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

Abstract:

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 $n \ge 1$, there exists a primary decomposition $I^{n} = Q_{1} \cap \dots \cap Q_{s}$ 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 unramified.