Filtrations, asymptotic and Prüferian closures, cancellation laws

Authors:
Henri Dichi and Daouda Sangare

Journal:
Proc. Amer. Math. Soc. **113** (1991), 617-624

MSC:
Primary 13B22; Secondary 13A15

DOI:
https://doi.org/10.1090/S0002-9939-1991-1064901-0

MathSciNet review:
1064901

Full-text PDF

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**, https://doi.org/10.1016/0022-4049(90)90095-Y**[2]**Wayne Bishop,*A theory of multiplicity for multiplicative filtrations*, J. Reine Angew. Math.**277**(1975), 8–26. MR**0382259**, https://doi.org/10.1515/crll.1975.277.8**[3]**Henri Dichi,*Integral dependence over a filtration*, J. Pure Appl. Algebra**58**(1989), no. 1, 7–18. MR**996171**, https://doi.org/10.1016/0022-4049(89)90049-2**[4]**Stephen McAdam,*Asymptotic prime divisors*, Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983. MR**722609****[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**, https://doi.org/10.1215/kjm/1250777054**[6]**J. S. Okon,*Prime divisors, analytic spread and filtrations*, Pacific J. Math.**113**(1984), no. 2, 451–462. MR**749548****[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**, https://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****[10]**D. Rees,*Semi-Noether filtrations. I*, J. London Math. Soc. (2)**37**(1988), no. 1, 43–62. MR**921745**, https://doi.org/10.1112/jlms/s2-37.121.43**[11]**Shiro Goto,*Integral closedness of complete-intersection ideals*, J. Algebra**108**(1987), no. 1, 151–160. MR**887198**, https://doi.org/10.1016/0021-8693(87)90128-1

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:
https://doi.org/10.1090/S0002-9939-1991-1064901-0

Article copyright:
© Copyright 1991
American Mathematical Society