Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if AssR(R/Ik) ⊆ AssR(R/Ik+1) for all positive integers k, where AssR(R/I ) denotes the set of associated prime ideals of I. In addition, an ideal I has the strong persistence property if (Ik+1: RI) = Ik for all positive integers k. Also, an ideal I is called normally torsion-free if AssR(R/Ik) ⊆ AssR(R/I) for all positive integers k. In this paper, we collect the latest results in associated primes of powers of monomial ideals in three concepts, i.e., the persistence property, strong persistence property, and normally torsion-freeness. Also, we present some classes of monomial ideals such that are none of edge ideals, cover ideals, and polymatroidal ideals, but satisfy the persistence property and strong persistence property. In particular, we study the Alexander dual of path ideals of unrooted starlike trees. Furthermore, we probe the normally torsion-freeness of the Alexander dual of some path ideals which are related to banana trees. We close this paper with exploring the normally torsion-freeness under some monomial operations.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 6, Issue 1) |
| DOI | 10.11648/j.ijtam.20200601.11 |
| Page(s) | 1-13 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Associated Prime Ideals, Powers of Ideals, Monomial Ideals, Persistence Property,Strong Persistence Property, Normally Torsion-free
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APA Style
Mehrdad Nasernejad. (2019). Associated Primes of Powers of Monomial Ideals: A Survey. International Journal of Theoretical and Applied Mathematics, 6(1), 1-13. https://doi.org/10.11648/j.ijtam.20200601.11
ACS Style
Mehrdad Nasernejad. Associated Primes of Powers of Monomial Ideals: A Survey. Int. J. Theor. Appl. Math. 2019, 6(1), 1-13. doi: 10.11648/j.ijtam.20200601.11
AMA Style
Mehrdad Nasernejad. Associated Primes of Powers of Monomial Ideals: A Survey. Int J Theor Appl Math. 2019;6(1):1-13. doi: 10.11648/j.ijtam.20200601.11
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Download@article{10.11648/j.ijtam.20200601.11,
author = {Mehrdad Nasernejad},
title = {Associated Primes of Powers of Monomial Ideals: A Survey},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {6},
number = {1},
pages = {1-13},
doi = {10.11648/j.ijtam.20200601.11},
url = {https://doi.org/10.11648/j.ijtam.20200601.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20200601.11},
abstract = {Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if AssR(R/Ik) ⊆ AssR(R/Ik+1) for all positive integers k, where AssR(R/I ) denotes the set of associated prime ideals of I. In addition, an ideal I has the strong persistence property if (Ik+1: RI) = Ik for all positive integers k. Also, an ideal I is called normally torsion-free if AssR(R/Ik) ⊆ AssR(R/I) for all positive integers k. In this paper, we collect the latest results in associated primes of powers of monomial ideals in three concepts, i.e., the persistence property, strong persistence property, and normally torsion-freeness. Also, we present some classes of monomial ideals such that are none of edge ideals, cover ideals, and polymatroidal ideals, but satisfy the persistence property and strong persistence property. In particular, we study the Alexander dual of path ideals of unrooted starlike trees. Furthermore, we probe the normally torsion-freeness of the Alexander dual of some path ideals which are related to banana trees. We close this paper with exploring the normally torsion-freeness under some monomial operations.},
year = {2019}
}
TY - JOUR T1 - Associated Primes of Powers of Monomial Ideals: A Survey AU - Mehrdad Nasernejad Y1 - 2019/12/30 PY - 2019 N1 - https://doi.org/10.11648/j.ijtam.20200601.11 DO - 10.11648/j.ijtam.20200601.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 1 EP - 13 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20200601.11 AB - Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if AssR(R/Ik) ⊆ AssR(R/Ik+1) for all positive integers k, where AssR(R/I ) denotes the set of associated prime ideals of I. In addition, an ideal I has the strong persistence property if (Ik+1: RI) = Ik for all positive integers k. Also, an ideal I is called normally torsion-free if AssR(R/Ik) ⊆ AssR(R/I) for all positive integers k. In this paper, we collect the latest results in associated primes of powers of monomial ideals in three concepts, i.e., the persistence property, strong persistence property, and normally torsion-freeness. Also, we present some classes of monomial ideals such that are none of edge ideals, cover ideals, and polymatroidal ideals, but satisfy the persistence property and strong persistence property. In particular, we study the Alexander dual of path ideals of unrooted starlike trees. Furthermore, we probe the normally torsion-freeness of the Alexander dual of some path ideals which are related to banana trees. We close this paper with exploring the normally torsion-freeness under some monomial operations. VL - 6 IS - 1 ER -
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