Czechoslovak Mathematical Journal, first online, pp. 1-18

Monomial ideals with tiny squares and Freiman ideals

Ibrahim Al-Ayyoub, Mehrdad Nasernejad

Received March 23, 2020.   Published online February 11, 2021.

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).
Keywords:  Freiman ideal; number of generator; power of ideal; Ratliff-Rush closure
Classification MSC:  13E15, 13F20, 05E40
DOI:  10.21136/CMJ.2021.0124-20

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[1] I. Al-Ayyoub: An algorithm for computing the Ratliff-Rush closure. J. Algebra Appl. 8 (2009), 521-532. DOI 10.1142/s0219498809003473 | MR 2555518 | Zbl 1174.13011
[2] I. Al-Ayyoub: On the reduction numbers of monomial ideals. J. Algebra Appl. 19 (2020), Article ID 2050201, 27 pages. DOI 10.1142/S0219498820502011 | MR 4140141 | Zbl 07272746
[3] I. Al-Ayyoub, M. Jaradat, K. Al-Zoubi: A note on the ascending chain condition of ideals. J. Algebra Appl. 19 (2020), Article ID 2050135, 19 pages. DOI 10.1142/S0219498820501352 | MR 4129182 | Zbl 07227845
[4] W. Decker, G-M. Greuel, G. Pfister, H. Schönemann: Singular 4-0-2: A computer algebra system for polynomial computations. Available at (2015). SW 00866
[5] S. Eliahou, J. Herzog, M. M. Saem: Monomial ideals with tiny squares. J. Algebra 514 (2018), 99-112. DOI 10.1016/j.jalgebra.2018.07.037 | MR 3853060 | Zbl 1403.13033
[6] G. A. Freiman: Foundations of a Structural Theory of Set Addition. Translations of Mathematical Monographs 37. American Mathematical Society, Providence (1973). DOI 10.1090/mmono/037 | MR 0360496 | Zbl 0271.10044
[7] J. Herzog, T. Hibi: Monomial Ideals. Graduate Text in Mathematics 206. Springer, London (2011). DOI 10.1007/978-0-85729-106-6 | MR 2724673 | Zbl 1206.13001
[8] J. Herzog, A. A. Qureshi, M. M. Saem: The fiber cone of a monomial ideal in two variables. J. Symb. Comput. 94 (2019), 52-69. DOI 10.1016/j.jsc.2018.06.022 | MR 3945057 | Zbl 1430.13047
[9] J. Herzog, M. M. Saem, N. Zamani: The number of generators of powers of an ideal. Int. J. Algebra Comput. 29 (2019), 827-847. DOI 10.1142/s0218196719500309 | MR 3978117 | Zbl 1423.13105
[10] J. Herzog, G. Zhu: Freiman ideals. Commun. Algebra 47 (2019), 407-423. DOI 10.1080/00927872.2018.1477948 | MR 3924789 | Zbl 1410.13007
[11] I. Swanson, C. Huneke: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series 336. Cambridge University Press, Cambridge (2006). MR 2266432 | Zbl 1117.13001

Affiliations:   Ibrahim Al-Ayyoub (corresponding author), Department of Mathematics, Sultan Qaboos University, P.O.Box 31, Muscat, Oman, Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O.Box 3030, Irbid 22110, Jordan, e-mail:; Mehrdad Nasernejad, Department of Mathematics, Khayyam University, Mashhad, Iran, e-mail:

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