Vol. 39, No. 1, 1971

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Zero divisors in differential rings

Howard Edwin Gorman

Vol. 39 (1971), No. 1, 163–171

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.

Mathematical Subject Classification 2000
Primary: 12H05
Received: 17 March 1970
Revised: 29 April 1971
Published: 1 October 1971
Howard Edwin Gorman