Maximal chains of prime ideals in integral extension domains. I
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- by L. J. Ratliff and S. McAdam PDF
- Trans. Amer. Math. Soc. 224 (1976), 103-116 Request permission
Abstract:
Let (R, M) be a local domain, let k be a positive integer, and let Q be a prime ideal in ${R_k} = R[{X_1}, \ldots ,{X_k}]$ such that $M{R_k} \subset Q$. 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 depth $z = n$. (3) There exists a minimal prime ideal w in the completion of ${({R_k})_Q}$ such that depth $w = n + k - {\text {depth}}\;Q$. (4) There exists an integral extension domain of ${({R_k})_Q}$ which has a maximal chain of prime ideals of length $n + k - {\text {depth}}\;Q$. (5) There exists a maximal chain of prime ideals of length $n + k - {\text {depth}}\;Q$ in ${({R_k})_Q}$. (6) There exists a maximal chain of prime ideals of length $n + 1$ in $R{[{X_1}]_{(M,{X_1})}}$.References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 224 (1976), 103-116
- MSC: Primary 13A15; Secondary 13B20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0437513-3
- MathSciNet review: 0437513