Geodesic
The nil radical of an Archimedean partially ordered ring with positive squares
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 2, pp. 231-238.

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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 $a_1a_2=a_2a_1=0$.
Classification : 06F25, 16N40, 16W80
Keywords: partially ordered ring; Archimedean; nil radical; nilpotent 
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     author = {Lavri\v{c}, Boris},
     title = {The nil radical of an {Archimedean} partially ordered ring with positive squares},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {231--238},
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Lavrič, Boris. The nil radical of an Archimedean partially ordered ring with positive squares. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 2, pp. 231-238. http://geodesic.mathdoc.fr/item/CMUC_1994__35_2_a2/