Every local ring is dominated by a one-dimensional local ring

Gilmer, Robert; Heinzer, William

doi:10.1090/S0002-9939-97-03847-1

Every local ring is dominated by a one-dimensional local ring
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by Robert Gilmer and William Heinzer

Proc. Amer. Math. Soc. 125 (1997), 2513-2520

DOI: https://doi.org/10.1090/S0002-9939-97-03847-1

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Abstract:

Let $(R, \mathbf {m})$ be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring $S$ of $R$ such that (i) $S$ dominates $R$, (ii) the residue field of $S$ is a finite purely transcendental extension of $R/ \mathbf {m}$, (iii) every associated prime of (0) in $S$ contracts in $R$ to an associated prime of (0), and (iv) $\dim (S) \le 1$. In addition, it is shown that $S$ can be obtained so that either $\mathbf {m} S$ is the maximal ideal of $S$ or $S$ is a localization of a finitely generated $R$-algebra.

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