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ON $\alpha $-LIKE RADICALS OF RINGS

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  12 December 2012

H. FRANCE-JACKSON ,
T. KHULAN  and
S. TUMURBAT
H. FRANCE-JACKSON*
Affiliation:
Department of Mathematics and Applied Mathematics, Summerstrand Campus (South), PO Box 77000, Nelson Mandela Metropolitan University, Port Elizabeth 6031, South Africa
T. KHULAN
Affiliation:
Department of Algebra, University of Mongolia, PO Box 75, Ulaan Baatar 20, Mongolia email hulangaaa@yahoo.com
S. TUMURBAT
Affiliation:
Department of Algebra, University of Mongolia, PO Box 75, Ulaan Baatar 20, Mongolia email stumurbat@hotmail.com
*
✉cbf@easterncape.co.uk
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Abstract

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Let $\alpha $ be any radical of associative rings. A radical $\gamma $ is called $\alpha $-like if, for every $\alpha $-semisimple ring $A$, the polynomial ring $A[x] $ is $\gamma $-semisimple. In this paper we describe properties of $\alpha $-like radicals and show how they can be used to solve some open problems in radical theory.

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 88 , Issue 2 , October 2013 , pp. 331 - 339
Copyright
Copyright ©2012 Australian Mathematical Publishing Association Inc. 

References

Andrunakievich, V. A. and Ryabukhin, Yu. M., Radicals of Algebra and Structure Theory (Nauka, Moscow, 1979) (in Russian).Google Scholar
Beidar, K. I., ‘Atoms in the “lattice” of radicals’, Mat. Issled. 85 (1985), 21–31 (in Russian).Google Scholar
France-Jackson, H., ‘*-rings and their radicals’, Quaest. Math. 8 (1985), 231–239.CrossRefGoogle Scholar
France-Jackson, H., ‘On atoms of the lattice of supernilpotent radicals’, Quaest. Math. 10 (1987), 251–255.CrossRefGoogle Scholar
France-Jackson, H., ‘Rings related to special atoms’, Quaest. Math. 24 (2001), 105–109.CrossRefGoogle Scholar
France-Jackson, H., ‘On supernilpotent radicals with the Amitsur property’, Bull. Aust. Math. Soc. 80 (2009), 423–429.CrossRefGoogle Scholar
France-Jackson, H., ‘On $\alpha $-like radicals’, Bull. Aust. Math. Soc. 84 (2011), 111–115.CrossRefGoogle Scholar
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. Shoji Kyuno) (Uchida Rokakuho, Tokyo, 1989), pp. 25–56.Google Scholar
Gardner, B. J. and Zhian, Liang, ‘Small and large radicals’, Comm. Algebra 20 (1992), 2533–2551.Google Scholar
Gardner, B. J. and Wiegandt, R., Radical Theory of Rings (Marcel Dekker, New York, 2004).Google Scholar
Khan, M. A. and Aslam, M., ‘Polynomial equation in radicals’, Kyungpook Math. J. 48 (2008), 545–551.CrossRefGoogle Scholar
Korolczuk, H., ‘A note on the lattice of special radicals’, Bull. Pol. Acad. Sci. Math. 29 (1981), 103–104.Google Scholar
Krempa, J., ‘Logical connections between some open problems concerning nil rings’, Fund. Math. 76 (1972), 121–130.CrossRefGoogle Scholar
Loi, N. V. and Wiegandt, R., ‘On the Amitsur property of radicals’, Algebra Discrete Math. 3 (2006), 92–100.Google Scholar
Puczylowski, E. R. and Smoktunowicz, Agata, ‘On maximal ideals and the Brown-McCoy radical of polynomial rings’, Comm. Algebra 26 (1968), 2473–2482.CrossRefGoogle Scholar
Sands, A. D., ‘On relations among radical properties’, Glasgow Math. J. 18 (1977), 17–23.CrossRefGoogle Scholar
Snider, R. L., ‘Lattices of radicals’, Pacific J. Math. 42 (1972), 207–220.CrossRefGoogle Scholar
Tumurbat, S. and France-Jackson, H., ‘On prime-like radicals’, Bull. Aust. Math. Soc. 82 (2010), 113–119.CrossRefGoogle Scholar
Tumurbat, S. and Wiegandt, R., ‘A note on special radicals and partitions of simple rings’, Comm. Algebra 30 (4) (2002), 1769–1777.CrossRefGoogle Scholar
Tumurbat, S. and Wiegandt, R., ‘Radicals of polynomial rings’, Soochow J. Math. 29 (4) (2003), 425–434.Google Scholar
Tumurbat, S. and Wiegandt, R., ‘On the matrix-extensibility of radicals’, J. Appl. Algebra Discrete Struct. 2 (2) (2004), 119–130.Google Scholar