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February 2022 On the embedded associated primes of monomial ideals
Mirsadegh Sayedsadeghi, Mehrdad Nasernejad, Ayesha Asloob Qureshi
Rocky Mountain J. Math. 52(1): 275-287 (February 2022). DOI: 10.1216/rmj.2022.52.275

Abstract

Let I be a square-free monomial ideal in a polynomial ring R=K[x1,,xn] over a field K, 𝔪=(x1,,xn) be the graded maximal ideal of R, and {u1,,uβ1(I)} be a maximal independent set of minimal generators of I such that 𝔪xiAss(R(Ixi)t) for all xii=1β1(I)ui and some positive integer t, where Ixi denotes the deletion of I at xi and β1(I) denotes the maximum cardinality of an independent set in I. We prove that if 𝔪Ass(RIt), then tβ1(I)+1. As an application, we verify that under certain conditions, every unmixed Kőnig ideal is normally torsion-free, and so has the strong persistence property. In addition, we show that every square-free transversal polymatroidal ideal is normally torsion-free. Next, we state some results on the corner elements of monomial ideals. In particular, we prove that if I is a monomial ideal in a polynomial ring R=K[x1,,xn] over a field K and z is an It corner element for some positive integer t such that 𝔪xiAss(Ixi)t for some 1in, then xi divides z.

Citation

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Mirsadegh Sayedsadeghi. Mehrdad Nasernejad. Ayesha Asloob Qureshi. "On the embedded associated primes of monomial ideals." Rocky Mountain J. Math. 52 (1) 275 - 287, February 2022. https://doi.org/10.1216/rmj.2022.52.275

Information

Received: 3 September 2020; Revised: 24 March 2021; Accepted: 30 March 2021; Published: February 2022
First available in Project Euclid: 19 April 2022

Digital Object Identifier: 10.1216/rmj.2022.52.275

Subjects:
Primary: 13B25 , 13F20

Keywords: Associated primes , corner elements , König ideals , normally torsion-free ideals , strong persistence property

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

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Vol.52 • No. 1 • February 2022
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