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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 7, Issue 4


Stanley depth of monomial ideals with small number of generators

Mircea Cimpoeaş
Published Online: 2009-10-31 | DOI: https://doi.org/10.2478/s11533-009-0037-0


For a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1 for any monomial ideal in I ⊂ K[x 1,x 2,x 3].

MSC: 13H10; 13P10

Keywords: Stanley depth; Monomial ideal

  • [1] Ahmad S., Popescu D., Sequentially Cohen-Macaulay monomial ideals of embedding dimension four, Bull. Math. Soc. Sci. Math. Roumanie, 2007, 50(98), 99–110 Google Scholar

  • [2] Anwar I., Janet’s algorithm, Bull. Math. Soc. Sci. Math. Roumanie, 2008, 51(99), 11–19 Google Scholar

  • [3] Anwar I., Popescu D., Stanley Conjecture in small embedding dimension, J. Algebra, 2007, 318, 1027–1031 http://dx.doi.org/10.1016/j.jalgebra.2007.06.005CrossrefWeb of ScienceGoogle Scholar

  • [4] Apel J., On a conjecture of R.P.Stanley, J. Algebraic Combin., 2003, 17, 36–59 Google Scholar

  • [5] Cimpoeas M., Stanley depth for monomial complete intersection, Bull. Math. Soc. Sci. Math. Roumanie, 2008, 51(99), 205–211 Google Scholar

  • [6] Cimpoeas M., Some remarks on the Stanley depth for multigraded modules, Le Mathematiche, 2008, LXIII, 165–171 Google Scholar

  • [7] Herzog J., Jahan A.S., Yassemi S., Stanley decompositions and partitionable simplicial complexes, J. Algebraic Combin., 2008, 27, 113–125 http://dx.doi.org/10.1007/s10801-007-0076-1CrossrefGoogle Scholar

  • [8] Herzog J., Vladoiu M., Zheng X., How to compute the Stanley depth of a monomial ideal, J. Algebra, doi:10:1016/j.jalgebra.2008.01.006, to appear Web of ScienceGoogle Scholar

  • [9] Jahan A.S., Prime filtrations of monomial ideals and polarizations, J. Algebra, 2007, 312, 1011–1032 http://dx.doi.org/10.1016/j.jalgebra.2006.11.002CrossrefGoogle Scholar

  • [10] Nasir S., Stanley decompositions and localization, Bull. Math. Soc. Sci. Math. Roumanie, 2008, 51(99), 151–158 Google Scholar

  • [11] Popescu D., Stanley depth of multigraded modules, J. Algebra, 2009, 321(10), 2782–2797 http://dx.doi.org/10.1016/j.jalgebra.2009.03.009CrossrefWeb of ScienceGoogle Scholar

  • [12] Rauf A., Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Soc. Sci. Math. Roumanie, 2007, 50(98), 347–354 Google Scholar

  • [13] Rauf A., Depth and Stanley depth of multigraded modules, Comm. Algebra, to appear Google Scholar

  • [14] Shen Y., Stanley depth of complete intersection monomial ideals and upper-discrete partitions, J. Algebra, 2009, 321, 1285–1292 http://dx.doi.org/10.1016/j.jalgebra.2008.11.010Web of ScienceCrossrefGoogle Scholar

About the article

Published Online: 2009-10-31

Published in Print: 2009-12-01

Citation Information: Open Mathematics, Volume 7, Issue 4, Pages 629–634, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-009-0037-0.

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© 2009 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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