No Access

Bounds related to projective equivalence classes of good filtrations

Abdoulaye Assane

UNA, UFR-SFA, Laboratoire de Mathématiques, 02 PB 801 Abidjan 02, Côte d’Ivoire, France

E-mail Address: abdoulassan2002@yahoo.fr

Search for more papers by this author

Damase Kamano

E.N.S. d’Abidjan, 08 BP 10 Abidjan 08, Côte d’Ivoire, France

E-mail Address: kamanodamase@yahoo.fr

Corresponding author.

Search for more papers by this author

, and

Eric Dago Akeke

UFHB de Cocody, UFRMI, 22 BP 582 Abidjan 22, Côte d’Ivoire, France

E-mail Address: ericdago@yahoo.fr

Search for more papers by this author

https://doi.org/10.1142/S1793557118500523Cited by:0 (Source: Crossref)

Abstract

Let $I$ $I$ be a regular ideal in noetherian ring $A$ $A$ . Mc Adam and Ratliff showed the existence of the unique minimal reduction number of $I$ $I$ , noted $e = e (I) \in ℕ$ , such that for every minimal reduction $Y$ of $I$ , $Y I^{e} = I^{e + 1}$ and $Y I^{e - 1} \neq I^{e}$ . They showed that the set of integers ${e (I^{n}), n \in ℕ}$ is bounded in terms of the analytic spread of $I$ . Here, we extend these results to good filtrations. Let $f = {(I_{n})}_{n \in ℕ}$ be a good filtration on $A$ , we show that the set of integers ${e (I_{n}), n \in ℕ}$ is bounded.

Communicated by V. A. Artamanov

Keywords:

AMSC: 13A15, 13A30, 13E05, 13C99

References

1. W. Bishop, J. W. Petro, L. J. Ratlifff and D. E. Rush, Note on noetherian filtrations, Comm. Algebra 17(2) (1989) 471–485. Crossref, Web of Science, Google Scholar
2. H. Dichi, Integral independance over a filtration, J. Pure Appl. Algebra 58 (1989) 7–18. Crossref, Web of Science, Google Scholar
3. H. Dichi and D. Sangaré, Filtrations, asymptotic and prüferian closures, cancellation laws, Proc. Amer. Math. Soc. 113(3) (1991) 617–624. Web of Science, Google Scholar
4. H. Dichi and D. Sangaré, Analytic spread of filtrations, asymptotic nature and some stability properties, Comm. Algebra 28(7) (2000) 3115–3124. Crossref, Web of Science, Google Scholar
5. H. Dichi, D. Sangaré and M. Soumaré, Filtrations, integral dependance, reduction, f-good filtrations, Comm. Algebra 20(8) (1992) 2393–2418. Crossref, Web of Science, Google Scholar
6. L. T. Hoa and S. Zarzuela, Reduction number and a-invariant of good filtrations, Comm. Algebra 22(14) (1994) 5635–5656. Crossref, Web of Science, Google Scholar
7. H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Vol. 8. Google Scholar
8. S. McAdam and L. J. Ratliff, Bounds related to projective equivalence classes of ideals, J. Algebra 119 (1988) 23–33. Crossref, Web of Science, Google Scholar
9. D. G. Northcott and D. Rees, Reduction of ideals in local rings, Proc. Cambridge. Philos. Soc. 50 (1954) 145–158. Crossref, Web of Science, Google Scholar
10. L. J. Ratliff and D. E. Rush, Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978) 929–934. Crossref, Web of Science, Google Scholar