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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Maximal chains of prime ideals in integral extension domains. I


Authors: L. J. Ratliff and S. McAdam
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
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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 $ {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})}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0437513-3
Keywords: Altitude formula, catenary ring, catenary chain conjecture, chain condition for prime ideals, chain conjecture, completion of a local ring, depth conjecture, first chain condition for prime ideals, integral extension, local ring, maximal chain of prime ideals, Noetherian ring, polynomial extension ring, quasi-unmixed local ring, second chain condition for prime ideals, semilocal ring, unmixed local ring, upper conjecture
Article copyright: © Copyright 1976 American Mathematical Society