Associated Primes of Powers of Monomial Ideals: A Survey

Mehrdad Nasernejad

doi:10.11648/j.ijtam.20200601.11

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Associated Primes of Powers of Monomial Ideals: A Survey

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International Journal of Theoretical and Applied Mathematics
Volume 6, Issue 1, February 2020, Pages: 1-13
Received: Oct. 21, 2019; Accepted: Nov. 12, 2019; Published: Dec. 30, 2019

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Author

Mehrdad Nasernejad, Department of Mathematics, Khayyam University, Mashhad, Iran

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Abstract

Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if Ass_R(R/I^k) ⊆ Ass_R(R/I^k+1) for all positive integers k, where Ass_R(R/I ) denotes the set of associated prime ideals of I. In addition, an ideal I has the strong persistence property if (I^k+1: _RI) = I^k for all positive integers k. Also, an ideal I is called normally torsion-free if Ass_R(R/I^k) ⊆ Ass_R(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.

Keywords

Associated Prime Ideals, Powers of Ideals, Monomial Ideals, Persistence Property,Strong Persistence Property, Normally Torsion-free

To cite this article

Mehrdad Nasernejad, Associated Primes of Powers of Monomial Ideals: A Survey, International Journal of Theoretical and Applied Mathematics. Vol. 6, No. 1, 2020, pp. 1-13. doi: 10.11648/j.ijtam.20200601.11

Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

[1]

Brodmann M. Asymptotic stability of Ass(M=IⁿM). Proc. Amer. Math. Soc. 1979; 74:16-18. doi: 10.2307/2042097.

[2]

Hoa L. T. Stability of associated primes of monomial ideals. Vietnam J. Math. 2006; 34: 473-487.

[3]

Khashyarmanesh K, Nasernejad M. On the stable set of associated prime ideals of monomial ideals and squarefree monomial ideals. Comm. Algebra. 2014; 42: 3753-3759. doi: 10.1080/00927872.2013.793696.

[4]

Khashyarmanesh K, Nasernejad M. Some results on the associated primes of monomial ideals. Southeast Asian Bull. Math. 2015; 39: 439-451.

[5]

McAdam S. Asymptotic Prime Divisors. Lecture Notes in Mathematics. 103. New York: Springer-Verlag; 1983. 120 p.

[6]

Herzog J, Hibi T. The depth of powers of an ideal. J. Algebra 2005; 291: 534-550. doi: 10.1016/j.jalgebra.2005.04.007.

[7]

Mart_inez-Bernal J, Morey S, Villarreal R. H. Associated primes of powers of edge ideals. J. Collect. Math. 2012; 63: 361-374. doi: 10.1007/s13348-011-0045-9.

[8]

Morey S, Villarreal R. H. Edge ideals: Algebraic and combinatorial properties, Progress in commutative Algebra. 2012; 1: 85-126.

[9]

Kaiser T, Stehl_ik M, _ Skrekovski R. Replication in critical graphs and the persistence of monomial ideals. J. Comb. Theory Ser. A. 2014; 123: 239-251. doi: 10.1016/j.jcta.2013.12.005.

[10]

Ratliff L. J. Jr. On prime divisors of In, n large. Michigan Math. J. 1976; 23: 337-352. doi: 10.1307/mmj/1029001769.

[11]

Reyes E, Toledo J. On the strong persistence property for monomial ideals. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 2017; 60(108): 293-305.

[12]

Herzog J, Rauf A, Vladoiu M. The stable set of associated prime ideals of a polymatroidal ideal. J. Algebraic Combin. 2013; 37: 289-312. doi: 10.1007/s10801-012-0367-z.

[13]

Francisco C. A, H`a H. T, Van Tuyl A. Colorings of hypergraphs, perfect graphs and associated primes of powers of monomial ideals. J. Algebra. 2011; 331: 224-242. doi: 10.1016/j.jalgebra.2010.10.025.

[14]

Simis A, Vasconcelos W, Villarreal R. H. On the ideal theory of graphs. J. Algebra. 1994; 167: 389-416. doi: 10.1006/jabr.1994.1192.

[15]

Gitler I, Reyes E, Villarreal R. H. Blowup algebras of ideals of vertex covers of bipartite graphs. Contemp. Math. 2005; 376: 273-279.

[16]

Hungerford T. W. Algebra,. Graduate Texts in Mathematics. 73. New York: Springer-Verlag, 1974. 504p.

[17]

Nasernejad M. A new proof of the persistence property for ideals in Dedekind rings and Pr¨ufer domains. J. Algebr. Syst. 2013; 1: 91-100. doi: 10.22044/jas.2014.229.

[18]

Nasernejad M. Persistence property for associated primes of a family of ideals. Southeast Asian Bull. Math. 2016; 40: 571-575.

[19]

Villarreal R. H. Cohen-Macaulay graphs. Manuscripta Math. 1990; 66: 277-293. doi: 10.1007/BF02568497.

[20]

Conca A, De Negri E. M-sequences, graph ideals, and ladder ideals of linear type. J. Algebra. 1999; 211: 599-624. doi:10.1006/jabr.1998.7740.

[21]

Nasernejad M. Asymptotic behaviour of assicated primes of monomial ideals with combinatorial applications. J. Algebra Relat. Topics. 2014; 2: 15-25.

[22]

Nasernejad M, Khashyarmanesh K. On the Alexander dual of the path ideals of rooted and unrooted trees. Comm. Algebra. 2017; 45: 1853-1864. doi: 10.1080/00927872.2016.1226855.

[23]

Bhat A, Biermann J, Van Tuyl A. Generalized cover ideals and the persistence property. J. Pure Appl. Algebra. 2014; 218: 1683-1695. doi: 10.1016/j.jpaa.2014.01.007.

[24]

Bayati S, Herzog J. Expansions of monomial ideals and multigraded modules. Rocky Mountain J. Math. 2014; 44: 1781-1804. doi: 10.1216/RMJ-2014-44-6-1781.

[25]

Nasernejad M. Persistence property for some classes of monomial ideals of a polynomial ring. J. Algebra Appl. 2017; 16: 1750105 (17 pages). doi: 10.1142/S0219498817501055.

[26]

Nasernejad M, Rajaee S. Detecting the maximalassociated prime ideal of monomial ideals. Bol. Soc. Mat. Mex. doi: 10.1007/s40590-018-0208-8.

[27]

Rajaee S, Nasernejad M, Al-Ayyoub I. Superficial ideals for monomial ideals. J. Algebra Appl. 2018; 17: 1850102 (28 pages). doi: 10.1142/S0219498818501025.

[28]

Nasernejad M, Khashyarmanesh K, Al-Ayyoub I. Associated primes of powers of cover ideals under graph operations. Comm. Algebra. doi: 10.1080/00927872.2018.1527920.

[29]

Sayedsadeghi M, Nasernejad M. Normally torsionfreeness of monomial ideals under monomial operators. Comm. Algebra. doi: 10.1080/00927872.2018.1469029.

[30]

Chen W. C, Lu H. I, and Yeh Y. N. Operations of Interlaced Trees and Graceful Trees. Southeast Asian Bull. Math. 1997; 21: 337-348.

[31]

Khashyarmanesh K, Nasernejad M. A note on the Alexander dual of path ideals of rooted trees. Comm. Algebra. 2018; 46: 283-289. doi: 10.1080/00927872.2017.1324867.

[32]

Herzog J, Qureshi A. A. Persistence and stability properties of powers of ideals. J. Pure Appl. Algebra. 2015; 219: 530-542. doi: 10.1016/j.jpaa.2014.05.011.

[33]

H`a H. T, Morey S. Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra. 2010; 214: 301-308. doi: 10.1016/j.jpaa.2009.05.002.

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