Let R be an ordered ring. Describe the order ring of a positive element a of R and the monomials x, x², x³ , · · ·, xⁿ. · · · in R[x] as we did in Example 25.6, but using the set P_high of Example 25.6 as set of positive elements of R[x].

Data from in Example 25.6

Let R be an ordered ring. It is illustrative to think what the orderings of R[x] given by P_low and P_high in Example 25.2 mean in terms of the relation < of Theorem 25.5. Taking P_low, we observe, for every a > 0 in R, that a - x is positive so x < a. Also, x = x - 0 is positive, so 0 < x. Thus 0 < x < a for every a ∈ R. We have (xⁱ - x^j) ∈ P_low when i < j, so x^j < xⁱ if i < j. Our monomials have the ordering 0 < · · · x⁶ < x⁵ < x⁴ < x³ < x² < x < a for any positive a ∈ R. Taking R = R, we see that in this ordering of R[x] there are infinitely many positive elements that are less than any positive real number!