  Motivated by the problem of the existence of bounds on degrees and orders in
checking primality of radical (partial) differential ideals, the nonstandard
methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings
over fields. A nonstandard approach.", Inventionnes Mathematicae, 76:77--91,
1984] are here extended to differential polynomial rings over differential
fields. Among the standard consequences of this work are: a partial answer to
the primality problem, the equivalence of this problem with several others
related to the Ritt problem, and the existence of bounds for characteristic
sets of minimal prime differential ideals and for the differential
Nullstellensatz.
