965, Seth Warner, Modern Algebra, Dover, 1990, Single-volume republication, page 217,
 * If $$\le$$ is an ordering on $$A$$ compatible with its ring structure, we shall say that $$(A,\ +,\ \cdot,\le)$$ is an ordered ring. An element $$x$$ of an ordered ring $$A$$ is positive if $$x\ge 0$$, 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 $$(A,\ +,\ \cdot,\le)$$ is an ordered ring and if $$\le$$ is a total ordering, we shall, of course, call $$(A,\ +,\ \cdot,\le)$$ a totally ordered ring; if $$(A,\ +,\ \cdot)$$ is a field, we shall call $$(A,\ +,\ \cdot,\le)$$ an ordered field, and if, moreover, $$\le$$ is a total ordering, we shal call $$(A,\ +\ \cdot,\le)$$ a totally ordered field.
 * 1) * 1990, P. M. Cohn, J. Howie (translators),, Algebra II: Chapters 4-7, [1981, N. Bourbaki, Algèbre, Chapitres 4 à 7, Masson], Springer, 2003, Softcover reprint, page 19,
 * D EFINITION 1. — Given a commutative ring $$A$$, we say that an ordering on $$A$$ is compatible with the ring structure on $$A$$ if it is compatible with the additive group structure of $$A$$, and if it satisfies the following axiom:
 * (OR) The relations $$x \ge 0$$ and $$y \ge 0$$ imply $$xy \ge 0$$.
 * 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 $$A^E$$ 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.
 * 1)  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.
 * 2) * 2014, Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger, Introduction to Abstract Algebra,, page 77,
 * Definition 3.5.4. A ring $$R$$ is an ordered ring if there exists a distinguished set $$R^+$$, $$R^+\subset R$$, called the set of positive elements, with the properties that:
 * (1) The set $$R^+$$ is closed under addition and multiplication.
 * (2) If $$x\in R$$ then exactly one of the following is true: (trichotomy law)
 * (a) $$x=0$$,
 * (b) $$x\in R^+$$,
 * (c) $$-x\in R^+$$.
 * 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 $$a\in R$$ is a positive element, then the set $$\{ na: n\in \N \} \subset R^+$$.
 * Theorem 3.5.2. An ordered ring must be infinite.
 * Lemma 3.5.9. If $$R$$ is an ordered ring and $$a\in R$$ is a positive element, then the set $$\{ na: n\in \N \} \subset R^+$$.
 * Theorem 3.5.2. An ordered ring must be infinite.
 * Theorem 3.5.2. An ordered ring must be infinite.

Usage notes

 * While the ring is, strictly speaking, not necessarily associative or commutative, it may be defined as either or both by authors working within an overarching theory.
 * The property $$\textsf{if}\ a\le b\ \textsf{and}\ 0\le c \ \textsf{then}\ ca\le cb\ \textsf{and}\ ac \le bc$$ in the definition is sometimes replaced by the equivalent $$\textsf{if}\ 0\le a\ \textsf{and}\ 0\le b\ \textsf{then}\ 0 \le ab$$.
 * The order is said to be compatible with the ring structure of $$R$$ (in the sense that order is preserved by addition and, to an extent, multiplication).
 * A partial order $$\le$$ is a total order if and only if the trichotomy condition holds: in other words, $$P\cup -P=R$$, where $$P =\left \{x:x\in R, 0\le x\right \}$$ is the of $$R$$ and $$-P =\left \{-x:x\in P\right \}$$.
 * Consequently, in the total order case, it makes sense to define an applicable to every element of $$R$$: $$|x| = \left\{\begin{array}{rl} x, & \textsf{if}\ 0 \leq x \\ -x, & \textsf{if}\ 0 > x. \end{array}\right.$$