A note on termination of the Baer construction of the prime radical |
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Authors: | M A Chebotar P -H Lee E R Puczyłowski |
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Institution: | 1. Department of Mathematical Sciences, Kent State University, Kent, OH, USA 2. Department of Mathematics, National Taiwan University, Taipei, Taiwan 3. National Center for Theoretical Sciences, Taipei Office, Taipei, Taiwan 4. Institute of Mathematics, University of Warsaw, Warsaw, Banacha 2, Poland
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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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