Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T05:43:39.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-socle ideals in Buchsbaum rings

Published online by Cambridge University Press:  11 January 2016

Shiro Goto,
Jun Horiuchi and
Hideto Sakurai
Shiro Goto
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-8571, Japan, goto@math.meiji.ac.jp
Jun Horiuchi
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-8571, Japan, jhoriuchi@math.meiji.ac.jp
Hideto Sakurai
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-8571, Japan, hsakurai@math.meiji.ac.jp
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.

Quasi-socle ideals, that is, ideals of the form I = Q: mq (q ≥ 2), with Q parameter ideals in a Buchsbaum local ring (A,m), are explored in connection to the question of when I is integral over Q and when the associated graded ring G(I) ⊕ n≥0In/In+1 of I is Buchsbaum. The assertions obtained by Wang in the Cohen-Macaulay case hold true after necessary modifications of the conditions on parameter ideals Q and integers q. Examples are explored.

Keywords

13H1013A3013B2213H15

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 200 , December 2010, pp. 93 - 106
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[CHV] Corso, A., Huneke, C., and Vasconcelos, W. V., On the integral closure of ideals, Manuscripta Math. 95 (1998), 331–347.CrossRefGoogle Scholar
[CP1] Corso, A. and Polini, C., Links of prime ideals and their Rees algebras, J. Algebra 178 (1995), 224–238.CrossRefGoogle Scholar
[CP2] Corso, A. and Polini, C., Reduction number of links of irreducible varieties, J. Pure Appl. Algebra 121 (1997), 29–43.CrossRefGoogle Scholar
[CPV] Corso, A., Polini, C., and Vasconcelos, W. V., Links of prime ideals, Math. Proc. Cambridge Philos. Soc. 115 (1994), 431–436.CrossRefGoogle Scholar
[G1] Goto, S., On the associated graded rings of parameter ideals in Buchsbaum rings, J. Algebra 85 (1983), 490–534.CrossRefGoogle Scholar
[G2] Goto, S., Integral closedness of complete intersection ideals, J. Algebra 108 (1987), 151–160.CrossRefGoogle Scholar
[GKM] Goto, S., Kimura, S., and Matsuoka, N., Quasi-socle ideals in Gorenstein numerical semigroup rings, J. Algebra 320 (2008), 276–293.CrossRefGoogle Scholar
[GKMP] Goto, S., Kimura, S., Matsuoka, N., and Phuong, T. T., Quasi-socle ideals in local rings with Gorenstein tangent cones, J. Commut. Algebra 1 (2009), 603–620.CrossRefGoogle Scholar
[GKPT] Goto, S., Kimura, S., Phuong, T. T., and Truong, H. L., Quasi-socle ideals and Goto numbers of parameters, J. Pure Appl. Algebra 214 (2010), 501–511.CrossRefGoogle Scholar
[GMT] Goto, S., Matsuoka, N., and Takahashi, R., Quasi-socle ideals in a Gorenstein local ring, J. Pure Appl. Algebra 212 (2008), 969–980.CrossRefGoogle Scholar
[GN] Goto, S. and Nishida, K., Hilbert coefficients and Buchsbaumness of associated graded rings, J. Pure Appl. Algebra 181 (2003), 61–74.CrossRefGoogle Scholar
[GO] Goto, S. and Ozeki, K., The structure of Sally modules—towards a theory of non- Cohen-Macaulay cases, preprint, 2009.CrossRefGoogle Scholar
[GSa1] Goto, S. and Sakurai, H., The equality I2 = QI in Buchsbaum rings, Rend. Sem. Mat. Univ. Padova 110 (2003), 25–56.Google Scholar
[GSa2] Goto, S. and Sakurai, H., The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two, J. Math. Soc. Japan 56 (2004), 1157-1168.CrossRefGoogle Scholar
[GSa3] Goto, S. and Sakurai, H., When does the equality I2 = QI hold true in Buchsbaum rings?, Lect. Notes Pure Appl. Math. 244 (2006), 115–139.Google Scholar
[PU] Polini, C. and Ulrich, B., Linkage and reduction numbers, Math. Ann. 310 (1998), 631–651.CrossRefGoogle Scholar
[Sch] Schenzel, P., Multiplizitäten in verallgemeinerten Cohen-Macaulay-Moduln, Math. Nachr. 88 (1979), 295–306.CrossRefGoogle Scholar
[SV] Stückrad, J. and Vogel, W., Toward a theory of Buchsbaum singularities, Amer. J. Math. 100 (1978), 727–746.CrossRefGoogle Scholar
[T] Trung, N. V., Toward a theory of generalized Cohen-Macaulay modules, Nagoya Math. J. 102 (1986), 1–49.CrossRefGoogle Scholar
[VV] Valabrega, P. and Valla, G., Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93–101.CrossRefGoogle Scholar
[Wan] Wang, H.-J., Links of symbolic powers of prime ideals, Math. Z. 256 (2007), 749–756.CrossRefGoogle Scholar