대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제61권1호
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  • Pages.147-160
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  • 2024
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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DEPTH AND STANLEY DEPTH OF TWO SPECIAL CLASSES OF MONOMIAL IDEALS

  • Xiaoqi Wei (School of Mathematics and Physics Jiangsu University of Technology)
  • 투고 : 2023.02.08
  • 심사 : 2023.08.02
  • 발행 : 2024.01.31
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초록

In this paper, we define two new classes of monomial ideals I𝑙,d and Jk,d. When d ≥ 2k + 1 and 𝑙 ≤ d - k - 1, we give the exact formulas to compute the depth and Stanley depth of quotient rings S/It𝑙,d for all t ≥ 1. When d = 2k = 2𝑙, we compute the depth and Stanley depth of quotient ring S/I𝑙,d. When d ≥ 2k, we also compute the depth and Stanley depth of quotient ring S/Jk,d.

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과제정보

The author thanks the referee for his or her carefully reading this manuscript. Also the author would like to thank Professor Shengli Tan and Dr. Hong Wang for their helpful comments.

참고문헌

  1. I. Anwar and D. Popescu, Stanley conjecture in small embedding dimension, J. Algebra 318 (2007), no. 2, 1027-1031. https://doi.org/10.1016/j.jalgebra.2007.06.005 
  2. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge Univ. Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511608681 
  3. M. Cimpoeas, Stanley depth of monomial ideals with small number of generators, Cent. Eur. J. Math. 7 (2009), no. 4, 629-634. https://doi.org/10.2478/s11533-009-0037-0 
  4. M. Cimpoea,s, On the Stanley depth of powers of some classes of monomial ideals, Bull. Iranian Math. Soc. 44 (2018), no. 3, 739-747. https://doi.org/10.1007/s41980-018-0049-2 
  5. A. M. Duval, B. Goeckner, C. J. Klivans, and J. L. Martin, A non-partitionable Cohen-Macaulay simplicial complex, Adv. Math. 299 (2016), 381-395. https://doi.org/10.1016/j.aim.2016.05.011 
  6. J. Herzog, A survey on Stanley depth, in Monomial ideals, computations and applications, 3-45, Lecture Notes in Math., 2083, Springer, Heidelberg. https://doi.org/10.1007/978-3-642-38742-5_1 
  7. J. Herzog, M. Vladoiu, and X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra 322 (2009), no. 9, 3151-3169. https://doi.org/10.1016/j.jalgebra.2008.01.006 
  8. Z. Iqbal, M. Ishaq, and M. Aamir, Depth and Stanley depth of the edge ideals of square paths and square cycles, Comm. Algebra 46 (2018), no. 3, 1188-1198. https://doi.org/10.1080/00927872.2017.1339068 
  9. Z. Iqbal, M. Ishaq, and M. A. Binyamin, Depth and Stanley depth of the edge ideals of the strong product of some graphs, Hacet. J. Math. Stat. 50 (2021), no. 1, 92-109. https://doi.org/10.15672/hujms.638033 
  10. M. T. Keller and S. J. Young, Combinatorial reductions for the Stanley depth of I and S/I, Electron. J. Combin. 24 (2017), no. 3, Paper No. 3.48, 19 pp. https://doi.org/10.37236/6783 
  11. R. Okazaki, A lower bound of Stanley depth of monomial ideals, J. Commut. Algebra 3 (2011), no. 1, 83-88. https://doi.org/10.1216/JCA-2011-3-1-83 
  12. D. Popescu and M. I. Qureshi, Computing the Stanley depth, J. Algebra 323 (2010), no. 10, 2943-2959. https://doi.org/10.1016/j.jalgebra.2009.11.025 
  13. A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra 38 (2010), no. 2, 773-784. https://doi.org/10.1080/00927870902829056 
  14. S. A. Seyed Fakhari, On the Stanley depth of powers of monomial ideals, Mathematics 7 (2019), no. 7, 607. https://doi.org/10.3390/math7070607 
  15. Y.-H. Shen, Stanley depth of complete intersection monomial ideals and upper-discrete partitions, J. Algebra 321 (2009), no. 4, 1285-1292. https://doi.org/10.1016/j.jalgebra.2008.11.010 
  16. R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175-193. https://doi.org/10.1007/BF01394054 
  17. R. H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, 238, Marcel Dekker, Inc., New York, 2001. 
  18. X. Wei and Y. Gu, Depth and Stanley depth of the facet ideals of some classes of simplicial complexes, Czechoslovak Math. J. 67 (2017), no. 3, 753-766. https://doi.org/10.21136/CMJ.2017.0172-16 
  19. G. J. Zhu, Depth and Stanley depth of the edge ideals of some m-line graphs and m-cyclic graphs with a common vertex, Rom. J. Math. Comput. Sci. 5 (2015), no. 2, 118-129.