Positive squares and cubic nilpotence in ordered rings

Let R be an associative partially ordered ring in which every square is nonnegative; an identity is not assumed. We prove x, r ≥ 0, x² = 0 ⇒ (xr)³ = xrxrx = 0. If the order is directed, the convex two-sided ideal generated by x has cube zero. Both cubic bounds are attained even with lattice order, in a rational algebra generated by two positive elements and having no nonzero idempotents. In the unital lattice-ordered case the generated ideal instead has square zero. A second, unital partially ordered example shows that the lattice assumption in this improvement is essential. The cubic theorem gives an affirmative answer to the question recorded by Diem in 1968: in a lattice-ordered ring with nonnegative squares whose nonzero lattice ideals have nonzero products, the product of two nonzero positive elements is nonzero. We also identify the lattice radical as the elements whose absolute values are nilpotent and give explicit nilpotence bounds for lattice ideals generated by finitely many positive nilpotents. In a partially ordered algebra over a totally ordered field, an algebraic middle factor gives the stronger identities xyx = (xy)² = (yx)² = 0 when x ≥ 0 and x² = 0. Further examples separate the order, ideal and derivation properties used in these arguments. GitHub repository: https://github.com/eulogica/positive-squares-cubic-nilpotence-ordered-rings