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We prove that each ideal of a locally formally equidimensional analytically unramified Noetherian integral domain is the contraction of an ideal of a one-dimensional semilocal 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 an earlier paper by the second author 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 In = Qi n ... Q, where each Qi 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 unramified. In this paper we prove that every ideal in a locally formally equidimensional analytically unramified Noetherian ring R is the contraction of an ideal of a onedimensional semilocal extension which is essentially of finite type over R. If R is a domain, the extension may be taken to be birational, i.e., with the same field of fractions as R. By passing to the extended Rees ring R[It, 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 fixed ideal in such a ring is obtained from just taking the powers of the primary components of the fixed 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]. 1. POWERS OF IDEALS AND PRIMARY DECOMPOSITIONS Let I be a proper ideal of a commutative Noetherian ring R. It is known that only finitely many prime ideals of R are associated primes of a power of I [Rat], and that all suitably large powers of I have the same associated primes [B]. In considering primary decompositions of the powers of I, it is natural to ask about the growth of the exponents of primary components of In, where the exponent of Received by the editors September 6, 1995. 1991 Mathematics Subject Classification. Primary 13C05, 13E05, 13H99. The authors thank Craig Huneke for helpful suggestions concerning this paper. (?)1997 American Mathematical Society

Ideals contracted from 1-dimensional overrings with an application to the primary decomposition of ideals