| Title:
|
Monomial ideals with tiny squares and Freiman ideals (English) |
| Author:
|
Al-Ayyoub, Ibrahim |
| Author:
|
Nasernejad, Mehrdad |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
847-864 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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). (English) |
| Keyword:
|
Freiman ideal |
| Keyword:
|
number of generator |
| Keyword:
|
power of ideal |
| Keyword:
|
Ratliff-Rush closure |
| MSC:
|
05E40 |
| MSC:
|
13E15 |
| MSC:
|
13F20 |
| idZBL:
|
07396202 |
| idMR:
|
MR4295250 |
| DOI:
|
10.21136/CMJ.2021.0124-20 |
| . |
| Date available:
|
2021-08-02T08:09:01Z |
| Last updated:
|
2023-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149061 |
| . |
| Reference:
|
[1] Al-Ayyoub, I.: An algorithm for computing the Ratliff-Rush closure.J. Algebra Appl. 8 (2009), 521-532. Zbl 1174.13011, MR 2555518, 10.1142/s0219498809003473 |
| Reference:
|
[2] Al-Ayyoub, I.: On the reduction numbers of monomial ideals.J. Algebra Appl. 19 (2020), Article ID 2050201, 27 pages. Zbl 07272746, MR 4140141, 10.1142/S0219498820502011 |
| Reference:
|
[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. Zbl 07227845, MR 4129182, 10.1142/S0219498820501352 |
| Reference:
|
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| Reference:
|
[5] Eliahou, S., Herzog, J., Saem, M. M.: Monomial ideals with tiny squares.J. Algebra 514 (2018), 99-112. Zbl 1403.13033, MR 3853060, 10.1016/j.jalgebra.2018.07.037 |
| Reference:
|
[6] Freiman, G. A.: Foundations of a Structural Theory of Set Addition.Translations of Mathematical Monographs 37. American Mathematical Society, Providence (1973). Zbl 0271.10044, MR 0360496, 10.1090/mmono/037 |
| Reference:
|
[7] Herzog, J., Hibi, T.: Monomial Ideals.Graduate Text in Mathematics 206. Springer, London (2011). Zbl 1206.13001, MR 2724673, 10.1007/978-0-85729-106-6 |
| Reference:
|
[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. Zbl 1430.13047, MR 3945057, 10.1016/j.jsc.2018.06.022 |
| Reference:
|
[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. Zbl 1423.13105, MR 3978117, 10.1142/s0218196719500309 |
| Reference:
|
[10] Herzog, J., Zhu, G.: Freiman ideals.Commun. Algebra 47 (2019), 407-423. Zbl 1410.13007, MR 3924789, 10.1080/00927872.2018.1477948 |
| Reference:
|
[11] Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules.London Mathematical Society Lecture Note Series 336. Cambridge University Press, Cambridge (2006). Zbl 1117.13001, MR 2266432 |
| . |
