Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T11:44:11.434Z Has data issue: false hasContentIssue false

RATLIFF–RUSH CLOSURE OF IDEALS IN INTEGRAL DOMAINS

Published online by Cambridge University Press:  01 September 2009

A. MIMOUNI*
Affiliation:
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, PO Box 5046, Dhahran 31261, Kingdom of Saudi Arabia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies the Ratliff–Rush closure of ideals in integral domains. By definition, the Ratliff–Rush closure of an ideal I of a domain R is the ideal given by Ĩ := ∪(In+1 :R In), and an ideal I is said to be a Ratliff–Rush ideal if Ĩ = I. We completely characterise integrally closed domains in which every ideal is a Ratliff–Rush ideal, and we give a complete description of the Ratliff–Rush closure of an ideal in a valuation domain.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Anderson, D. D., Huckaba, J. A. and Papick, I. J., A notes on stable domains, Houston J. Math. 13 (1) (1987), 1317.Google Scholar
2.Barucci, V., Strongly divisorial ideals and complete integral closure of an integral domain, J. Algebra 99 (1986), 132142.CrossRefGoogle Scholar
3.Barucci, V. and Houston, E., On the prime spectrum of a Mori domain, Comm. Algebra 24 (11) (1996), 35993622.CrossRefGoogle Scholar
4.Bastida, E. and Gilmer, R., Overrings and divisorial ideals of rings of the form D+M, Michigan Math. J. 20 (1992), 7995.Google Scholar
5.Dobbs, D. and Fedder, R., Conducive integral domains, J. Algebra 86 (1984), 494510.CrossRefGoogle Scholar
6.Dobbs, D., Houston, E., Lucas, T., Roitman, M. and Zafrullah, M., On t-linked overrings, Comm. Algebra 20 (1992), 14631488.CrossRefGoogle Scholar
7.Dobbs, D., Houston, E., Lucas, T. and Zafrullah, M., t-linked overrings and Prüfer v-multiplication domains, Comm. Algebra 17 (1989), 28352852.CrossRefGoogle Scholar
8.Fangui, W. and McCasland, R. L., On strong Mori domains, J. Pure Appl. Algebra 135 (1999), 155165.Google Scholar
9.Fontana, M., Huckaba, J. and Papick, I., Domains satisfying the trace property, J. Algebra 107 (1987), 169182.CrossRefGoogle Scholar
10.Fontana, M., Huckaba, J. and Papick, I., Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics, vol. 203 (Marcel Dekker, New York, 1997).Google Scholar
11.Gilmer, R., Multiplicative ideal theory, Pure and Applied Mathematics, no. 12. (Marcel Dekker, New York, 1972).Google Scholar
12.Heinzer, W., Lantz, Johnston D. and Shah, K., The Ratliff–Rush ideals in a Noetherian ring: A survey in methods in module theory, vol. 140 (Marcel Dekker, New York, 1992), 149159.Google Scholar
13.Heinzer, W., Lantz, D. and Shah, K., The Ratliff–Rush ideals in a Noetherian ring, Comm. Algebra 20 (1992), 591622.CrossRefGoogle Scholar
14.Heinzer, W. and Papick, I., The radical trace property, J. Algebra 112 (1988), 110121.CrossRefGoogle Scholar
15.Huckaba, J. A. and Papick, I. J., When the dual of an ideal is a ring, Manuscripta Math. 37 (1982), 67–85.CrossRefGoogle Scholar
16.Kaplansky, I., Commutative rings (University of Chicago Press, Chicago, 1974).Google Scholar
17.Liu, J. C., Ratliff–Rush closures and coefficient modules, J. Algebra 201 (1998), 584603.Google Scholar
18.Northcoot, D. G. and Rees, D., Reductions of ideals in local rings, Proc. Camb. Phil. Soc. 50 (1954), 145158.Google Scholar
19.Olberding, B., Globalizing local properties of Prüfer domains, J. Algebra 205 (1998), 480504.CrossRefGoogle Scholar
20.Olberding, B., On the classification of stable domains, J. Algebra 243 (2001), 177197.CrossRefGoogle Scholar
21.Olberding, B., Stability of ideals and its applications, in Ideal theoretic methods in commutative algebra (Anderson, D. D. and Papick, I. J., Editors), Lecture Notes in Pure and Applied Mathematics, vol. 220 (Marcel Dekker, New York, 2001), 319341.Google Scholar
22.Olberding, B., On the structure of stable domains, Comm. algebra 30 (2) (2002), 877895.CrossRefGoogle Scholar
23.Ratliff, L. J. Jr, and Rush, D. E., Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), 929934.CrossRefGoogle Scholar
24.Rossi, and Swanson, I., Notes on the behavior of the Ratliff–Rush filtration, Commutative Algebra, Contemporary Mathematics, vol. 331 (American Mathematical Society, Providence RI, 2003), 313328.Google Scholar
25.Sally, J. D. and Vasconcelos, W. V., Stable rings and a problem of Bass, Bull. Amer. Math. Soc. 79 (1973), 574576.CrossRefGoogle Scholar