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Radicals and one-sided ideals

Published online by Cambridge University Press:  14 November 2011

A.D. Sands
A.D. Sands
Affiliation:
Department of Mathematical Sciences, The University, Dundee, DD1 4HN, U.K.
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Extract

The correspondence between radicals of associative rings and A-radicals is studied. It is shown that corresponding to each A-radical there is an interval of radicals and that each radical belongs to exactly one such interval. The question of the nature of the radical of a one-sided ideal is considered. It is shown that the radicals such that the radical of a one-sided ideal is always a one-sided ideal are those which contain their associated A-radicals. Radicals such that the radical of a one-sided ideal always equals the intersection of a left ideal and a right ideal are described, as are those A-radicals such that every associated radical has this property.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 3-4 , 1986 , pp. 241 - 251
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Anderson, T., Divinsky, N. and Sulinski, A.. Hereditary radicals in associative and alternative rings. Canad. J. Math. 17 (1965), 594–603.CrossRefGoogle Scholar
2Andrunakievic, V. A.. Radicals of associative rings, II. Mat. Sb. 55 (1961), 329–346. (Amer. Math. Soc. Translations 52 (1966), 129–150.)Google Scholar
3Andrunakievic, V. A. and Rjabuhin, Y. M.. Rings without nilpoteni elements and completely simple ideals. Dokl. Akad. Nauk. SSSR 180 (1968), 9–11. (Soviet Math. Dokl. 9 (1968), 565–568.)Google Scholar
4Beidar, K. I.. Examples of rings and radicals. (Radical Theory, Eger, Hungary, 1982.) Colloq. Math. Soc. János Bolyai 38 (1985), 19–46.Google Scholar
5Divinsky, N. J.. Rings and Radicals (Toronto: Toronto Univ. Press, 1965).Google Scholar
6Gardner, B. J.. Radicals of abelian groups and associative rings. Ada Math. Acad. Sci. Hungar. 29 (1973), 259–268.CrossRefGoogle Scholar
7Gardner, B. J.. Sub-prime radical classes determined by zerorings. Bull. Austral. Math. Soc. 12 (1975), 95–97.CrossRefGoogle Scholar
8Gardner, B. J. and Stewart, P. N.. Reflected radical classes. Ada Math. Acad. Sci. Hungar. 28 (1976), 293–298.CrossRefGoogle Scholar
9Jaegermann, M.. Morita contexts and radicals. Bull. Acad. Polon. Sci. Sér. Sci. Math. 20 (1972), 619–625.Google Scholar
10Jaegermann, M. and Sands, A. D.. On normal radicals, N-radicals and A-radicals. J. Algebra 50 (1978), 337–349.CrossRefGoogle Scholar
11Puczylowski, E. R.. Remarks on stable radicals. Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 11–16.Google Scholar
12Puczylowski, E. R.. A note on hereditary radicals. Ada Sci. Math. (Szeged) 44 (1982), 133–135.Google Scholar
13Puczylowski, E. R.. On Sands' questions concerning strong and hereditary radicals. Glasgow Math. J. 28 (1986), 1–3.CrossRefGoogle Scholar
14Sands, A. D.. On normal radicals. J. London Math. Soc. 11 (1975), 361–365.CrossRefGoogle Scholar
15Sands, A. D.. Strong upper radicals. Quart. J. Math. Oxford Ser.2 27 (1976), 21–24.CrossRefGoogle Scholar
16Sands, A. D.. On relations among radical properties. Glasgow Math. J. 18 (1977), 17–23.CrossRefGoogle Scholar
17Stewart, P. N.. Strict radical classes of associative rings. Proc. Amer. Math. Soc. 39 (1973), 273–278.CrossRefGoogle Scholar
18Szasz, F. and Wiegandt, R.. On hereditary radicals. Period. Math. Hungar. 3 (1973), 235–241.CrossRefGoogle Scholar
19Tangeman, R. and Kreiling, D.. Lower radicals in nonassociative rings. J. Austral. Math. Soc. 14 (1972), 419–423.CrossRefGoogle Scholar
20Thierrin, G.. Sur les ideaux complement premiers d'un anneau quelconque. Bull. Acad. Roy. Belg. 43 (1957), 124–132.Google Scholar
21Wiegandt, R.. Radical and semisimple classes of rings (Kingston, Ontario: Queen's University, 1974).Google Scholar