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ON α-LIKE RADICALS

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  16 June 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)
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Abstract

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A radical ρ is called prime-like if for every prime ring A, the polynomial ring A[x] is ρ-semisimple. Let α be a radical satisfying the polynomial equation α(A[x])=(α(A))[x] for every ring A. A radical γ is called α-like if for every α-semisimple ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of α-like radicals. We show that α-likeness is a generalization of prime-likeness and extend some results concerning prime-like radicals. This allows us easily to find distinct special radicals which coincide on simple rings and on polynomial rings, which answers a question put by Ferrero.

Keywords

prime-like radicalα-like radicalspecial radicalAmitsur property of radicalspolynomially extensible radicals

MSC classification

Secondary: 16N80: General radicals and rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 111 - 115
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Andrunakievich, V. A. and Ryabukhin, Yu. M., Radicals of Algebra and Structure Theory (Nauka, Moscow, 1979) (in Russian).Google Scholar
[2]France-Jackson, H., ‘*-rings and their radicals’, Quaest. Math. 8 (1985), 231–239.CrossRefGoogle Scholar
[3]France-Jackson, H., ‘On atoms of the lattice of supernilpotent radicals’, Quaest. Math. 10 (1987), 251–255.CrossRefGoogle Scholar
[4]France-Jackson, H., ‘Rings related to special atoms’, Quaest. Math. 24 (2001), 105–109.CrossRefGoogle Scholar
[5]France-Jackson, H., ‘On supernilpotent radicals with the Amitsur property’, Bull. Aust. Math. Soc. 80 (2009), 423–429.CrossRefGoogle Scholar
[6]Gardner, B. J., ‘Some recent results and open problems concerning special radicals’, Radical Theory, Proceedings of the 1988 Sendai Conference, Sendai, 24–30 July 1988 (ed. S. Kyuno) (Uchida Rokakuho, Tokyo, 1989), pp. 25–56.Google Scholar
[7]Gardner, B. J. and Wiegandt, R., Radical Theory of Rings (Marcel Dekker Inc., New York, 2004).Google Scholar
[8]Khan, M. A. and Aslam, M., ‘Polynomial equation in radicals’, Kyungpook Math. J. 48 (2008), 545–551.CrossRefGoogle Scholar
[9]Korolczuk, H., ‘A note on the lattice of special radicals’, Bull. Pol. Acad. Sci. Math. 29 (1981), 103–104.Google Scholar
[10]Tumurbat, S., ‘On special radicals coinciding on simple rings and on polynomial rings’, J. Algebra Appl. 2(1) (2003), 51–56.CrossRefGoogle Scholar
[11]Tumurbat, S. and France-Jackson, H., ‘On prime-like radicals’, Bull. Aust. Math. Soc. 82 (2010), 113–119.CrossRefGoogle Scholar
[12]Tumurbat, S. and Wiegandt, R., ‘A note on special radicals and partitions of simple rings’, Comm. Algebra 30(4) (2002), 1769–1777.CrossRefGoogle Scholar
[13]Tumurbat, S. and Wiegandt, R., ‘Radicals of polynomial rings’, Soochow J. Math. 29(4) (2003), 425–434.Google Scholar
[14]Tumurbat, S. and Wiegandt, R., ‘On radicals with Amitsur property’, Comm. Algebra 32(3) (2004), 1219–1227.CrossRefGoogle Scholar