Geodesic
On Sands' questions concerning strong and hereditary radicals
Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 1-3

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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). 
Puczyłowski, E. R. On Sands' questions concerning strong and hereditary radicals. Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 1-3. doi: 10.1017/S0017089500006248
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[1] 1.Beidar, K. I., Examples of rings and radicals, Radical Theory, Colloq. Math. Soc. Jdnos Bolyai 38 (1982). Google Scholar

[2] 2.Divinsky, N. J., Rings and radicals (George Allen and Unwin, 1965). Google Scholar

[3] 3.Divinsky, N. J., Krempa, J. and Sulinski, A., Strong radical properties of alternative and associative rings, J. Algebra 17 (1971), 369–388. Google Scholar | DOI

[4] 4.Groenewald, N. J. and Heyman, G. A. P., Certain classes of ideals in group rings II, Comm. Algebra 9 (1981), 137–148. Google Scholar | DOI

[5] 5.Handelman, D. and Lawrence, J., Strongly prime rings, Trans. Amer. Math. Soc. 211 (1975), 209–223. Google Scholar | DOI

[6] 6.Parmenter, M. M., Passman, D. S. and Stewart, P. N., The strongly prime radical of crossed products, Comm. Algebra 12 (1984), 1099–1113. Google Scholar | DOI

[7] 7.Sands, A. D., On relations among radical properties, Glasgow Math. J. 18 (1977), 17–23. Google Scholar | DOI

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