Geodesic
Monomial ideals with tiny squares and Freiman ideals
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 847-864.

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We provide a construction of monomial ideals in $R=K[x,y]$ such that $\mu (I^{2}) \nobreak \mu (I)$, 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).
DOI : 10.21136/CMJ.2021.0124-20
Classification : 05E40, 13E15, 13F20
Keywords: Freiman ideal; number of generator; power of ideal; Ratliff-Rush closure 
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Al-Ayyoub, Ibrahim; Nasernejad, Mehrdad. Monomial ideals with tiny squares and Freiman ideals. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 847-864. doi : 10.21136/CMJ.2021.0124-20. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2021.0124-20/

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