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

Boris Lavrič

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 2, page 231-238
  • ISSN: 0010-2628

Abstract

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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 1 a 2 = a 2 a 1 = 0 .

How to cite

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Lavrič, Boris. "The nil radical of an Archimedean partially ordered ring with positive squares." Commentationes Mathematicae Universitatis Carolinae 35.2 (1994): 231-238. <http://eudml.org/doc/247636>.

@article{Lavrič1994,
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 $a_1a_2=a_2a_1=0$.},
author = {Lavrič, Boris},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {partially ordered ring; Archimedean; nil radical; nilpotent; Archimedean partially ordered ring; nilpotent elements; nil radical},
language = {eng},
number = {2},
pages = {231-238},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The nil radical of an Archimedean partially ordered ring with positive squares},
url = {http://eudml.org/doc/247636},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Lavrič, Boris
TI - The nil radical of an Archimedean partially ordered ring with positive squares
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 2
SP - 231
EP - 238
AB - 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 $a_1a_2=a_2a_1=0$.
LA - eng
KW - partially ordered ring; Archimedean; nil radical; nilpotent; Archimedean partially ordered ring; nilpotent elements; nil radical
UR - http://eudml.org/doc/247636
ER -

References

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  1. Bernau S.J., Huijsmans C.B., Almost f -algebras and d -algebras, Proc. Cambridge Philos. Soc. 107 (1990), 287-308. (1990) Zbl0707.06009MR1027782
  2. Birkhoff G., Pierce R.S., Lattice-ordered rings, An. Acad. Brasil Ciênc. 28 (1956), 41-69. (1956) Zbl0070.26602MR0080099
  3. Diem J.E., A radical for lattice-ordered rings, Pacific J. Math. 25 (1968), 71-82. (1968) Zbl0157.08004MR0227068
  4. Divinsky N., Rings and Radicals, Allen, London, 1965. Zbl0138.26303MR0197489
  5. Fuchs L., Partially Ordered Algebraic Systems, Pergamon Press, Oxford-London-New YorkParis, 1963. Zbl0137.02001MR0171864
  6. Hayes A., A characterization of f -rings without non-zero nilpotents, J. London Math. Soc. 39 (1964), 706-707. (1964) Zbl0126.06502MR0167501
  7. Jacobson N., Structure of Rings, Colloquium Publication 37, Amer. Math. Soc., Providence, 1956. Zbl0098.25901MR0081264
  8. Steinberg S.A., On lattice-ordered rings in which the square of every element is positive, J. Austral. Math. Soc. Ser. A 22 (1976), 362-370. (1976) Zbl0352.06017MR0427198
  9. Szász F.A., Radicals of Rings, Akademiai Kiado - John Wiley & Sons, Budapest-ChichesterNew York-Brisbane-Toronto, 1981. MR0636787

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