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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
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[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
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