Geodesic
A Note on Hypernilpotent Radical Properties for Associative Rings
Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 447-448

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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). 
Dickson, S. E. A Note on Hypernilpotent Radical Properties for Associative Rings. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 447-448. doi: 10.4153/CJM-1967-037-3
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[1] 1. Amitsur, S. A., A general theory of radicals II, Amer. J. Math., 76 (1954), 100–125. Google Scholar

[2] 2. Anderson, T., Divinsky, N., and Sulinsky, A., Lower radical properties for associative and alternative rings (to appear). Google Scholar

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