We provide a construction of monomial ideals in R = K[x, y] such that μ(I²) < 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).

我们在R = K [ x，y ]中提供单项式理想的构造，以使μ（I ²）< μ（I），其中μ表示生成器的最少数量。该构造概括了S.Eliahou，J.Herzog，M.Mohammadi Saem（2018）的主要结果。在R环中，我们概括了弗莱曼理想的定义，该定义在J.Herzog，G.Zhu（2019）中引入，然后我们对这种理想进行了完整的描述。这种表征的特殊情况导致对μ（I ^k）概括了S.Eliahou，J.Herzog，M.Mohammadi Saem（2018），J.Herzog，M.Mohammadi Saem，N.Zamani（2019）和J.Herzog，A.Asloob Qureshi，M。 Mohammadi Saem（2019）。