Hostname: page-component-7dd5485656-j7khd Total loading time: 0 Render date: 2025-10-31T09:48:33.442Z Has data issue: false hasContentIssue false
  • English
  • Français

ON BASE RADICAL OPERATORS FOR CLASSES OF FINITE ASSOCIATIVE RINGS

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  18 July 2018

R. G. MCDOUGALL Open the ORCID record for R. G. MCDOUGALL [Opens in a new window]  and
L. K. THORNTON Open the ORCID record for L. K. THORNTON [Opens in a new window]
R. G. MCDOUGALL
Affiliation:
Faculty of Science, Health, Education and Engineering, University of the Sunshine Coast, Maroochydore, Queensland 4588, Australia email rmcdouga@usc.edu.au
L. K. THORNTON*
Affiliation:
Faculty of Science, Health, Education and Engineering, University of the Sunshine Coast, Maroochydore, Queensland 4588, Australia email lauren.thornton@research.usc.edu.au
*
✉lauren.thornton@research.usc.edu.au
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.

In this paper, class operators are used to give a complete listing of distinct base radical and semisimple classes for universal classes of finite associative rings. General relations between operators reveal that the maximum order of the semigroup formed is 46. In this setting, the homomorphically closed semisimple classes are precisely the hereditary radical classes and hence radical–semisimple classes, and the largest homomorphically closed subclass of a semisimple class is a radical–semisimple class.

Keywords

base radicalbase semisimplefinite associative ring

MSC classification

Primary: 16N80: General radicals and rings

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 239 - 250
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Amitsur, S. A., ‘A general theory of radicals II, radicals in rings and bicategories’, Amer. J. Math. 76 (1954), 100–125.Google Scholar
Arhangel’skiĭ, A. V. and Wiegandt, R., ‘Connectednesses and disconnectednesses in topology’, Topology Appl. 5 (1975), 9–33.Google Scholar
Buys, A., Groenewald, N. J. and Veldsman, S., ‘Radical and semisimple classes in categories’, Quaest. Math. 4 (1981), 205–220.Google Scholar
Gardner, B. J., ‘There isn’t much duality in radical theory’, Algebra Discrete Math. 3 (2007), 59–66.Google Scholar
Gardner, B. J. and Wiegandt, R., Radical Theory of Rings (Marcel Dekker, New York–Basel, 2004).Google Scholar
Krempa, J. and Malinowska, I. A., ‘On Kurosh–Amitsur radicals of finite groups’, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 19(1) (2011), 175–190.Google Scholar
Kurosh, A. G., ‘Radicals of rings and algebras’, Mat. Sb. 33 (1953), 13–26.Google Scholar
Márki, L., Mlitz, R. and Wiegandt, R., ‘A general Kurosh–Amitsur radical theory’, Comm. Algebra 16 (1988), 249–305.Google Scholar
McConnell, N. R., McDougall, R. G. and Stokes, T., ‘On base radical and semisimple classes defined by class operators’, Acta Math. Hungar. 138 (2013), 307–328.Google Scholar
McDougall, R. G., ‘A generalisation of the lower radical class’, Bull. Aust. Math. Soc. 59 (1999), 139–146.Google Scholar
Puczyłowski, E. R., ‘A note on hereditary radicals’, Acta Sci. Math. 44 (1982), 133–135.Google Scholar
Sands, A. D., ‘Strong upper radicals’, Q. J. Math. (Oxford) 27 (1976), 21–24.Google Scholar
Stewart, P. N., ‘Semi-simple radical classes’, Pacific J. Math. 32 (1970), 249–254.Google Scholar
Thornton, L. K., ‘On base radical and semisimple operators for a class of finite algebras’, Beiträge Algebra Geom. 59(2) (2018), 361–374.Google Scholar
Wedderburn, J. H. M., ‘On hypercomplex numbers’, Proc. Lond. Math. Soc. (2) 6 (1908), 77–118.Google Scholar
Wiegandt, R., Radical and Semisimple Classes, Queen’s Papers in Pure and Applied Mathematics, 37 (Queen’s University, Kingston, ON, 1974).Google Scholar
Wiegandt, R., ‘A condition in general radical theory and its meaning for rings, topological spaces and graphs’, Acta Math. Acad. Sci. Hungar. 26 (1975), 233–240.Google Scholar
Wiegandt, R., ‘Radicals of rings defined by means of ring elements’, Sitzungsber. Österreich. Akad. Wiss. Math.-Natur. Kl. 184 (1975), 117–125.Google Scholar