  This paper investigates the doubly exponential ideal membership property of
the Mayr-Meyer ideals from the point of view of their associated primes. A
doubly-exponential upper bound on the set of associated primes is proved.
  In the paper a new family of ideals emerges which also has the doubly
exponential ideal membership property. More on the new family can be found in
author's paper "A new family of ideals with the doubly exponential ideal
membership property".
