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The Hereditary Property in the Lower Radical Construction

Published online by Cambridge University Press:  20 November 2018

E. P. Armendariz  and
W. G. Leavitt
E. P. Armendariz
Affiliation:
University of Texas, Austin, Texas 78712
W. G. Leavitt
Affiliation:
University of Texas, Austin, Texas 78712
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All rings considered are associative. We show that if a homomorphically closed class P1 of rings is hereditary in the sense that every ideal of a ring in P1 is also in P1, then the lower Kurosh radical construction terminates at P3. This is an improvement on the result of Anderson, Divinsky, and Sulinski (3) showing that the lower radical construction terminates at P2 provided P1 is homomorphically closed, hereditary, and contains all zero rings. Examples are given to show that the third step is actually attained in some constructions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 474 - 476
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Amitsur, S. A., A general theory of radicals, II. Radicals in rings and bicategories, Amer. J. Math., 76 (1954), 100–125.Google Scholar
2. Anderson, T., Divinsky, N., and Sulinski, A., Hereditary radicals in associative and alternative rings, Can. J. Math., 17 (1965), 594–603.Google Scholar
3. Anderson, T., Divinsky, N., and Sulinski, A., Lower radical properties for associative and alternative rings, J. London Math. Soc, 41 (1966), 417–424.Google Scholar
4. Armendariz, E. P. and Leavitt, W. G., Non-hereditary semisimple classes, Proc. Amer. Math. Soc. 18 (1967), 1114–1117.Google Scholar
5. Hoffman, A. E. and Leavitt, W. G., Properties inherited by the lower radical (to be published).Google Scholar
6. Koethe, G., Die Strukur der Ringe, deren Restklassenring nach dem Radikal vollstandig reduzibel ist, Math. Z., 32 (1930), 161–186.Google Scholar
7. Kurosh, A. G., Radicals of rings and algebras, Mat. Sb. (N.S.), 33 (75) (1953), 13–26 in Russian).Google Scholar