Associated Primes of Powers of Monomial Ideals: A Survey
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 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.
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
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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.
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