Let R be a commutative
ordinary differential ring with 1. Let A be a commutative differential R-algebra
satisfying the ascending chain condition on radical differential ideals. Let M be a
differentially finitely generated R-module. We obtain the following results on
the zero divisors of A and M in R. (i) If R satisfies the ascending chain
condition on radical differential ideals and if A has zero nilradical, then
the assassinator of A in R is finite and consists of differential ideals; it is
contained in the support of A in R, and the minimal members of each set
comprise exactly the minimal prime ideals which contain the annihilator
of A in R; (ii) If R ⊆ A and I is a radical differential ideal of A, then we
obtain the assassinator of A∕I in R from the assassinator of A∕I in A by
intersecting with R; (iii) If R is noetherian, then the set of zero divisors of
M in R is a unique union of prime differential ideals of R, each of which
is maximal among annihilators in R of nonzero elements of M; (iv) If I
is the annihilator or power annihilator of M in R, then any prime ideal
of R minimal over I is the annihilator of a nonzero element of M. In the
above, (iii) and (iv) require an additional hypothesis to be made explicit
later.
