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Radicals in Categories

Published online by Cambridge University Press:  20 January 2009

Michael Holcombe  and
Roland Walker
Michael Holcombe
Affiliation:
The Queen's University of Belfast, Belfast BT7 Inn
Roland Walker
Affiliation:
Temple Moor High SchoolField End Grove Selby Road Leeds Ls15 0PT
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The study of radicals in general categories has followed several lines of development. The problem of defining radical properties in general categories has been considered by Kurosh and Shul'geifer, see (7). Under mild conditions on their categories they obtain sufficient conditions for the existence of radical functors which are closely related to radical properties. Another approach is by Maranda (5) and Dickson (3) who studied idempotent radical functors and torsion theories in abelian categories. Our aim has been to study radical functors in as general a category as possible. To this end we introduce the concept of an R-category. The categories of rings, modules, near-rings, groups and Jordan algebras are all examples of R-categories.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 21 , Issue 2 , September 1978 , pp. 111 - 128
Copyright
Copyright © Edinburgh Mathematical Society 1978

References

REFERENCES

(1) Betsch, G., Struktursatze für Fastringe, Doctoral Dissertation Eberhard-Karls. Universität zu Tübingen (1963).Google Scholar
(2) Carreau, F., Sous-categories réflexives et la théorie générate des radicaux, Fund. Math. 71 (1971), 223–242.CrossRefGoogle Scholar
(3) Dickson, S. E., A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235.CrossRefGoogle Scholar
(4) Divinsky, N. J., Rings and radicals, (George Allen and Unwin Ltd, 1965).Google Scholar
(5) Maranda, J. M., Injective structures, Trans. Amer. Math. Soc. 110 (1964), 98–135.CrossRefGoogle Scholar
(6) Mitchell, B., Theory of categories (Academic Press, 1965).Google Scholar
(7) Shul'geifer, E. G., Functor characterisation of strict radicals in categories, Sib. Matem. Zh. 7, No. 6, (1966), 1412–1421.Google Scholar
(8) Wiegandt, R., Radical and semi-simplicity in categories, Ada. Math. Acad. Sc. Hung. 19 (1968), 345–364.Google Scholar