  This paper gives a complete primary decomposition of the first, that is, the
smallest, Mayr-Meyer ideal, its radical, and the intersection of its minimal
components. The particular membership problem which makes the Mayr-Meyer
ideals' complexity doubly exponential in the number of variables is here
examined also for the radical and the intersection of the minimal components.
It is proved that for the first Mayr-Meyer ideal the complexity of this
membership problem is the same as for its radical. This problem was motivated
by a question of Bayer, Huneke and Stillman.
