Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative

Orin Chein; Edgar G. Goodaire

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 1, page 1-8
  • ISSN: 0010-2628

Abstract

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We investigate finite Moufang loops with a unique nonidentity commutator which are not associative, but all of whose proper subloops are associative. Curiously, perhaps, such loops turn out to be ``ring alternative'', in the sense that their loop rings are alternative rings.

How to cite

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Chein, Orin, and Goodaire, Edgar G.. "Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative." Commentationes Mathematicae Universitatis Carolinae 43.1 (2002): 1-8. <http://eudml.org/doc/248960>.

@article{Chein2002,
abstract = {We investigate finite Moufang loops with a unique nonidentity commutator which are not associative, but all of whose proper subloops are associative. Curiously, perhaps, such loops turn out to be ``ring alternative'', in the sense that their loop rings are alternative rings.},
author = {Chein, Orin, Goodaire, Edgar G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Moufang loops; RA loops; alternative rings; minimal nonassociativity; Moufang loops; RA loops; alternative rings; minimal nonassociativity},
language = {eng},
number = {1},
pages = {1-8},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative},
url = {http://eudml.org/doc/248960},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Chein, Orin
AU - Goodaire, Edgar G.
TI - Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 1
SP - 1
EP - 8
AB - We investigate finite Moufang loops with a unique nonidentity commutator which are not associative, but all of whose proper subloops are associative. Curiously, perhaps, such loops turn out to be ``ring alternative'', in the sense that their loop rings are alternative rings.
LA - eng
KW - Moufang loops; RA loops; alternative rings; minimal nonassociativity; Moufang loops; RA loops; alternative rings; minimal nonassociativity
UR - http://eudml.org/doc/248960
ER -

References

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  9. Goodaire E.G., Jespers E., Polcino Milies C., Alternative loop rings, North-Holland Math. Studies, vol. 184, Elsevier, Amsterdam, 1996. Zbl0878.17029MR1433590
  10. Goodaire E.G., Parmenter M.M., Semi-simplicity of alternative loop rings, Acta Math. Hungar. 50 (1987), 3-4 241-247. (1987) Zbl0634.17014MR0918159
  11. Jespers E., Leal G., Polcino Milies C., Classifying indecomposable RA loops, J. Algebra 176 (1995), 5057-5076. (1995) MR1351625
  12. Miller G.A., Moreno H.C., Nonabelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc. 4 (1903), 398-404. (1903) MR1500650
  13. Pflugfelder H.O., Quasigroups and Loops: Introduction, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767

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