Let R be an ordered ring and consider the ring R[x, y] of polynomials in two variables with coefficients in R. Example 25.2 describes two ways in which we can order R[x], and for each of these, we can continue on and order (R[x])[y] in the analogous two ways, giving four ways of arriving at an ordering of R[x, y]. There are another four ways of arriving at an ordering of R[x, y] if we first order R[y] and then (R[y])[x]. Show that all eight of these orderings of R[x, y] are different.

Data from Example 25.2

Let R be an ordered ring with set P of positive elements. There are two natural ways to define an ordering of the polynomial ring R[x]. We describe two possible sets, P_low and P_high, of positive elements. A nonzero polynomial in R[x] can be written in the form f (x) = a_rX^r + a_r+1x^r+1 + · · · + a_nXⁿ where ar ≠ 0 and a_n ≠ 0, so that a_rx^r and a_nxⁿ are the nonzero terms of lowest and highest degree, respectively. Let P_low be the set of all such f(x) for which a_r ∈ P, and let P_high be the set of all such f(x) for which a_n ∈ P. The closure and trichotomy requirements that P_low and P_high must satisfy to give orderings of R [x] follow at once from those same properties for P and the definition of addition and multiplication in R[x]. Illustrating in Z[x], with ordering given by P_low, the polynomial f(x) = -2x + 3x⁴ would not be positive because -2 is not positive in Z. With ordering given by P_high, this same polynomial would be positive because 3 is positive in Z.