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Extra info for Differential Algebra and Related Topics: Proceedings of the International Workshop, Newark Campus of Rutgers, The State University of New Jersey, 2-3 November 2000
It is easy to see t h a t if A has the strong Rosenfeld property, then it has the Rosenfeld property since the initials I A are partially reduced with respect to A . If A has the Rosenfeld property, then it is possible to answer certain questions about a differential ideal by answering similar questions about an ideal. For the differential ideals a # and as, we shall be interested in t h e properties of being prime, radical, and zero-reduced. 3, an autoreduced subset of a proper differential ideal a is a characteristic set if o is zero-reduced with respect to it.
His lemma then states that a coherent autoreduced set has the Rosenfeld property. Kolchin  (Chapter III, Section 8, p. 135) generalized this to L-coherence relative to an ideal L (not necessarily differential, but having a set of generators partially reduced with respect to A) and over differential domains of arbitrary characteristic. Morrison  recently introduced new 10 This notion is closely related to, and probably implies, the involutiveness, formal integrability, or other completeness properties of the system defined by A .
1 For any v 6 @Y, let A(„) be the set of all differential polynomials 6A with 6 € ©, A G A and 9UA < v. We say an autoreduced set A is coherent (resp. subcoherent11, resp. strongly coherent, resp. strongly subcoherent) if for all pairs A, A' £ A whose leaders UA,UA> have a common derivative, say v = QUA = O'UA', the differential polynomial A(A,A',v) = SA>0A — SAO'A', which has lower rank than v, belongs to the ideal (A(„)): H°° (resp. (A U A ( „)): H°°, resp. (A ( „ } ): 5°°, resp. (A U A ( „)): 5°°) in the polynomial ring 01.