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A Note on Hypernilpotent Radical Properties for Associative Rings

Published online by Cambridge University Press:  20 November 2018

S. E. Dickson
S. E. Dickson*
Affiliation:
University of Nebraska, Lincoln, Nebraska
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We work entirely in the category of associative rings. We show that if P1 is a homomorphically closed class which contains the zero rings, then the lower Kurosh radical P of P1 is the class P2 of all rings R such that every non-zero homomorphic image of R has non-zero ideals in P1, provided that P1 is closed under extensions by zero rings (i.e., if I is a P1-ideal of R and (R/I)2 = 0, then R ∈ P1). The latter assumption replaces the hypothesis that P1 be hereditary for ideals in a similar result of Anderson-Divinsky-Sulinsky in (2). This leads to a brief proof that the lower radical construction of Kurosh terminates at Pω0 (where ω0 is the first infinite ordinal) when P1 is a homomorphically closed class of associative rings containing the zero rings. This was proved for arbitrary homomorphically closed classes P1 of associative rings in (2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 447 - 448
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Amitsur, S. A., A general theory of radicals II, Amer. J. Math., 76 (1954), 100–125.Google Scholar
2. Anderson, T., Divinsky, N., and Sulinsky, A., Lower radical properties for associative and alternative rings (to appear).Google Scholar