Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-20T17:04:35.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of semisimple classes

Published online by Cambridge University Press:  09 April 2009

L. C. A. van Leeuwen ,
C. Roos  and
R. Wiegandt
L. C. A. van Leeuwen
Affiliation:
Department of Mathematics, The University of Groningen, Groningen, The Netherlands.
C. Roos
Affiliation:
Department of Mathematics, Institute of Technology, Delft, The Netherlands.
R. Wiegandt
Affiliation:
Mathematical Institute of the Hung. Acad. of Sci., Reáltanoda u. 13–15, H-1053 Budapest, Hungary.
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.

Two characterizations of semisimple classes of associative and alternative rings (and semigroups with 0) are given:

(i) A class is a semisimple class if and only if it is hereditary, closed under extensions and subdirect sums;

(ii) A class is a semisimple class if and only if it is hereditary, closed under extensions, and has the co-inductive property.

The first characterization sharpens Armendariz's (1968) result proved for associative rings, the second one is categorically dual to a characterization of radical classes due to Amitsur (1954).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 2 , March 1977 , pp. 172 - 182
Copyright
Copyright © Australian Mathematical Society 1977

References

Amitsur, S. A. (1954), ‘A general theory of radicals’, II., Amer. J. Math., 76, 100–125.CrossRefGoogle Scholar
Anderson, T., Divinsky, N. and Suliński, A. (1965), ‘Hereditary radicals in associative and alternative rings’, Canadian J. Math., 17, 594–603.CrossRefGoogle Scholar
Armendariz, E. P. (1968), ‘Closure properties in radical theory’, Pacific J. Math., 26, 1–7.CrossRefGoogle Scholar
Divinsky, N. (1973), ‘Duality between radical and semisimple classes of associative rings’, Scripta Math., 29, 409–416.Google Scholar
Ahmed, Eqbal and Wiegandt, R. (1973), ‘On lower radicals of semigroups’, Math. Nachr., 57, 163–167.CrossRefGoogle Scholar
Grigor, R. S. (1971), ‘To the theory of radicals for semigroups’, Mat. Issled., 6 vyp. 4 (22), 37–55 (in Russian).Google Scholar
Grigor, R. S. (1973), ‘To the theory of radicals for semigroups’, II., Mat. Issled., 8 vyp. 4 (30), 28–46 (in Russian).Google Scholar
Leavitt, W. G. and Armendariz, E. P. (1967), ‘Nonhereditary semisimple classes’, Proc. Amer. Math. Soc., 18, 1114–1117.CrossRefGoogle Scholar
Leavitt, W. G. and Lee, Yu-lee (1969), ‘A radical coinciding with the lower radical in associative and alternative rings’, Pacific J. Math., 30, 459–462.CrossRefGoogle Scholar
Yu-lee, Lee (1969), ‘On the construction of the lower radical properties’, Pacific J. Math., 28, 393–395.CrossRefGoogle Scholar
van Leeuwen, L. C. A. (to appear), ‘Properties of semisimple classes’, J. Nat. Sci. and Math. (Lahore).Google Scholar
Hao, Trân` Vân (Chang Wang Hao) (1962), ‘On semisimple classes of groups’, Sib. Mat. Zh., 3, 943–949 (in Russian).Google Scholar
Watters, J. F. (1969), ‘Lower radicals in associative rings’, Canadian J. Math. 21, 466–476.CrossRefGoogle Scholar
Wiegandt, R. (1972), ‘On the structure of lower radical semigroups’, Czechoslovak Math. J., 22, 1–6.CrossRefGoogle Scholar
Wiegandt, R. (1974), Radical and semisimple classes of rings, (Queen's papers in pure and appl. math., No. 37, Kingston, 1974).Google Scholar