A note on termination of the Baer construction of the prime radical

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 alpha there is a ring for which the construction stops precisely at alpha. In this paper we construct such examples with some extra properties. This allows us to construct, for every countable non-limit ordinal number alpha, an affine algebra for which the construction terminates precisely at alpha. Such an example was known due to Bergman for alpha = 2.