The nil radical of an Archimedean partially ordered ring with positive squares

Let R be an Archimedean partially ordered ring in which the square of every element is positive, and N(R) the set of all nilpotent elements of R. It is shown that N(R) is the unique nil radical of R, and that N(R) is locally nilpotent and even nilpotent with exponent at most 3 when R is 2-torsion-free. R is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator. The results are applied on partially ordered rings in which every element a is expressed as a = a1 − a2 with positive a1, a2 satisfying a1a2 = a2a1 = 0.