Three theorems on form rings
Author:
Louis J. Ratliff
Journal:
Proc. Amer. Math. Soc. 99 (1987), 432436
MSC:
Primary 13E05; Secondary 13A17, 13C15
MathSciNet review:
875376
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Abstract: Three theorems concerning the form ring (= associated graded ring) of an ideal in a Noetherian ring are proved. The first characterizes, for a primary ideal in a locally quasiunmixed ring, when is an integral domain in terms of when is an integral domain. For an aribtrary Noetherian ring the second gives a somewhat similar characterization for to have only one prime divisor of zero for some ideal that is projectively equivalent to . And the third characterizes unmixed semilocal rings in terms of the existence of an open ideal such that the zero ideal in is isobathy.
 [1]
Craig
Huneke, On the associated graded ring of an ideal, Illinois J.
Math. 26 (1982), no. 1, 121–137. MR 638557
(83d:13029)
 [2]
Daniel
Katz and Louis
J. Ratliff Jr., 𝑢essential prime divisors and sequences
over an ideal, Nagoya Math. J. 103 (1986),
39–66. MR
858471 (87j:13002)
 [3]
Stephen
McAdam, Asymptotic prime divisors, Lecture Notes in
Mathematics, vol. 1023, SpringerVerlag, Berlin, 1983. MR 722609
(85f:13018)
 [4]
Stephen
McAdam and L.
J. Ratliff Jr., Essential sequences, J. Algebra
95 (1985), no. 1, 217–235. MR 797664
(87a:13016), http://dx.doi.org/10.1016/00218693(85)901024
 [5]
Masayoshi
Nagata, On the chain problem of prime ideals, Nagoya Math. J.
10 (1956), 51–64. MR 0078974
(18,8e)
 [6]
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)
 [7]
L.
J. Ratliff Jr., On prime divisors of the integral closure of a
principal ideal, J. Reine Angew. Math. 255 (1972),
210–220. MR 0311638
(47 #200)
 [8]
L.
J. Ratliff Jr., On the prime divisors of zero in form rings,
Pacific J. Math. 70 (1977), no. 2, 489–517. MR 0491648
(58 #10857)
 [9]
L.
J. Ratliff Jr., Powers of ideals in locally unmixed Noetherian
rings, Pacific J. Math. 107 (1983), no. 2,
459–472. MR
705759 (85c:13010)
 [10]
L.
J. Ratliff Jr., On linearly equivalent ideal topologies, J.
Pure Appl. Algebra 41 (1986), no. 1, 67–77. MR 844465
(87i:13004), http://dx.doi.org/10.1016/00224049(86)901015
 [11]
L.
J. Ratliff Jr., The topology determined by the symbolic powers of
primary ideals, Comm. Algebra 13 (1985), no. 9,
2073–2104. MR 795491
(86h:13003), http://dx.doi.org/10.1080/00927878508823265
 [12]
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
 [13]
D.
Rees, A note on form rings and ideals, Mathematika
4 (1957), 51–60. MR 0090588
(19,835c)
 [14]
Motoyoshi
Sakuma and Hiroshi
Okuyama, On a criterion for analytically unramification of a local
ring, J. Gakugei Tokushima Univ. 15 (1966),
36–38. MR
0200290 (34 #189)
 [15]
Peter
Schenzel, Symbolic powers of prime ideals and
their topology, Proc. Amer. Math. Soc.
93 (1985), no. 1,
15–20. MR
766518 (86e:13011), http://dx.doi.org/10.1090/S00029939198507665189
 [1]
 C. Huneke, On the associated graded ring of an ideal, Illinois J. Math. 26 (1982), 121137. MR 638557 (83d:13029)
 [2]
 D. Katz and L. J. Ratliff, Jr., essential prime divisors and sequences over an ideal, Nagoya J. Math. 103 (1986), 3966. MR 858471 (87j:13002)
 [3]
 S. McAdam, Asymptotic prime divisors, Lecture Notes in Math., vol. no. 1023, SpringerVerlag, Berlin and New York, 1983. MR 722609 (85f:13018)
 [4]
 S. McAdam and L. J. Ratliff, Jr., Essential sequences, J. Algebra 95 (1985), 271235. MR 797664 (87a:13016)
 [5]
 M. Nagata, On the chain problem of prime ideals, Nagoya Math. J. 10 (1956), 5164. MR 0078974 (18:8e)
 [6]
 , Local rings, Interscience Tracts 13, Interscience, New York, 1962. MR 0155856 (27:5790)
 [7]
 L. J. Ratliff, Jr., On prime divisors of the integral closure of a principal ideal, J. Reine Angew. Math. 255 (1972), 210220. MR 0311638 (47:200)
 [8]
 , On the prime divisors of zero in form rings, Pacific J. Math. 70 (1977), 489517. MR 0491648 (58:10857)
 [9]
 , Powers of ideals in locally unmixed Noetherian rings, Pacific J. Math. 107 (1983), 459472. MR 705759 (85c:13010)
 [10]
 , On linearly equivalent ideal topologies, J. Pure Appl. Algebra 41 (1986), 6777. MR 844465 (87i:13004)
 [11]
 , The topology determined by the symbolic powers of primary ideals, Comm. Algebra 13 (1985), 20732104. MR 795491 (86h:13003)
 [12]
 , Five notes on asymptotic prime divisors, Math. Z. 190 (1985), 567581. MR 808923 (87c:13001)
 [13]
 D. Rees, A note on form rings and ideals, Mathematika 4 (1957), 5160. MR 0090588 (19:835c)
 [14]
 M. Sakuma and H. Okuyama, On a criterion for analytically unramification of a local ring, J. Gakugei Tokushima Univ. 15 (1966), 3638. MR 0200290 (34:189)
 [15]
 P. Schenzel, Symbolic powers of prime ideals and their topology, Proc. Amer. Math. Soc. 93 (1985), 1520. MR 766518 (86e:13011)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198708753765
PII:
S 00029939(1987)08753765
Keywords:
Analytically unramified semilocal ring,
asymptotic prime divisor,
analytic spread of an ideal,
essential prime divisor,
form ring,
integral closure of an ideal,
isobathy ideal,
Noetherian ring,
normal ideal,
projectively equivalent ideals,
quasiunmixed local ring,
Rees ring,
unmixed local ring,
essential prime divisor
Article copyright:
© Copyright 1987
American Mathematical Society
