Abstract
The well known Baer construction of the prime radical shows that the prime radical of an arbitrary ring is the union of the chain of ideals of the ring, constructed by transfinite induction, which starts with 0 and repeats the procedure of taking the sum of ideals that are nilpotent modulo ideals in the chain already constructed. Amitsur showed that for every ordinal number α there is a ring for which the construction stops precisely at α. In this paper we construct such examples with some extra properties. This allows us to construct, for every countable non-limit ordinal number α, an affine algebra for which the construction terminates precisely at α. Such an example was known due to Bergman for α = 2.
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References
S. A. Amitsur, Nil radicals, historical notes and some new results, Rings, Modules and Radicals (Proc. Internat. Colloq., Keszthely, 1971), 47–65. Colloq. Math. Soc. Janos Bolyai, 6, North-Holland, Amsterdam, 1973.
M. Chebotar, P.-H. Lee, and E. R. Puczyłowski, On Andrunakievich chain and Koethe’s problem, Israel J. Math., to appear.
Markov V.T.: Some examples of finitely generated algebr as. (Russian) Uspekhi Mat. Nauk 36, 185–186 (1981)
D. S. Passman, The algebraic structure of group rings. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.
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E. R. Puczyłowski was supported by MNiSW Grant Nr N N201 268435 and National Center for Theoretical Sciences, Taipei Office, Taiwan.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Chebotar, M.A., Lee, P.H. & Puczyłowski, E.R. A note on termination of the Baer construction of the prime radical. Arch. Math. 95, 325–332 (2010). https://doi.org/10.1007/s00013-010-0172-7
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DOI: https://doi.org/10.1007/s00013-010-0172-7