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
Abstract 
References 
Similar Articles 
Additional Information
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 .
 [1]
L. Dechene, Adjacent extensions of rings, Ph. D. Dissertation, Univ. of California, Riverside, 1978.
 [2]
E.
Graham Evans Jr., A generalization of Zariski’s
main theorem, Proc. Amer. Math. Soc. 26 (1970), 45–48. MR 0260716
(41 #5340), http://dx.doi.org/10.1090/S00029939197002607168
 [3]
D.
Ferrand and J.P.
Olivier, Homomorphisms minimaux d’anneaux, J. Algebra
16 (1970), 461–471 (French). 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]
Stephen
McAdam, Asymptotic prime divisors, Lecture Notes in
Mathematics, vol. 1023, SpringerVerlag, Berlin, 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]
Masayoshi
Nagata, On the chain problem of prime ideals, Nagoya Math. J.
10 (1956), 51–64. MR 0078974
(18,8e)
 [10]
Masayoshi
Nagata, Local rings, Interscience Tracts in Pure and Applied
Mathematics, No. 13, Interscience Publishers a division of John Wiley &
Sons New YorkLondon, 1962. MR 0155856
(27 #5790)
 [11]
L.
J. Ratliff Jr., On quasiunmixed local domains, the altitude
formula, and the chain condition for prime ideals. II, Amer. J. Math.
92 (1970), 99–144. MR 0265339
(42 #249)
 [12]
L.
J. Ratliff Jr., Independent elements, integrally closed ideals, and
quasiunmixedness, J. Algebra 73 (1981), no. 2,
327–343. MR
640040 (82m:13006), http://dx.doi.org/10.1016/00218693(81)903252
 [13]
L.
J. Ratliff Jr., Five notes on asymptotic prime divisors, Math.
Z. 190 (1985), no. 4, 567–581. MR 808923
(87c:13001), http://dx.doi.org/10.1007/BF01214755
 [14]
Oscar
Zariski and Pierre
Samuel, Commutative algebra, Volume I, The University Series
in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New
Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
(19,833e)
 [15]
Oscar
Zariski and Pierre
Samuel, Commutative algebra. Vol. II, The University Series in
Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.
J.TorontoLondonNew York, 1960. MR 0120249
(22 #11006)
 [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)
 [11]
 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)
 [12]
 , 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
13A17,
13E05
Retrieve articles in all journals
with MSC:
13A17,
13E05
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
