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Zero divisors in differential rings

1971; Mathematical Sciences Publishers; Volume: 39; Issue: 1 Linguagem: Inglês

10.2140/pjm.1971.39.163

1945-5844

Howard Gorman,

Polynomial and algebraic computation

Let R be a commutative ordinary differential ring with 1.Let A be a commutative differential i?-algebra satisfying the ascending chain condition on radical differential ideals.Let M be a differentially finitely generated iϋ-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 Q A and / is a radical differential ideal of A, then we obtain the assassinator of A/1 in R from the assassinator of All 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 / 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.These results (except (ii)) are well known for finite modules over noetherian rings. 2* Preliminaries*In what follows, all rings are commutative and all modules are unitary.R will always be a differential ring with 1, with fixed derivation denoted by " ' ".By a differential module M over R, one means an i?-module M together with an additive map from M to M, again denoted by " ' ", which satisfies (rm)' = r'm + rm' for each reR and me M. If xe M, the successive derivatives of x will be denoted by x',x", •• ,x (%) , ••• .By a differential algebra A over R, one means a differential module A which is a ring and for which the module derivation is a ring derivation.By an ideal of A, we always mean an algebra ideal.Let M be any jR-module and T g M a subset.We denote the zero divisors of T in R by %Έ(Ί) and the annihilator of T in R by jχf R {T).The assassinator of M in R, written Ass^ M, is the set of prime ideals of R which are the annihilators of nonzero elements of M. The support of M in R, written Supp^ M, is the set of prime ideals P of R such that M P Φ 0. Now let R be a differential ring and M a differential iϋ-module.