Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-18T15:20:37.452Z Has data issue: false hasContentIssue false
  • English
  • Français

ON PRIME-LIKE RADICALS

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  02 March 2010

S. TUMURBAT  and
H. FRANCE-JACKSON
S. TUMURBAT
Affiliation:
Department of Algebra, University of Mongolia, PO Box 75, Ulaan Baatar 20, Mongolia (email: [email protected])
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: [email protected])
*
✉For correspondence; e-mail: [email protected]
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 radical γ is prime-like if, for every prime ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of prime-like radicals. In particular, we give necessary and sufficient conditions for a radical γ containing the prime radical β to be prime-like. This allows us to easily find distinct special radicals that coincide on simple rings and on polynomial rings, which answers a question put by Ferrero. It also allows us to reformulate a long-standing open problem of Gardner in terms of prime-like radicals.

Keywords

prime-like radicalspecial radicalAmitsur property of radicalspolynomially extensible radicalssubidempotent radicals

MSC classification

Secondary: 16N80: General radicals and rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 113 - 119
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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’, Quaestiones Math. 8 (1985), 231–239.Google Scholar
[3]France-Jackson, H., ‘On atoms of the lattice of supernilpotent radicals’, Quaestiones Math. 10 (1987), 251–255.Google Scholar
[4]France-Jackson, H., ‘Rings related to special atoms’, Quaestiones Math. 24 (2001), 105–109.Google Scholar
[5]France-Jackson, H., ‘On supernilpotent radicals with the Amitsur property’, Bull. Aust. Math. Soc. 80 (2009), 423–429.Google Scholar
[6]Gardner, B. J., ‘Sub-prime radicals determined by zerorings’, Bull. Aust. Math. Soc. 12 (1975), 95–97.Google Scholar
[7]Gardner, B. J., ‘Some recent results and open problems concerning special radicals’, in: Radical Theory (Proceedings of the 1988 Sendai Conference, Sendai, 24–30 July 1988), (ed. Kyuno, S.) (Uchida Rokakuho Pub. Co. Ltd, Tokyo, Japan, 1989), pp. 25–56.Google Scholar
[8]Gardner, B. J. and Wiegandt, R., Radical Theory of Rings (Marcel Dekker Inc., New York, 2004).Google Scholar
[9]Korolczuk, H., ‘A note on the lattice of special radicals’, Bull. Pol. Acad. Sci. Math. 29 (1981), 103–104.Google Scholar
[10]Loi, N. V. and Wiegandt, R., ‘On the Amitsur property of radicals’, Algebra Discrete Math. 3 (2006), 92–100.Google Scholar
[11]Tumurbat, S., ‘On special radicals coinciding on simple rings and on polynomial rings’, J. Algebra Appl. 2(1) (2003), 51–56.Google Scholar
[12]Tumurbat, S. and Wiegandt, R., ‘Principally left hereditary and principally left strong radicals’, Algebra Colloq. 8(4) (2001), 409–418.Google Scholar
[13]Tumurbat, S. and Wiegandt, R., ‘A note on special radicals and partitions of simple rings’, Comm. Algebra 30(4) (2002), 1769–1777.Google Scholar
[14]Tumurbat, S. and Wiegandt, R., ‘Radicals of polynomial rings’, Soochow J. Math. 29(4) (2003), 425–434.Google Scholar
[15]Tumurbat, S. and Wiegandt, R., ‘On radicals with Amitsur property’, Comm. Algebra 32(3) (2004), 1219–1227.Google Scholar