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Functorial radicals and non-abelian torsion

Published online by Cambridge University Press:  20 January 2009

Shalom Feigelstock  and
Aaron Klein
Aaron Klein
Affiliation:
Bar-Ilan UniversityRamat-Gan, Israel
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Radicals appear in many algebraic contents. For modules over a ring, they give rise to pre-torsion and torsion theories, Goldman (5), Lambek (14). In the category of groups, Kurosh, Plotkin and others have introduced radicals (6), (13), (21), but unlike the radicals in module theory these radicals are not necessarily functorial, as for example the nil radical and the Hirsch-Plotkin radical (6). The functorial method in module theory has been extended to abelian categories, Dickson (2), to the category of nilpotent groups, Hilton (8), Warfield (25), and to the category of groups, Plotkin (22), and to general categories, Wiegandt (26), Holcombe and Walker (10).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 23 , Issue 3 , October 1980 , pp. 317 - 329
Copyright
Copyright © Edinburgh Mathematical Society 1980

References

REFERENCES

(1)Barr, M., Non-abelian torsion theories, Can J. Math. 25 (1973), 1224–1237.CrossRefGoogle Scholar
(2)Dickson, S. E., A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235.CrossRefGoogle Scholar
(3)Feigelstock, S. and Klein, A., Functorial radicals and non-abelian torsion, II (in preparation).Google Scholar
(4)Fröhlich, A., On groups over a d.g. near-ring, I, II, Quart. J. Math. 11 (1960), 193–228.CrossRefGoogle Scholar
(5)Goldman, O., Rings and modules of quotients, J. Algebra 13 (1969), 10–47.CrossRefGoogle Scholar
(6)Hall, P., On Frattini subgroups of finitely-generated groups, J. London Math. Soc. 24 (1949), 353–384.Google Scholar
(7)Higman, G., Neumann, B. H. and Neumann, H., Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254.CrossRefGoogle Scholar
(8)Hilton, P. J., Localization and cohomology of nilpotent groups, Math. Zeit. 132 (1973), 263–286.CrossRefGoogle Scholar
(9)Hirsch, K., On infinite soluble groups II, Proc. London Math. Soc. (2) 44 (1938), 336–344.CrossRefGoogle Scholar
(10)Holcombe, M. and Walker, R., Radicals in categories, Proc. Edinburgh Math. Soc. 21 (1978), 111–128.CrossRefGoogle Scholar
(11)Katz, R. A. and Magnus, W., Residual properties of free groups, Comm. Pure & Appl. Math. 22 (1969), 1–13.CrossRefGoogle Scholar
(12)Klein, A., Injectives and simple objects, J. Pure App. Algebra 15 (1979), 243–245.CrossRefGoogle Scholar
(13)Kurosh, A. G. and Chernikov, S. N., Solvable and nilpotent groups, Uspehi 2–3 (1947), 18–59.Google Scholar
(14)Lambek, J., Torsion theories, additive semantics and rings of quotients (Lecture Notes 177, Springer, 1971).CrossRefGoogle Scholar
(15)Lambek, J. and Rattray, B., Localization and codensity triples, Comm. Alg. 1 (1974), 145–164.CrossRefGoogle Scholar
(16)Lambek, J. and Rattray, B., Localization and sheaf reflectors. Trans. Amer. Math. Soc. 210 (1975), 279–293.CrossRefGoogle Scholar
(17)Lambek, J. and Rattray, B., Localization and duality in additive categories, Houston J. Math. 1 (1975), 87–100.Google Scholar
(18)Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory (Interscience, New York and London, 1966).Google Scholar
(19)Meldrum, J. D. P., The group d.g. near-ring, Proc. London Math. Soc. 32 (1976), 323–346.CrossRefGoogle Scholar
(20)Neumann, H., Varieties of Groups (Springer Verlag, New York, 1967).CrossRefGoogle Scholar
(21)Plotkin, B. I., On the nil-radical of a group, Doklady 98 (1954), 341–343.Google Scholar
(22)Plotkin, B. I., Functorials, radicals, co-radicals, Mat. Zap. Sverdlovsk 7, 3 (1970), 150–182.Google Scholar
(23)Rattray, B., Torsion theories in non-additive categories, Manuscripta Math. 12 (1974), 285–305.CrossRefGoogle Scholar
(24)Scott, W. R., Group Theory (Prentice-Hall, N.J., 1974).Google Scholar
(25)Warfield, R. B. Jr., Nilpotent groups (Lecture Notes 513, Springer, 1976).CrossRefGoogle Scholar
(26)Wiegandt, R., Radicals and semi-simplicity in categories, Acta Math. Ac. Sc. Hung. 19 (3–4) (1968), 345–364.CrossRefGoogle Scholar