Let R be a
commutative rin g,I an ideal in R, and A an R-module. We always have
0 ⊆ 0s⊆ I⋂n=1∞InA ⊆⋂n=1∞InA where S is the multiplicatively closed set
{1 − i|i ∈ I} and 0s= 0s∩ A = {a ∈ A|∃s ∈ S ∋ sa = 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⋂n=1∞InA =⋂n=1∞InA.
Since ∩∞n = 1InA is finitely generated, ⋂n=1∞InA = 0s. Thus if I ⊆ rad (R),
the Jacobson radical of R, or R is a domain and A is torsion-free, we have
⋂n=1∞InA = 0. In this note we show that for a Prüfer domain R and
a torsion-free R-module A,I⋂i=1∞InA =⋂i=1∞InA. We also consider
the condition (∗);⋂n=1∞In= 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
(∗).
