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Journal Article

# Maximal Chains of Prime Ideals in Integral Extension Domains. I

L. J. Ratliff, Jr. and S. McAdam
Transactions of the American Mathematical Society
Vol. 224, No. 1 (Nov., 1976), pp. 103-116
DOI: 10.2307/1997418
Stable URL: http://www.jstor.org/stable/1997418
Page Count: 14
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## Abstract

Let (R, M) be a local domain, let k be a positive integer, and let Q be a prime ideal in Rk = R[ X1,..., Xk ] such that \$MR_k \subset Q\$. Then the following statements are equivalent: (1) There exists an integral extension domain of R which has a maximal chain of prime ideals of length n. (2) There exists a minimal prime ideal z in the completion of R such that \$\operatorname{depth}z = n\$. (3) There exists a minimal prime ideal w in the completion of (Rk)Q such that \$\operatorname{depth}w = n + k - \operatorname{depth}Q\$. (4) There exists an integral extension domain of (Rk)Q which has a maximal chain of prime ideals of \$\operatorname{length}n + k - \operatorname{depth}Q\$. (5) There exists a maximal chain of prime ideals of \$\operatorname{length}n + k - \operatorname{depth}Q\$ in (Rk)Q. (6) There exists a maximal chain of prime ideals of \$\operatorname{length}n + 1\$ in R[ X1 ](M,X1).

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