Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-21T21:07:00.805Z Has data issue: false hasContentIssue false
  • English
  • Français

On Sands' questions concerning strong and hereditary radicals

Published online by Cambridge University Press:  18 May 2009

E. R. Puczyłowski
E. R. Puczyłowski
Affiliation:
Institute of Mathematics, University of Warsaw Pkin, 00–901 Warsaw, Poland
Rights & Permissions [Opens in a new window]

Extract

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.

[7] Sands raised the following questions:

(1) Must a hereditary radical which is right strong be left strong?

(2) Must a right hereditary radical be left hereditary?

(3) (Example 6) Does there exist a right strong radical containing the prime radical β which is not left strong or hereditary?

Negative answers to questions (1) and (2) were given by Beidar [1].

In this paper we present different examples to answer (1) and (2), and we answer (3). We prove that the strongly prime radical defined in [4, 5] is right but not left strong. In the proof we use an example given by Parmenter, Passman and Stewart [6]. The same example and the strongly prime radical are used to answer (2) and (3).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 1 - 3
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Beidar, K. I., Examples of rings and radicals, Radical Theory, Colloq. Math. Soc. Jdnos Bolyai 38 (1982).Google Scholar
2.Divinsky, N. J., Rings and radicals (George Allen and Unwin, 1965).Google Scholar
3.Divinsky, N. J., Krempa, J. and Sulinski, A., Strong radical properties of alternative and associative rings, J. Algebra 17 (1971), 369–388.CrossRefGoogle Scholar
4.Groenewald, N. J. and Heyman, G. A. P., Certain classes of ideals in group rings II, Comm. Algebra 9 (1981), 137–148.CrossRefGoogle Scholar
5.Handelman, D. and Lawrence, J., Strongly prime rings, Trans. Amer. Math. Soc. 211 (1975), 209–223.CrossRefGoogle Scholar
6.Parmenter, M. M., Passman, D. S. and Stewart, P. N., The strongly prime radical of crossed products, Comm. Algebra 12 (1984), 1099–1113.CrossRefGoogle Scholar
7.Sands, A. D., On relations among radical properties, Glasgow Math. J. 18 (1977), 17–23.CrossRefGoogle Scholar