Ideals Contracted from 1-dimensional Overrings with an Application to the Primary Decomposition of Ideals

We prove that each ideal of a locally formally equidimensional analytically un-ramiied Noetherian integral domain is the contraction of an ideal of a one-dimensional semilo-cal birational extension domain. We give an application to a problem concerning the primary decomposition of powers of ideals in Noetherian rings. It is shown in S2] that for each ideal I in a Noetherian commutative ring R there exists a positive integer k such that, for all n 1, there exists a primary decomposition I n = Q 1 \ \ Q s where each Q i contains the nk-th power of its radical. We give an alternate proof of this result in the special case where R is locally at each prime ideal formally equidimensional and analytically unramiied. In this paper we prove that every ideal in a locally formally equidimensional analytically unramiied Noetherian ring R is the contraction of an ideal of a one-dimensional semilocal extension which is essentially of nite type over R. If R is a domain, the extension may be taken to be birational, i.e., with the same eld of fractions as R. By passing to the extended Rees ring RIt; t ?1 ] of an ideal I of R, these contraction properties give a type of uniform primary decomposition for the powers of I. This is based on the fact that the primary decomposition of a height-one ideal in a one-dimensional semilocal ring is unique, and the primary decomposition for powers of a xed ideal in such a ring is obtained from just taking the powers of the primary components of the xed ideal. Furthermore, contracting primary decompositions from an overring gives a primary decomposition for the contracted ideal. Our interest in establishing this result was motivated by a question, recently answered in S2], concerning the primary decompositions of powers of an ideal. All rings we consider are commutative and our notation is as in AM] and M].