Abstract: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 $a=a_1-a_2$ with positive $a_1$, $a_2$ satisfying $a_1a_2=a_2a_1=0$.
Keywords: partially ordered ring, Archimedean, nil radical, nilpotent
AMS Subject Classification: 06F25, 16N40