Unions of Chains of Primes
  The union of an ascending chain of prime ideals is not always prime. We show that this property is independent of the parallel property for semiprimes. We also show that the PI-class is a tight bound on the number of non-prime unions of subchains in a chain of primes in a PI-algebra. 
Be'eri Greenfeld, Louis H. Rowen and Uzi Vishne
math.RA