Sentence:  Definition 3.5.4. "A ring R is an <WSD>ordered ring</WSD> 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 <WSD>ordered ring</WSD> and a\in R is a positive element, then the set \{ na: n\in \N \} \subset R^+." / ... / Theorem 3.5.2. "An <WSD>ordered ring</WSD> must be infinite." 
Meaning of the word marked with <WSD> tags:  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".