Two notes on imbedded prime divisors
Author:
L. J. Ratliff
Journal:
Proc. Amer. Math. Soc. 101 (1987), 395402
MSC:
Primary 13A17; Secondary 13E05
MathSciNet review:
908637
Fulltext PDF Free Access
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Abstract: The first note shows that if are any two Noetherian rings, then there exists a Noetherian ring between and which has a maximal ideal such that and is a maximal ideal. The second note shows that if is a Noetherian ring, then there exists a free quadratic integral extension ring of such that and such that if is any regular ideal in and are prime ideals in containing , then there exists an ideal in integrally dependent on such that the prime ideals corresponding to the are prime divisors of for all .
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 [1]
 L. Dechene, Adjacent extensions of rings, Ph. D. Dissertation, Univ. of California, Riverside, 1978.
 [2]
 E. G. Evans, Jr., A generalization of Zariski's Main Theorem, Proc. Amer. Math. Soc. 26 (1970), 4548. MR 0260716 (41:5340)
 [3]
 D. Ferrand and J.P. Oliver, Homomorphismes minimaux d'anneaux, J. Algebra 16 (1970), 461471. MR 0271079 (42:5962)
 [4]
 A. Grothendieck, Elements de geometrie algebrique. IV (Premiere Partie), Inst. Hautes Études Sci., Paris, 1964.
 [5]
 D. Katz, S. McAdam, J. Okon, and L. J. Ratliff, Jr., Essential prime divisors and projectively equivalent ideals, J. Algebra (to appear).
 [6]
 S. McAdam, Asymptotic prime divisors, Lecture Notes in Math., vol. 1023, SpringerVerlag, Berlin and New York, 1983. MR 722609 (85f:13018)
 [7]
 S. McAdam and L. J. Ratliff, Jr., Persistent primes and projective extensions of ideals, Rocky Mountain J. Math (to appear).
 [8]
 M. L. Modica, Maximal subrings, Ph. D. Dissertation, Univ. of Chicago, 1975.
 [9]
 M. Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 5164. MR 0078974 (18:8e)
 [10]
 , Local rings, Interscience Tracts 13, Interscience, New York, 1962. MR 0155856 (27:5790)
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 L. J. Ratliff, Jr., On quasiunmixed local domains, the altitude formula, and the chain condition for prime ideals (II), Amer. J. Math. 92 (1970), 99144. MR 0265339 (42:249)
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 , Independent elements, integrally closed ideals, and quasiunmixedness, J. Algebra 73 (1981), 327343. MR 640040 (82m:13006)
 [13]
 , Five notes on asymptotic prime divisors, Math. Z. 190 (1985), 567581. MR 808923 (87c:13001)
 [14]
 O. Zariski and P. Samuel, Commutative algebra, Vol. I, Van Nostrand, New York, 1958. MR 0090581 (19:833e)
 [15]
 , Commutative algebra, Vol. II, Van Nostrand, New York, 1960. MR 0120249 (22:11006)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198709086371
PII:
S 00029939(1987)09086371
Keywords:
CohenMacaulay ring,
flat extension ring,
grade of an ideal,
integral closure of an ideal,
integral extension ring,
Notherian ring,
prime divisor semilocal ring
Article copyright:
© Copyright 1987
American Mathematical Society
