The Krull Intersection Theorem

Let R be a commutative ring, / an ideal in R, and A an i?-module. We always have 0 £= 0 £ I(\~=1 I A £ f|ϊ=i JM. where S is the multiplicatively closed set {1 — i\ie 1} and 0 = 0s Π A = {α G A13S 6 S 3 sα = 0}. It is of interest to know when some containment can be replaced by equality. The Krull intersection theorem states that for R Noetherian and A finitely generated I Π*=i I A = Π~=i IA. Since Π »=i /"A is finitely generated, f|*=i IA = 0. Thus if I £ rad (i?), the Jacobson radical of ϋ?, or R is a domain and A is torsionfree, we have (]n=i IA. = 0. In this note we show that for a Priif er domain R and a torsion-free ϋί-module A, I f|Γ=i ^A = nΓ=i/-A We also consider the condition (*): Πn=J = 0 for every ideal I in the commutative ring R. It is shown that a polynomial ring in any set of indeterminants over a Noetherian domain and the integral closure of a Noetherian domain satisfy (*).