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Special (p;q) Radicals

Published online by Cambridge University Press:  20 November 2018

J. D. McKnight Jr.  and
Gary L. Musser
J. D. McKnight Jr.
Affiliation:
University of Miami, Coral Gables, Florida
Gary L. Musser
Affiliation:
Northern Illinois University, Dekalb, Illinois
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In [3], the study of (p;q) radicals was initiated. In this paper, the integral polynomials p(x) and q(x) which determine the Jacobson radical are characterized and the Jacobson radical is shown to be the only semiprime (p;q) radical for which all fields are semisimple. Also, it is observed that the prime, nil, and Brown-McCoy radicals are not (p;q) radicals. To show that the semiprime (p;q) radicals are special and that they can be determined by subclasses of the class of primitive rings, a classification theorem for (p;q)-regular primitive rings is given. Finally, it is shown that the collection of semiprime (p;q) radicals and the collection of semiprime (p;1) radicals coincide.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 38 - 44
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Andrunakievič, V., Radicals of associative rings. I, Mat. Sb. 44 (1958), 179–212.Google Scholar
2. Divinsky, N. J., Rings and radicals (Univ. of Toronto Press, Toronto, 1965).Google Scholar
3. Musser, G. L., Linear semiprime (p;q) radicals, Pacific J. Math. 37 (1971), 749–757.Google Scholar
4. Snider, R. L., Lattices of radicals, Pacific J. Math. 39 (1971).Google Scholar