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Radicals related to the Brown-McCoy radical in some varieties of algebras

Part of: General nonassociative rings

Published online by Cambridge University Press:  09 April 2009

B. J. Gardner
B. J. Gardner
Affiliation:
University of TasmaniaHobart, Australia
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Abstract

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The Brown-McCoy radical is the upper radical defined by the class of simple rings with identities. For associative or alternative rings the Brown-McCoy radical is hereditary, and its semi-simple class consists of all subdirect products of simple rings with identities. In this paper we present some classes of simple non-associative algebras whose upper radicals behave similarly. Classifications are then obtained of ‘most’ semi-simple radical classes of (γ, δ) and right alternative rings.

MSC classification

Secondary: 17A30: Algebras satisfying other identities
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 3 , November 1979 , pp. 283 - 294
Copyright
Copyright © Australian Mathematical Society 1979

References

Andrunakievič, V. A. and Rjabuhin, Ju. M. (1968), ‘On the existence of the Brown-McCoy radical in Lie algebras’, Soviet Math. Dokl. 9, 373–376.Google Scholar
[Russian original: DAN SSSR 179, 503–506.]Google Scholar
Brown, B. and McCoy, N. H. (1947), ‘Radicals and subdirect sums’, Amer. J. Math. 69, 46–58.CrossRefGoogle Scholar
Fiedorowicz, Z. (1974), ‘The structure of autodistributive algebras’, J. Algebra 31, 427–436.CrossRefGoogle Scholar
Freidman, P. A. (1965), ‘O kol'tsakh, assotsiativnykh po modulyu svoevo radikala’, Ural'sk. Gos. Univ. Mat. Zap. 5, 67–71.Google Scholar
Gardner, B. J. (1975), ‘Semi-simple radical classes of algebras and attainability of identities’, Pacific J. Math. 61, 401–416.CrossRefGoogle Scholar
Gardner, B. J. (1979a), ‘Some degeneracy and pathology in non-associative radical theory’, Annales Univ. Set. Budapest. Sect. Math. (to appear).Google Scholar
Gardner, B. J. (1979b), ‘Radical properties defined locally by polynomial identities II’, J. Austral. Math. Soc. (Ser. A) 27, 274–283.CrossRefGoogle Scholar
Gardner, B. J. (1979c), ‘Multiple radical theories’, Colloq. Math. (to appear).CrossRefGoogle Scholar
Gardner, B. J. and Stewart, P. N. (1975), ‘On semi-simple radical classes’, Bull. Austral. Math. Soc. 13, 349–353.CrossRefGoogle Scholar
Hentzel, I. R. and Cattaneo, G. M. P. (1977), ‘Simple (γ, δ) algebras are associative’, J. Algebra 47, 52–76.CrossRefGoogle Scholar
Kepka, T. (1977), ‘On a class of non-associative rings’, Comment. Math. Univ. Carolinae 18, 531–540.Google Scholar
Kleinfeld, E. (1953), ‘Right alternative rings’, Proc. Amer. Math. Soc. 4, 939–944.CrossRefGoogle Scholar
Leavitt, W. G. (preprint), ‘A minimally embeddable ring’.Google Scholar
Mal'tsev, A. I. (1967), ‘Ob umnozhenii klassov algebraicheskikh sistem’, Sibirsk. Mat. Zhurnal 8, 346–365.Google Scholar
Mikheev, I. M. (1969), ‘One identity in right-alternative rings’, Algebra and Logic 8, 204–211.CrossRefGoogle Scholar
[Russian original: Algebra i Logika 8, 357–366.]Google Scholar
Schafer, R. D. (1966), An introduction to nonassociative algebras (Academic Press, New York and London).Google Scholar
Smiley, M. F. (1950), ‘Application of a radical of Brown and McCoy to non-associative rings’, Amer. J. Math. 72, 93–100.CrossRefGoogle Scholar
Stewart, P. N. (1970), ‘Semi-simple radical classes’, Pacific J. Math. 32, 249–254.CrossRefGoogle Scholar
Suliński, A. (1958), ‘Nekotorye voprosy obshchei teorii radikalov’, Mat. Sb. 44, 273–286.Google Scholar
Suliński, A. (1966), ‘The Brown-McCoy radical in categories’, Fund. Math. 59, 23–41.CrossRefGoogle Scholar
Tamura, T. (1966), ‘Attainability of systems of identities on semigroups’, J. Algebra 3, 261–276.CrossRefGoogle Scholar
Wiegandt, R. (1974), Radical and semisimple classes of rings (Queen's University, Kingston, Ontario).Google Scholar
Zaks, A. (1968), ‘Simple modules and hereditary rings’, Pacific J. Math. 26, 627–630.CrossRefGoogle Scholar