Vol. 57, No. 1, 1975

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The Krull intersection theorem

Daniel D. Anderson

Vol. 57 (1975), No. 1, 11–14

Let R be a commutative rin g,I an ideal in R, and A an R-module. We always have 0 0s I n=1InA n=1InA 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=1InA = n=1InA. Since ∩∞n = 1InA is finitely generated, n=1InA = 0s. Thus if I rad (R), the Jacobson radical of R, or R is a domain and A is torsion-free, we have n=1InA = 0. In this note we show that for a Prüfer domain R and a torsion-free R-module A,I i=1InA = i=1InA. We also consider the condition (); n=1In = 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 ().

Mathematical Subject Classification 2000
Primary: 13C05
Received: 26 September 1974
Published: 1 March 1975
Daniel D. Anderson