Filtrations, asymptotic and Prüferian closures, cancellation laws
Authors:
Henri Dichi and Daouda Sangare
Journal:
Proc. Amer. Math. Soc. 113 (1991), 617624
MSC:
Primary 13B22; Secondary 13A15
MathSciNet review:
1064901
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a commutative ring. For any filtration on the ring let (resp. be the asymptotic (resp. prüferian, integral) closure of the filtration . Then we have (*) In this paper several examples to show that each relation in (*) may be an equality or a strict inequality even in noetherian rings, are given. Some transfer properties (such as the property that a filtration be AP or strongly AP) between the filtrations , and are also given and negative answers are illustrated by some examples. This paper is closed by studying some cancellation laws concerning the prüferian closure of filtrations. In particular it is shown in the main theorem that if are filtrations on the noetherian ring such that , if and if is strongly AP then we have . In this theorem the hypothesis " strongly AP" cannot be weakened to " AP" as shown in Example 2.3(3).
 [1]
Philippe
Ayégnon and Daouda
Sangare, Generalized Samuel numbers and A.P filtrations, J.
Pure Appl. Algebra 65 (1990), no. 1, 1–13. MR 1065058
(91f:13001), http://dx.doi.org/10.1016/00224049(90)90095Y
 [2]
Wayne
Bishop, A theory of multiplicity for multiplicative
filtrations, J. Reine Angew. Math. 277 (1975),
8–26. MR
0382259 (52 #3144)
 [3]
Henri
Dichi, Integral dependence over a filtration, J. Pure Appl.
Algebra 58 (1989), no. 1, 7–18. MR 996171
(90d:13009), http://dx.doi.org/10.1016/00224049(89)900492
 [4]
Stephen
McAdam, Asymptotic prime divisors, Lecture Notes in
Mathematics, vol. 1023, SpringerVerlag, Berlin, 1983. MR 722609
(85f:13018)
 [5]
Masayoshi
Nagata, Note on a paper of Samuel concerning asymptotic properties
of ideals, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math.
30 (1957), 165–175. MR 0089836
(19,727c)
 [6]
J.
S. Okon, Prime divisors, analytic spread and filtrations,
Pacific J. Math. 113 (1984), no. 2, 451–462. MR 749548
(85e:13006)
 [7]
J.
S. Okon and L.
J. Ratliff Jr., Filtrations, closure operations and prime
divisors, Math. Proc. Cambridge Philos. Soc. 104
(1988), no. 1, 31–46. MR 938450
(90a:13016), http://dx.doi.org/10.1017/S0305004100065221
 [8]
J. W. Petro, Some results in the theory of pseudo valuations, Ph.D., State Univ. of Iowa, Graduate College, 1961.
 [9]
L.
J. Ratliff Jr., Notes on essentially powers filtrations,
Michigan Math. J. 26 (1979), no. 3, 313–324. MR 544599
(81a:13007)
 [10]
D.
Rees, SemiNoether filtrations. I, J. London Math. Soc. (2)
37 (1988), no. 1, 43–62. MR 921745
(88k:13001), http://dx.doi.org/10.1112/jlms/s237.121.43
 [11]
Shiro
Goto, Integral closedness of completeintersection ideals, J.
Algebra 108 (1987), no. 1, 151–160. MR 887198
(88d:13015), http://dx.doi.org/10.1016/00218693(87)901281
 [1]
 P. Ayegnon and D. Sangare, Generalized Samuel numbers and AP filtrations, J. Pure Appl. Algebra 65 (1990), 113. MR 1065058 (91f:13001)
 [2]
 W. Bishop, A theory of multiplicity for multiplicative filtrations, J. Reine Angew. Math. 277 (1975), 826. MR 0382259 (52:3144)
 [3]
 H. Dichi, Integral dependence over a filtration, J. Pure Appl. Algebra 58 (1989), 718. MR 996171 (90d:13009)
 [4]
 S. McAdam, Asymptotic prime divisors, Lecture Notes in Math., vol. 1023, SpringerVerlag, Berlin, 1983. MR 722609 (85f:13018)
 [5]
 M. Nagata, Note on a paper of Samuel concerning asymptotic properties of ideals, Mem. Coll. Sci. Kyoto Univ. Ser. A, vol. 30, Mathematics, No. 2, 1957, pp. 165175. MR 0089836 (19:727c)
 [6]
 J. S. Okon, Prime divisors, analytic spread and filtrations, Pacific J. Math. 113 (1984), 451462. MR 749548 (85e:13006)
 [7]
 J. S. Okon and L. J. Ratliff, Jr., Filtrations, closure operations and prime divisors, Math. Proc. Cambridge Philos. Soc. 104 (1988), 3146. MR 938450 (90a:13016)
 [8]
 J. W. Petro, Some results in the theory of pseudo valuations, Ph.D., State Univ. of Iowa, Graduate College, 1961.
 [9]
 L. J. Ratliff, Jr., Notes on essentially powers filtrations, Michigan Math. J. 26 (1979), 313324. MR 544599 (81a:13007)
 [10]
 D. Rees, Semi noether filtrations: I, J. London Math. Soc. (2) 37 (1988), 4362. MR 921745 (88k:13001)
 [11]
 Shiro Goto, Integral closedness of complete intersection ideals, J. Algebra 108 (1987), 151160. MR 887198 (88d:13015)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
13B22,
13A15
Retrieve articles in all journals
with MSC:
13B22,
13A15
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110649010
PII:
S 00029939(1991)10649010
Article copyright:
© Copyright 1991
American Mathematical Society
