Monomial ideals with tiny squares and Freiman ideals

Ibrahim Al-Ayyoub; Mehrdad Nasernejad

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 847-864
  • ISSN: 0011-4642

Abstract

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We provide a construction of monomial ideals in R = K [ x , y ] such that μ ( I 2 ) < μ ( I ) , where μ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring R , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on μ ( I k ) that generalize some results of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018), J. Herzog, M. Mohammadi Saem, N. Zamani (2019), and J. Herzog, A. Asloob Qureshi, M. Mohammadi Saem (2019).

How to cite

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Al-Ayyoub, Ibrahim, and Nasernejad, Mehrdad. "Monomial ideals with tiny squares and Freiman ideals." Czechoslovak Mathematical Journal 71.3 (2021): 847-864. <http://eudml.org/doc/297870>.

@article{Al2021,
abstract = {We provide a construction of monomial ideals in $R=K[x,y]$ such that $\mu (I^\{2\})< \mu (I)$, where $\mu $ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$, we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $\mu (I^\{k\})$ that generalize some results of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018), J. Herzog, M. Mohammadi Saem, N. Zamani (2019), and J. Herzog, A. Asloob Qureshi, M. Mohammadi Saem (2019).},
author = {Al-Ayyoub, Ibrahim, Nasernejad, Mehrdad},
journal = {Czechoslovak Mathematical Journal},
keywords = {Freiman ideal; number of generator; power of ideal; Ratliff-Rush closure},
language = {eng},
number = {3},
pages = {847-864},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Monomial ideals with tiny squares and Freiman ideals},
url = {http://eudml.org/doc/297870},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Al-Ayyoub, Ibrahim
AU - Nasernejad, Mehrdad
TI - Monomial ideals with tiny squares and Freiman ideals
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 847
EP - 864
AB - We provide a construction of monomial ideals in $R=K[x,y]$ such that $\mu (I^{2})< \mu (I)$, where $\mu $ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$, we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $\mu (I^{k})$ that generalize some results of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018), J. Herzog, M. Mohammadi Saem, N. Zamani (2019), and J. Herzog, A. Asloob Qureshi, M. Mohammadi Saem (2019).
LA - eng
KW - Freiman ideal; number of generator; power of ideal; Ratliff-Rush closure
UR - http://eudml.org/doc/297870
ER -

References

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  1. Al-Ayyoub, I., 10.1142/s0219498809003473, J. Algebra Appl. 8 (2009), 521-532. (2009) Zbl1174.13011MR2555518DOI10.1142/s0219498809003473
  2. Al-Ayyoub, I., 10.1142/S0219498820502011, J. Algebra Appl. 19 (2020), Article ID 2050201, 27 pages. (2020) Zbl07272746MR4140141DOI10.1142/S0219498820502011
  3. Al-Ayyoub, I., Jaradat, M., Al-Zoubi, K., 10.1142/S0219498820501352, J. Algebra Appl. 19 (2020), Article ID 2050135, 19 pages. (2020) Zbl07227845MR4129182DOI10.1142/S0219498820501352
  4. Decker, W., Greuel, G-M., Pfister, G., Schönemann, H., Singular 4-0-2: A computer algebra system for polynomial computations, Available at http://www.singular.uni-kl.de (2015). (2015) MR1413182
  5. Eliahou, S., Herzog, J., Saem, M. M., 10.1016/j.jalgebra.2018.07.037, J. Algebra 514 (2018), 99-112. (2018) Zbl1403.13033MR3853060DOI10.1016/j.jalgebra.2018.07.037
  6. Freiman, G. A., 10.1090/mmono/037, Translations of Mathematical Monographs 37. American Mathematical Society, Providence (1973). (1973) Zbl0271.10044MR0360496DOI10.1090/mmono/037
  7. Herzog, J., Hibi, T., 10.1007/978-0-85729-106-6, Graduate Text in Mathematics 206. Springer, London (2011). (2011) Zbl1206.13001MR2724673DOI10.1007/978-0-85729-106-6
  8. Herzog, J., Qureshi, A. A., Saem, M. M., 10.1016/j.jsc.2018.06.022, J. Symb. Comput. 94 (2019), 52-69. (2019) Zbl1430.13047MR3945057DOI10.1016/j.jsc.2018.06.022
  9. Herzog, J., Saem, M. M., Zamani, N., 10.1142/s0218196719500309, Int. J. Algebra Comput. 29 (2019), 827-847. (2019) Zbl1423.13105MR3978117DOI10.1142/s0218196719500309
  10. Herzog, J., Zhu, G., 10.1080/00927872.2018.1477948, Commun. Algebra 47 (2019), 407-423. (2019) Zbl1410.13007MR3924789DOI10.1080/00927872.2018.1477948
  11. Swanson, I., Huneke, C., Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series 336. Cambridge University Press, Cambridge (2006). (2006) Zbl1117.13001MR2266432

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