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General Heredity and Strength for Radical Classes

Published online by Cambridge University Press:  20 November 2018

A. D. Sands  and
P. N. Stewart
A. D. Sands
Affiliation:
The University of Dundee, Dundee, Scotland
P. N. Stewart
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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An H-relation, as introduced by Rossa and Tangeman [4], is a relation σ on the class of associative rings with their subrings satisfying the following conditions:

(1) IσR implies that I is a subring of R;

(2) if IσR and f is a homomorphism of R, then (If)σ(Rf);

(3) if IσR and J is an ideal of R, then (I∩J)σJ.

Puczylowski [3] imposes also the condition

(4) if J is an ideal of R, then JσR.

A further condition satisfied by many familiar H-relations is the following:

(5) if f is a homomorphism from a ring R onto a ring S and BσS, then there exists a subring A of R such that AσR and Af = B.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 6 , 01 December 1988 , pp. 1410 - 1421
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Divinsky, N. J., Rings and radicals (University of Toronto Press, Toronto, 1965).Google Scholar
2. Puczylowski, E. R., A note on hereditary radicals, Acta. Sci. Math. 44 (1982), 133–135.Google Scholar
3. Puczylowski, E. R., On semisimple classes of associative rings, Proc. Edinburgh Math. Soc. 27 (1984), 1–5.Google Scholar
4. Rossa, R. F. and Tangeman, R. L., General heredity for radical theory, Proc. Edinburgh Math. Soc. 20 (1977), 333–337.Google Scholar
5. Sands, A. D., Strong upper radicals, Quart. J. Math. (Oxford) (2) 27 (1976), 21–24.Google Scholar
6. Sands, A. D., On relations among radical properties, Glasgow Math. J. 18 (1977), 17–23.Google Scholar
7. Sands, A. D., A characterization of semisimple classes, Proc. Edinburgh Math. Soc. 24 (1981), 5–7.Google Scholar
8. Stewart, P. N., Strict radical classes of associative rings, Proc. Amer. Math. Soc. 39 (1973), 273–278.Google Scholar
9. Tangeman, R. and Kreiling, D., Lower radicals in nonassociative rings, J. Austral. Math. Soc. 14 (1972), 419–23.Google Scholar
10. Wiegandt, R., Radical and semisimple classes of rings, Queen's Papers in Pure and Applied Mathematics 37 (Queen's University, Kingston, 1974).Google Scholar
11. Wiegandt, R., Semisimple classes and H-relations, Studia Sci. Math. Hung. 13 (1978), 181–185.Google Scholar