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Kurosh Radicals of Rings with Operators

Published online by Cambridge University Press:  20 November 2018

N. Divinsky  and
A. Suliński
N. Divinsky
Affiliation:
University of British Columbia and Institute of Mathematics, Polish Academy of Sciences
A. Suliński
Affiliation:
University of British Columbia and Institute of Mathematics, Polish Academy of Sciences
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If A is an algebra over a commutative ring with unity, Φ, then the Jacobson radical of the algebra A is equal to the Jacobson radical of A, thought of as a ring (1, p. 18, Theorem 1). The present note extends this result to all radical properties (in the sense of Kurosh 2) and allows ϕ to be any set of operators on A.

If A is a ring and Φ is an arbitrary set, we say that Φ is a set of operators for A if for any α in Φ and any x in A, the composition ax is defined and is an element of A, and if this composition satisfies the following two conditions:

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 278 - 280
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Jacobson, N., Structure of rings, Amer. Math. Soc. Coll. Publ., vol. 37 (1956).Google Scholar
2. Kurosh, A., Radicals of rings and algebras, Mat. Sbornik, 33 (75) (1953), 13–26.Google Scholar