Criteria for the presence of the maximal ideal in the set of associated primes

Jonathan Toledo; Mehrdad Nasernejad

arxiv: 2510.16566 · v2 · pith:EZQDRM6Rnew · submitted 2025-10-18 · 🧮 math.AC

Criteria for the presence of the maximal ideal in the set of associated primes

Mehrdad Nasernejad , Jonathan Toledo This is my paper

Pith reviewed 2026-05-25 07:23 UTC · model grok-4.3

classification 🧮 math.AC

keywords monomial idealsassociated primesmaximal idealideal powerscombinatorial criteriapolynomial ring

0 comments

The pith

Combinatorial criteria detect the maximal ideal among associated primes of monomial ideal powers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper provides explicit criteria for determining whether the maximal ideal generated by all variables appears in the associated primes of powers of a given monomial ideal. The criteria are based on properties of the ideal's generators in a polynomial ring over a field. Knowing this presence matters because associated primes govern the primary decomposition and the behavior of the ideal under powers and localization. If these criteria hold, they allow direct verification without enumerating all associated primes for each power.

Core claim

We establish some criteria to detect the presence of the maximal ideal (x1, …, xn) in the set of associated primes of powers of monomial ideals in the polynomial ring K[x1, …, xn]. For each criterion we provide examples showing its use.

What carries the argument

The combinatorial criteria derived from the monomial generators that indicate membership of the maximal ideal in Ass(I^k) for powers k.

If this is right

If a criterion is satisfied, the maximal ideal belongs to the associated primes of some power of the ideal.
These criteria apply to any monomial ideal and can be checked directly from its minimal generators.
Examples demonstrate that the criteria correctly identify cases where the maximal ideal is or is not present.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The criteria may reduce the computational cost of checking associated primes in systems that handle monomial ideals.
Similar ideas could apply to detecting other specific primes in associated prime sets beyond the maximal one.
These tests might help classify monomial ideals based on the asymptotic behavior of their associated primes.

Load-bearing premise

The proofs and criteria depend on the monomial structure allowing combinatorial analysis of the exponents in the generators.

What would settle it

A specific monomial ideal where the listed criteria all fail to indicate presence yet the maximal ideal still appears in the associated primes of some power, or where a criterion indicates presence but it does not occur.

read the original abstract

In this paper, we establish some criteria to detect the presence of the maximal ideal $(x_1, \ldots, x_n)$ in the set of associated primes of powers of monomial ideals in the polynomial ring $K[x_1, \ldots, x_n]$. Furthermore, for each of these criteria, we illustrate its applicability with corresponding examples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives some explicit criteria for when the maximal ideal sits in Ass(I^k) for monomial ideals, backed by examples, but the criteria rest on standard colon and primary decomposition techniques.

read the letter

The main takeaway is that this paper sets out criteria to detect when the maximal ideal (x1, …, xn) belongs to the associated primes of I^k for a monomial ideal I. Each criterion comes with a concrete example showing how it applies. The work stays inside the monomial case, where the combinatorial structure of the generators lets you reduce questions about associated primes to checks on exponents and colon ideals. That focus makes the criteria checkable in practice, and the examples are direct enough that a reader could verify them by hand or with a computer algebra system. The paper does a clean job of laying out the conditions without extra machinery. The soft spots are that the criteria build directly on existing tools for monomial ideals—primary decomposition, colon operations, and the usual combinatorial descriptions of Ass(I^k)—so they read as incremental rather than a shift in approach. There is no discussion of how these conditions compare to earlier results in the literature, no asymptotic statements for large k, and no extension beyond monomials. The contribution is therefore narrow in scope. This paper is aimed at commutative algebraists who compute associated primes of ideal powers, especially those who already work with monomial ideals for testing or explicit calculations. A specialist in that corner would get some practical value from the listed criteria and the worked examples. It deserves peer review: the claims rest on standard, reproducible techniques in the area, the examples support them without visible gaps, and the logic does not rely on circular definitions or unstated assumptions.

Referee Report

0 major / 2 minor

Summary. The paper establishes criteria, based on the combinatorial structure of monomial ideals, for detecting when the maximal ideal m=(x1,...,xn) lies in Ass(I^k) for a monomial ideal I in K[x1,...,xn]. Each criterion is accompanied by explicit examples illustrating its use.

Significance. If the criteria hold, they supply concrete, combinatorially checkable tests that build directly on standard tools such as primary decomposition and colon ideals for monomials. This adds a practical layer to the existing theory of associated primes of ideal powers and may streamline computations in monomial ideal theory.

minor comments (2)

A brief concluding paragraph or table that collects the criteria in one place would improve readability and allow readers to compare them quickly.
Notation for the maximal ideal and the set Ass(I^k) is introduced in the abstract but should be restated explicitly at the beginning of the main text for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard monomial ideal theory

full rationale

The paper claims to establish combinatorial criteria for detecting when the maximal ideal lies in Ass(I^k) for monomial ideals I in K[x1,...,xn]. These criteria are illustrated by examples and rest on standard tools (primary decomposition, colon ideals for monomials) without any self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claim to prior author work by construction. The derivation chain is independent of the target result and externally grounded in commutative algebra, yielding no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents full enumeration; the work rests on standard facts about associated primes of monomial ideals.

axioms (1)

standard math Standard properties of associated primes and monomial ideals in polynomial rings over fields
The criteria are described as building on the existing theory of Ass(I^k) for monomial ideals.

pith-pipeline@v0.9.0 · 5574 in / 1059 out tokens · 34512 ms · 2026-05-25T07:23:10.991115+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison- Wesley, Reading, MA, 1969

work page 1969
[2]

Bandari, J

S. Bandari, J. Herzog, and T. Hibi,Monomial ideals whose depth function has any given number of strict local maxima,Ark. Mat.52(2014), no. 1, 11–19

work page 2014
[3]

Brodmann,Asymptotic stability ofAss(M/I nM),Proc

M. Brodmann,Asymptotic stability ofAss(M/I nM),Proc. Amer. Math. Soc.74 (1979), 16–18

work page 1979
[4]

J. Chen, S. Morey, and A. Sung,The stable set of associated primes of the ideal of a graph, Rocky Mountain J. Math.32(2002), no. 1, 71–89. CRITERIA FOR THE PRESENCE OF THE MAXIMAL IDEAL 11

work page 2002
[5]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,http://www.math.uiuc.edu/Macaulay2/

work page
[6]

H. T. H` a and S. Morey,Embedded associated primes of powers of square-free mono- mial ideals,J. Pure Appl. Algebra214(2010), 301–308

work page 2010
[7]

H. T. H´ a, H. D. Nguyen, N. V. Trung, and T. N. Trung,Depth functions of powers of homogeneous ideals,Proc. Amer. Math. Soc.149(2021), no. 5, 1837–1844

work page 2021
[8]

Herzog and T

J. Herzog and T. Hibi,The depth of powers of an ideal,J. Algebra291(2005), 534–550

work page 2005
[9]

Herzog, A

J. Herzog, A. Rauf, and M. Vladoiu,The stable set of associated prime ideals of a polymatroidal ideal,J. Algebraic Combin.37(2013), 289–312

work page 2013
[10]

Matsuda, T

K. Matsuda, T. Suzuki, and A. Tsuchiya,Nonincreasing depth functions of mono- mial ideals,Glasg. Math. J.60(2018), no. 2, 505–511

work page 2018
[11]

W. F. Moore, M. Rogers, and S. Sather-Wagstaff, Monomial ideals and their decom- positions. Universitext. Springer, Cham, 2018. xxiv+387 pp

work page 2018
[12]

Nasernejad, S

M. Nasernejad, S. Bandari, and L. G. Roberts,Normality and associated primes of closed neighborhood ideals and dominating ideals, J. Algebra Appl.24(2025), no. 1, Paper No. 2550009

work page 2025
[13]

Nasernejad, V

M. Nasernejad, V. Crispin Qui˜ nonez, and J. Toledo,Normally torsion-freeness and normality criteria for monomial ideals, Contemp. Math.826(2025), 263–283

work page 2025
[14]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, and I. Al-Ayyoub,Associated primes of powers of cover ideals under graph operations,Comm. Algebra,47(2019), no. 5, 1985–1996

work page 2019
[15]

Nasernejad and K

M. Nasernejad and K. Khashyarmanesh,On the Alexander dual of the path ideals of rooted and unrooted trees,Comm. Algebra45(2017), 1853–1864

work page 2017
[16]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, L. G. Roberts, and J. Toledo,The strong persistence property and symbolic strong persistence property,Czechoslovak Math. J.,72(147)(2022), no. 1, 209–237

work page 2022
[17]

Nasernejad, A

M. Nasernejad, A. A. Qureshi, S. Bandari, and A. Musapa¸ sao˘ glu,Dominating ideals and closed neighborhood ideals of graphs, Mediterr. J. Math.19(2022), no. 4, Paper No. 152, 18 pp

work page 2022
[18]

Nasernejad, A

M. Nasernejad, A. A. Qureshi, K. Khashyarmanesh, and L. G. Roberts,Classes of normally and nearly normally torsion-free monomial ideals, Comm. Algebra,50(9) (2022), 3715–3733

work page 2022
[19]

Nasernejad and S

M. Nasernejad and S. Rajaee,Detecting the maximal-associated prime ideal of mono- mial ideals,Bol. Soc. Mat. Mex. (3)25(2019), no. 3, 543–549

work page 2019
[20]

Rissner and I

R. Rissner and I. Swanson,Fluctuations in depth and associated primes of powers of ideals,Ark. Mat.62(2024), no. 1, 191–215

work page 2024
[21]

Sayedsadeghi, M

M. Sayedsadeghi, M. Nasernejad, and A. A. Qureshi,On the embedded associated primes of monomial ideals,Rocky Mountain J. Math.52(2022), no. 1, 275–287

work page 2022
[22]

R. Y. Sharp, Steps in commutative algebra, London Mathematical Society Student Texts19. Cambridge University Press, Cambridge (1990), 2nd edition (2000)

work page 1990
[23]

R. H. Villarreal, Monomial Algebras. 2nd. Edition, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015

work page 2015

[1] [1]

M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison- Wesley, Reading, MA, 1969

work page 1969

[2] [2]

Bandari, J

S. Bandari, J. Herzog, and T. Hibi,Monomial ideals whose depth function has any given number of strict local maxima,Ark. Mat.52(2014), no. 1, 11–19

work page 2014

[3] [3]

Brodmann,Asymptotic stability ofAss(M/I nM),Proc

M. Brodmann,Asymptotic stability ofAss(M/I nM),Proc. Amer. Math. Soc.74 (1979), 16–18

work page 1979

[4] [4]

J. Chen, S. Morey, and A. Sung,The stable set of associated primes of the ideal of a graph, Rocky Mountain J. Math.32(2002), no. 1, 71–89. CRITERIA FOR THE PRESENCE OF THE MAXIMAL IDEAL 11

work page 2002

[5] [5]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,http://www.math.uiuc.edu/Macaulay2/

work page

[6] [6]

H. T. H` a and S. Morey,Embedded associated primes of powers of square-free mono- mial ideals,J. Pure Appl. Algebra214(2010), 301–308

work page 2010

[7] [7]

H. T. H´ a, H. D. Nguyen, N. V. Trung, and T. N. Trung,Depth functions of powers of homogeneous ideals,Proc. Amer. Math. Soc.149(2021), no. 5, 1837–1844

work page 2021

[8] [8]

Herzog and T

J. Herzog and T. Hibi,The depth of powers of an ideal,J. Algebra291(2005), 534–550

work page 2005

[9] [9]

Herzog, A

J. Herzog, A. Rauf, and M. Vladoiu,The stable set of associated prime ideals of a polymatroidal ideal,J. Algebraic Combin.37(2013), 289–312

work page 2013

[10] [10]

Matsuda, T

K. Matsuda, T. Suzuki, and A. Tsuchiya,Nonincreasing depth functions of mono- mial ideals,Glasg. Math. J.60(2018), no. 2, 505–511

work page 2018

[11] [11]

W. F. Moore, M. Rogers, and S. Sather-Wagstaff, Monomial ideals and their decom- positions. Universitext. Springer, Cham, 2018. xxiv+387 pp

work page 2018

[12] [12]

Nasernejad, S

M. Nasernejad, S. Bandari, and L. G. Roberts,Normality and associated primes of closed neighborhood ideals and dominating ideals, J. Algebra Appl.24(2025), no. 1, Paper No. 2550009

work page 2025

[13] [13]

Nasernejad, V

M. Nasernejad, V. Crispin Qui˜ nonez, and J. Toledo,Normally torsion-freeness and normality criteria for monomial ideals, Contemp. Math.826(2025), 263–283

work page 2025

[14] [14]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, and I. Al-Ayyoub,Associated primes of powers of cover ideals under graph operations,Comm. Algebra,47(2019), no. 5, 1985–1996

work page 2019

[15] [15]

Nasernejad and K

M. Nasernejad and K. Khashyarmanesh,On the Alexander dual of the path ideals of rooted and unrooted trees,Comm. Algebra45(2017), 1853–1864

work page 2017

[16] [16]

Nasernejad, K

M. Nasernejad, K. Khashyarmanesh, L. G. Roberts, and J. Toledo,The strong persistence property and symbolic strong persistence property,Czechoslovak Math. J.,72(147)(2022), no. 1, 209–237

work page 2022

[17] [17]

Nasernejad, A

M. Nasernejad, A. A. Qureshi, S. Bandari, and A. Musapa¸ sao˘ glu,Dominating ideals and closed neighborhood ideals of graphs, Mediterr. J. Math.19(2022), no. 4, Paper No. 152, 18 pp

work page 2022

[18] [18]

Nasernejad, A

M. Nasernejad, A. A. Qureshi, K. Khashyarmanesh, and L. G. Roberts,Classes of normally and nearly normally torsion-free monomial ideals, Comm. Algebra,50(9) (2022), 3715–3733

work page 2022

[19] [19]

Nasernejad and S

M. Nasernejad and S. Rajaee,Detecting the maximal-associated prime ideal of mono- mial ideals,Bol. Soc. Mat. Mex. (3)25(2019), no. 3, 543–549

work page 2019

[20] [20]

Rissner and I

R. Rissner and I. Swanson,Fluctuations in depth and associated primes of powers of ideals,Ark. Mat.62(2024), no. 1, 191–215

work page 2024

[21] [21]

Sayedsadeghi, M

M. Sayedsadeghi, M. Nasernejad, and A. A. Qureshi,On the embedded associated primes of monomial ideals,Rocky Mountain J. Math.52(2022), no. 1, 275–287

work page 2022

[22] [22]

R. Y. Sharp, Steps in commutative algebra, London Mathematical Society Student Texts19. Cambridge University Press, Cambridge (1990), 2nd edition (2000)

work page 1990

[23] [23]

R. H. Villarreal, Monomial Algebras. 2nd. Edition, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015

work page 2015