| Title:
|
Minimally nonassociative Moufang loops with a unique nonidentity commutator are ring alternative (English) |
| Author:
|
Chein, Orin |
| Author:
|
Goodaire, Edgar G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
1 |
| Year:
|
2002 |
| Pages:
|
1-8 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Moufang loops |
| Keyword:
|
RA loops |
| Keyword:
|
alternative rings |
| Keyword:
|
minimal nonassociativity |
| MSC:
|
17D05 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1068.20069 |
| idMR:
|
MR1903302 |
| . |
| Date available:
|
2009-01-08T19:18:54Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119295 |
| . |
| Reference:
|
[Bru58] Bruck R.H.: A survey of binary systems.Ergeb. Math. Grenzgeb., vol. 20, Springer-Verlag, 1958. Zbl 0141.01401, MR 0093552 |
| Reference:
|
[CG86] Chein O., Goodaire E.G.: Loops whose loop rings are alternative.Comm. Algebra 14 (1986), 2 293-310. Zbl 0582.17015, MR 0817047 |
| Reference:
|
[CG90a] Chein O., Goodaire E.G.: Loops whose loop rings in characteristic $2$ are alternative.Comm. Algebra 18 (1990), 3 659-688. Zbl 0718.20034, MR 1052760 |
| Reference:
|
[CG90b] Chein O., Goodaire E.G.: Moufang loops with a unique nonidentity commutator (associator, square).J. Algebra 130 (1990), 2 369-384. Zbl 0695.20040, MR 1051308 |
| Reference:
|
[Che74] Chein O.: Moufang loops of small order I.Trans. Amer. Math. Soc. 188 (1974), 31-51. Zbl 0286.20088, MR 0330336 |
| Reference:
|
[Che78] Chein O.: Moufang loops of small order.Mem. Amer. Math. Soc. 13 (1978), 197 1-131. Zbl 0378.20053, MR 0466391 |
| Reference:
|
[CR72] Chein O., Robinson D.A.: An ``extra'' law for characterizing Moufang loops.Proc. Amer. Math. Soc. 33 (1972), 29-32. Zbl 0215.40302, MR 0292987 |
| Reference:
|
[Fen68] Fenyves F.: Extra loops I.Publ. Math. Debrecen 15 (1968), 235-238. Zbl 0172.02401, MR 0237695 |
| Reference:
|
[GJM96] Goodaire E.G., Jespers E., Polcino Milies C.: Alternative loop rings.North-Holland Math. Studies, vol. 184, Elsevier, Amsterdam, 1996. Zbl 0878.17029, MR 1433590 |
| Reference:
|
[GP87] Goodaire E.G., Parmenter M.M.: Semi-simplicity of alternative loop rings.Acta Math. Hungar. 50 (1987), 3-4 241-247. Zbl 0634.17014, MR 0918159 |
| Reference:
|
[JLM95] Jespers E., Leal G., Polcino Milies C.: Classifying indecomposable RA loops.J. Algebra 176 (1995), 5057-5076. MR 1351625 |
| Reference:
|
[MM03] Miller G.A., Moreno H.C.: Nonabelian groups in which every subgroup is abelian.Trans. Amer. Math. Soc. 4 (1903), 398-404. MR 1500650 |
| Reference:
|
[Pfl90] Pflugfelder H.O.: Quasigroups and Loops: Introduction.Heldermann Verlag, Berlin, 1990. Zbl 0715.20043, MR 1125767 |
| . |
