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A note on upper radicals in rings

Published online by Cambridge University Press:  09 April 2009

W. G. Leavitt
W. G. Leavitt
Affiliation:
University of Nebraska-Lincoln, Lincoln, Nebraska 68508, U.S.A.
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Abstract

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A class (called c-radicals) is defined such that, given a c-radical P, there is in any class M′ a certain internal criterion that its upper radical UM′ = P, For P a non-c-radical (called a q-radical) there exists no smallest class M such that UM = P, and P is a q-radical if and only if for some M with P = UM there exists 0 ≠ R ∈ M such that when an image ¯ of R has a non-zero image in M there exists an infinite chain of epimorphisms ¯ → R1 → R2 → … with all R1 ∈ M and no Ri, the image of any Rj, with j > i. Several examples of such rings are constructed including a ring all of whose images are primitive. Thus all radicals contained in the Jacobson radical are q-radicals.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 3 , November 1976 , pp. 264 - 273
Copyright
Copyright © Australian Mathematical Society 1976

References

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