Geodesic
A Note on Lower Radical Constructions for Associative Rings
Canadian mathematical bulletin, Tome 11 (1968) no. 1, pp. 23-30

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In [2], a construction for the lower radical class R∘ (η) with respect to a class η of rings was given as the union of an inductively defined ascending transfinite chain of classes of rings. It was shown there that this construction terminates, for associative rings, at ω∘, the first infinite ordinal, in the sense that if {ηα: α an ordinal} is the chain, then R∘ (η) =ηω∘. Also, examples of classes η for which R∘ (η) = η1, η2, η3 were given. 
Heinicke, A.G. A Note on Lower Radical Constructions for Associative Rings. Canadian mathematical bulletin, Tome 11 (1968) no. 1, pp. 23-30. doi: 10.4153/CMB-1968-003-x
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[1] 1. Divinsky, N., Rings and Radicals. Toronto: University of Toronto Press, 1965. Google Scholar

[2] 2. Sulinski, A., Anderson, R. and Divinsky, N., Lower radical properties for associative and alternative rings. J. Lond. Math. Soc., 41 (1966), 417-424.10.1112/jlms/s1-41.1.417 Google Scholar

[3] 3. Zariski, O. and Samuel, P., Commutative algebra I. Princeton: D. Van Nostrand Company, Inc., 1958. Google Scholar

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