Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T16:42:31.950Z Has data issue: false hasContentIssue false
  • English
  • Français

ON BAD SUPERNILPOTENT RADICALS

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  15 December 2011

H. FRANCE-JACKSON
H. FRANCE-JACKSON*
Affiliation:
Department of Mathematics and Applied Mathematics, Nelson Mandela Metropolitan University, Summerstrand Campus (South), PO Box 77000, Port Elizabeth 6031, South Africa (email: cbf@easterncape.co.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A supernilpotent radical α is called bad if the class π(α) of all prime and α-semisimple rings consists of the one-element ring 0 only. We construct infinitely many bad supernilpotent radicals which form a generalization of Ryabukhin’s example of a supernilpotent nonspecial radical. We show that the family of all bad supernilpotent radicals is a sublattice of the lattice of all supernilpotent radicals and give examples of supernilpotent radicals that are not bad.

Keywords

supernilpotent radicalspecial radicalprime radicalBoolean ringsprime essential rings

MSC classification

Secondary: 16N80: General radicals and rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 2 , April 2012 , pp. 271 - 274
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Andrunakievich, V. A., ‘Radicals of associative rings I’, Mat. Sb. 44 (1958), 179–212 (in Russian).Google Scholar
[2]Andrunakievich, V. A. and Ryabukhin, Yu. M., Radicals of Algebras and Structure Theory (Nauka, Moscow, 1979), (in Russian).Google Scholar
[3]Beidar, K. I. and Salavova, K., ‘Some examples of supernilpotent nonspecial radicals’, Acta Math. Hungar. 40 (1982), 109–112.Google Scholar
[4]Beidar, K. I. and Wiegandt, R., ‘Radicals induced by the total of rings’, Beiträge Algebra Geom. 38 (1997), 149–159.Google Scholar
[5]France-Jackson, H., ‘On atoms of the lattice of supernilpotent radicals’, Quaest. Math. 10 (1987), 251–256.CrossRefGoogle Scholar
[6]France-Jackson, H., ‘On prime essential rings’, Bull. Aust. Math. Soc. 47 (1993), 287–290.CrossRefGoogle Scholar
[7]Gardner, B. J. and Stewart, P. N., ‘Prime essential rings’, Proc. Edinb. Math. Soc. 34 (1991), 241–250.CrossRefGoogle Scholar
[8]Gardner, B. J. and Wiegandt, R., Radical Theory of Rings (Marcel Dekker, New York, 2004).Google Scholar
[9]Krachilov, K. K., ‘Coatoms of the lattice of special radicals’, Mat. Issled. Kishinev 49 (1979), 80–86 (in Russian).Google Scholar
[10]McCoy, N. H., The Theory of Rings (Chelsea, New York, 1973).Google Scholar
[11]Ryabukhin, Yu. M., ‘On overnilpotent and special radicals’, Issled. Alg. Mat. Anal., Kishinev (1965), 65–72 (in Russian).Google Scholar
[12]Ryabukhin, Yu. M., ‘Supernilpotent and special radicals’, Mat. Issled. Kishinev 48 (1978), 80–83 (in Russian).Google Scholar
[13]Snider, R. L., ‘Lattices of radicals’, Pac. J. Math. 40 (1972), 207–220.CrossRefGoogle Scholar
[14]Tumurbat, S., ‘Some issues in the theory of supernilpotent radicals’, Bul. Acad. Stiinte Rep. Mold. 1(38) (2002), 3–43.Google Scholar
[15]van Leeuwen, L. C. A. and Jenkins, T. L, ‘A supernilpotent non-special radical class’, Bull. Aust. Math. Soc. 9 (1973), 343–348.CrossRefGoogle Scholar