Sentence:  The positive elements in an <WSD>ordered ring</WSD> 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 4>2 because 4-2>0. We can extend this idea to any <WSD>ordered ring</WSD>. If R is an <WSD>ordered ring</WSD> and a,b\in R, then we know by trichotomy that exactly one of the following must be true: a-b>0, a-b=0, or -(a-b)>0. 
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".