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The hereditariness of the upper radical

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Summary

Starting from a regular classM, one can construct the upper radicalU M of the classM in a category which is like that of associative, alternative or not necessarily associative rings, or that of Lie rings. It turns out that in quite a few cases the upper radical is hereditary. (cf.Suliński [7], Rjabuhin [6], Armendariz [2], Szász—Wiegandt [8]).W. G. Leavitt has suggested the problem: Give a necessary and sufficient condition to be satisfied by the regular classM so that the upper radical classU M ofM is hereditary. In the present paper we shall give such a necessary and sufficient condition. If the classM satisfies an even stronger condition, then theU M-semisimple objects are subdirectly embeddable in a (direct) product ofM-objects. Also a necessary and sufficient condition is given which assures that eachU M-semisimple object can be subdirectly embedded in a (direct) product ofM-objects.

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References

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Authors and Affiliations

  1. Institute of Mathematics, University of Islamabad, Islamabad, Pakistan

    M. A. Rashid

  2. Mathematical Institute, Hungarian Academy of Sciences, Realtanoda u. 13-15, 1053, Budapest

    R. Wiegandt

Authors
  1. M. A. Rashid
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  2. R. Wiegandt
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Additional information

This work was done when the second named author was in the University of Islamabad under UNESCO-UNDP Special Fund Pak. 47.

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Rashid, M.A., Wiegandt, R. The hereditariness of the upper radical. Acta Mathematica Academiae Scientiarum Hungaricae 24, 343–347 (1973). https://doi.org/10.1007/BF01958045

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  • DOI: https://doi.org/10.1007/BF01958045

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