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The singular ideal and radicals

Published online by Cambridge University Press:  09 April 2009

Miguel Ferrero
Affiliation:
Instituto de Matemática Universidade Federal do Rio Grande do Sul91509-900 Porto AlegreBrazil email: ferrero@mat.ufrgs.br
Edmund R. Puczyłowski
Affiliation:
Institute of Mathematics University of Warsaw02-097 Warsaw, Banacha 2Poland email: edmundp@mimuw.edu.p1
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Abstract

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Some properties of the singular ideal are established. In particular its behaviour when passing to one-sided ideals is studied. Obtained results are applied to study some radicals related to the singular ideal. In particular a radical S such that for every ring R, S(R) and R/S(R) are close to being a singular ring and a non-singular ring, respectively, is constructed.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Chatters, A. W. and Hajamavis, C. R., Rings with chain conditions (Pitman, Boston, 1980).Google Scholar
[2]Divinsky, N. J., Rings and radicals (Allen & Unwin, 1965).Google Scholar
[3]Ferrero, M., ‘Centred bimodules over prime rings: Closed submodules and applications to ring extensions’, J. Algebra 172 (1995), 470485.CrossRefGoogle Scholar
[4]Ferrero, M. and Tömer, G., ‘Rings with chain annihilator conditions and right distributive rings’, Schriftenreihe des FB Mathematik, 190, Universität Duisburg, 1991.Google Scholar
[5]Fisher, J. W., ‘On the nilpotency of nil subrings’, Canad. J. Math. 23 (1970), 12111216.CrossRefGoogle Scholar
[6]Golod, E. S., ‘On nil algebras and finitely approximable groups’, Izv. Akad. Nauk. SSSR Ser. Mat. 28 (1985), 273276 (in Russian).Google Scholar
[7]Goodearl, K. R., Ring theory, nonsingular rings and modules (Marcel Dekker, New York, 1976).Google Scholar
[8]Handelman, D. E. and Lawrence, J., ‘Strongly prime rings’, Trans. Amer. Math. Soc. 211 (1975), 209233.CrossRefGoogle Scholar
[9]Johns, B., ‘Chain conditions and nil ideals’, J. Algebra 73 (1981), 287294.CrossRefGoogle Scholar
[10]Parmenter, M. M., Stewart, P. N. and Wiegandt, R., ‘On the Groenewald-Heyman strongly prime radical’, Quaestiones Math. 7 (1984), 225240.CrossRefGoogle Scholar
[11]Puczylowski, E. R., ‘Hereditariness of strong and stable radicals’, Glasgow Math. J. 23 (1982), 8590.CrossRefGoogle Scholar
[12]Puczylowski, E. R., ‘On Sands' questions concerning strong and hereditary radicals’, Glasgow Math. J. 28 (1986), 13.CrossRefGoogle Scholar
[13]Puczylowski, E. R., ‘On the left hereditariness of pseudocomplement of hereditary radicals’, Studia Sci. Math. Hungar. 27 (1992), 201205.Google Scholar
[14]Sands, A. D., ‘Radicals and Morita contexts’, J. Algebra 24 (1973), 335345.CrossRefGoogle Scholar
[15]Snider, R. L., ‘Lattices of radicals’, Pacific. J. Math. 40 (1972), 207220.CrossRefGoogle Scholar
[16]Stenström, W., Rings of quotients (Springer, Berlin, 1975).CrossRefGoogle Scholar
[17]Wiegandt, R., Radical and semisimple classes of rings, Queen's Papers in Pure and Appl. Math. No. 37 (Kingston, Ontario, 1974).Google Scholar
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