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 such that . 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 . (3) There exists a minimal prime ideal *w* in the completion of such that depth . (4) There exists an integral extension domain of which has a maximal chain of prime ideals of length . (5) There exists a maximal chain of prime ideals of length in . (6) There exists a maximal chain of prime ideals of length in .

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Additional Information

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