Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T01:33:03.432Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterisation of semisimple classes

Published online by Cambridge University Press:  20 January 2009

A. D. Sands
A. D. Sands
Affiliation:
Department of MathematicsThe UniversityDundee DD1 4HN
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.

Throughout this paper we shall work in the class of associative rings. In (4) it was shown that a class of rings is a semisimple class if and only if it is closed under extensions and ideals and is coinductive. This establishes a duality between radical classes and semisimple classes. This result has been proved also for classes of alternative rings in (2). In the original work by Kuros (1) on this subject two conditions were used for semisimple classes, one of which was weaker than the assumption that the class is closed under ideals. This condition is that every non-zero ideal of a ring in the class should have a non-zero homomorphic image in the class. It is natural to ask whether in the above set of conditions the condition of being closed under ideals can be replaced by this weaker condition. This question is raised in (3) and in (5) but it is suggested there that, in order to compensate, the coinductive condition be replaced by the stronger condition that the class is closed under subdirect sums. In fact we shall show that the weaker condition may be used without needing to replace the coinductive condition. We also give examples to show independence relations among these conditions.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 24 , Issue 1 , February 1981 , pp. 5 - 7
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

REFERENCES

(1)Kuros, A. G., Radicals of rings and algebras, Mat. Sb. 33, (1953), 13–26 (Russian) (English Translation: Rings, Modules and Radicals, Colloq. Math. Soc. Janos Bolyai, 6 (1973), 297–314).Google Scholar
(2)Van Leeuwen, L. C. A., Roos, C. and Wiegandt, R., Characterizations of semi-simple classes, J. Austral. Math. Soc. (Series A) 23 (1977), 172–182.CrossRefGoogle Scholar
(3)van Leeuwen, L. C. A. and Wiegandt, R., Radicals, semisimple classes and torsion theories, Acta. Math. Acad. Sci. Hunger. (to appear).Google Scholar
(4)Sands, A. D., Strong upper radicals, Quart. J. Math. (Oxford) (2) 27 (1976), 21–24.CrossRefGoogle Scholar
(5)Wiegandt, R., List of Problems-Kolloquium über Algebra, Vienna, (1978), 6.Google Scholar