Geodesic
Monomial ideals with tiny squares and Freiman ideals
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 847-864
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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).
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 
@article{10_21136_CMJ_2021_0124_20,
     author = {Al-Ayyoub, Ibrahim and Nasernejad, Mehrdad},
     title = {Monomial ideals with tiny squares and {Freiman} ideals},
     journal = {Czechoslovak Mathematical Journal},
     pages = {847--864},
     year = {2021},
     volume = {71},
     number = {3},
     doi = {10.21136/CMJ.2021.0124-20},
     mrnumber = {4295250},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2021.0124-20/}
}
TY  - JOUR
AU  - Al-Ayyoub, Ibrahim
AU  - Nasernejad, Mehrdad
TI  - Monomial ideals with tiny squares and Freiman ideals
JO  - Czechoslovak Mathematical Journal
PY  - 2021
SP  - 847
EP  - 864
VL  - 71
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2021.0124-20/
DO  - 10.21136/CMJ.2021.0124-20
LA  - en
ID  - 10_21136_CMJ_2021_0124_20
ER  - 
%0 Journal Article
%A Al-Ayyoub, Ibrahim
%A Nasernejad, Mehrdad
%T Monomial ideals with tiny squares and Freiman ideals
%J Czechoslovak Mathematical Journal
%D 2021
%P 847-864
%V 71
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2021.0124-20/
%R 10.21136/CMJ.2021.0124-20
%G en
%F 10_21136_CMJ_2021_0124_20
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

[1] Al-Ayyoub, I.: An algorithm for computing the Ratliff-Rush closure. J. Algebra Appl. 8 (2009), 521-532. | DOI | MR | JFM

[2] Al-Ayyoub, I.: On the reduction numbers of monomial ideals. J. Algebra Appl. 19 (2020), Article ID 2050201, 27 pages. | DOI | MR | JFM

[3] Al-Ayyoub, I., Jaradat, M., Al-Zoubi, K.: A note on the ascending chain condition of ideals. J. Algebra Appl. 19 (2020), Article ID 2050135, 19 pages. | DOI | MR | JFM

[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). | MR

[5] Eliahou, S., Herzog, J., Saem, M. M.: Monomial ideals with tiny squares. J. Algebra 514 (2018), 99-112. | DOI | MR | JFM

[6] Freiman, G. A.: Foundations of a Structural Theory of Set Addition. Translations of Mathematical Monographs 37. American Mathematical Society, Providence (1973). | DOI | MR | JFM

[7] Herzog, J., Hibi, T.: Monomial Ideals. Graduate Text in Mathematics 206. Springer, London (2011). | DOI | MR | JFM

[8] Herzog, J., Qureshi, A. A., Saem, M. M.: The fiber cone of a monomial ideal in two variables. J. Symb. Comput. 94 (2019), 52-69. | DOI | MR | JFM

[9] Herzog, J., Saem, M. M., Zamani, N.: The number of generators of powers of an ideal. Int. J. Algebra Comput. 29 (2019), 827-847. | DOI | MR | JFM

[10] Herzog, J., Zhu, G.: Freiman ideals. Commun. Algebra 47 (2019), 407-423. | DOI | MR | JFM

[11] Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series 336. Cambridge University Press, Cambridge (2006). | MR | JFM

Cité par Sources :