Czechoslovak Mathematical Journal, Vol. 71, No. 3, pp. 847-864, 2021
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
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: iayyoub@just.edu.jo; Mehrdad Nasernejad, Department of Mathematics, Khayyam University, Mashhad, Iran, e-mail: m.nasernejad@khayyam.ac.ir
