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 4>2 because 4-2>0. We can extend this idea to any ordered ring. If R is an ordered ring and a,b∈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.