Filtrations, asymptotic and Prüferian closures, cancellation laws

Dichi, Henri; Sangare, Daouda

doi:10.1090/S0002-9939-1991-1064901-0

Filtrations, asymptotic and Prüferian closures, cancellation laws
HTML articles powered by AMS MathViewer

by Henri Dichi and Daouda Sangare PDF

Proc. Amer. Math. Soc. 113 (1991), 617-624 Request permission

Abstract:

Let $A$ be a commutative ring. For any filtration $f = ({I_n})$ on the ring $A$ let $\bar f$ (resp. $P(f),f’)$ be the asymptotic (resp. prüferian, integral) closure of the filtration $f$. Then we have (*) $f \leq f’ \leq P(f) \leq \bar f.$ 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 $f,P(f)$, and $\bar f$ 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 $f,g,h$ are filtrations on the noetherian ring $A$ such that $\sqrt f \subseteq \sqrt h$, if $P(fh) \leq P(gh)$ and if $h$ is strongly AP then we have $P(f) \leq P(g)$. In this theorem the hypothesis "$h$ strongly AP" cannot be weakened to "$h$ AP" as shown in Example 2.3(3).

References

Philippe Ayégnon and Daouda Sangare, Generalized Samuel numbers and A.P filtrations, J. Pure Appl. Algebra 65 (1990), no. 1, 1–13. MR 1065058, DOI 10.1016/0022-4049(90)90095-Y
Wayne Bishop, A theory of multiplicity for multiplicative filtrations, J. Reine Angew. Math. 277 (1975), 8–26. MR 382259, DOI 10.1515/crll.1975.277.8
Henri Dichi, Integral dependence over a filtration, J. Pure Appl. Algebra 58 (1989), no. 1, 7–18. MR 996171, DOI 10.1016/0022-4049(89)90049-2
Stephen McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983. MR 722609, DOI 10.1007/BFb0071575
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 89836, DOI 10.1215/kjm/1250777054
J. S. Okon, Prime divisors, analytic spread and filtrations, Pacific J. Math. 113 (1984), no. 2, 451–462. MR 749548
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, DOI 10.1017/S0305004100065221

Some results in the theory of pseudo valuations

L. J. Ratliff Jr., Notes on essentially powers filtrations, Michigan Math. J. 26 (1979), no. 3, 313–324. MR 544599
D. Rees, Semi-Noether filtrations. I, J. London Math. Soc. (2) 37 (1988), no. 1, 43–62. MR 921745, DOI 10.1112/jlms/s2-37.121.43
Shiro Goto, Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), no. 1, 151–160. MR 887198, DOI 10.1016/0021-8693(87)90128-1