Powers of monomial ideals and the Ratliff-Rush operation

Oleksandra Gasanova

Powers of monomial ideals and the Ratliff-Rush operation

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For m-primary monomial ideals obeying the box decomposition principle, the Ratliff–Rush closure equals the intersection of n one-dimensional stabilizations, and this yields an algorithm.

desk verdict New and sound formula for Ratliff-Rush closure of good monomial ideals; worth publishing after fixing a false converse in Section 9 and a wording slip in Proposition 10.6. read the letter →

arxiv 1908.10185 v1 pith:YIUMC7D7 submitted 2019-08-27 math.AC

Oleksandra Gasanova This is my paper · ORCID

classification math.AC MSC 13B2213A3013F20

keywords monomialidealsRatliff–Rushclosuregoodboxdecompositionprinciplepowersofm-primaryFreimanHilbertpolynomial

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An $\mathfrak{m}$-primary monomial ideal is one generated by monomials that includes a positive power of every variable. This paper singles out the class of “good” ideals, those whose minimal generators of every power $I^l$ lie in the rectangular boxes indexed by $l-1$, and proves that for every good ideal the Ratliff–Rush closure $\tilde I=\bigcup_k(I^{k+1}:I^k)$ is obtained from just $n$ coordinate-axis stabilizations. The Ratliff–Rush closure is the largest ideal containing $I$ with the same Hilbert polynomial, so computing it has direct meaning for Hilbert functions. The paper’s theorem turns a generally expensive limiting construction into a finite, explicit intersection and gives an algorithm for polynomial and formal power series rings.

What carries the argument

The machinery is the box decomposition principle together with the finite cone coloring it induces. A box $B_{a_1,\dots,a_n}$ is a product of intervals cut out by the corner exponents $d_i$; for a good ideal each minimal generator of $I^l$ falls in a box with $a_1+\cdots+a_n=l-1$, which forces the translated ideals $I_{a_1,\dots,a_n}=I^{l}:\langle\mu_1^{a_1}\cdots\mu_n^{a_n}\rangle$ to grow monotonically. Theorem 5.8 then gives a finite decomposition of $\mathbb{N}^n$ into cones on which $I_{a_1,\dots,a_n}$ is constant. This cone coloring is what reduces a potentially infinite family of colon ideals to finitely many regions, and it is the tool used to factor $\mu_i^Q$ out of large minimal generators in Lemma 6.1.

What would settle it

Compute the axis-stabilized intersection from Section 7 for every good ideal in a small search space, and compare it with the fully stabilized colon ideal $I^{N+1}:I^N$ for $N$ beyond the bound $L+nq-n+2$ used in Lemma 6.1; a single good ideal where the two differ would disprove the theorem.

Watch

Extended reading notes

Core claim

The paper’s central result is Theorem 6.2: if $I$ is a good $\mathfrak{m}$-primary monomial ideal, then $$\tilde I = I_{q_1,0,\dots,0}\cap I_{0,q_2,\dots,0}\cap \cdots \cap I_{0,\dots,0,q_n},$$ where $I_{0,\dots,q_i,\dots,0}$ is the stable ideal of the chain obtained by moving the box decomposition along the $i$-th coordinate axis. The proof shows that for large powers of $I$, every minimal generator can be factored by a pure power $\mu_i^{q_i}$ of one of the corner monomials $\mu_i=x_i^{d_i}$, so only the axis directions matter. Consequently the usual description of $\tilde I$ as a union of colon ideals $I^{k+1}:I^k$ over all $k$ is captured, for this class, by $n$ one-dimensional limits, each reached after finitely many steps.

Load-bearing premise

The load-bearing premise is that every ideal satisfying the box decomposition principle gives a finite cone coloring of $\mathbb{N}^n$ on which the ideals $I_{a_1,\dots,a_n}$ are constant (Theorem 5.8); if that structural stabilization fails, the axis-intersection formula for $\tilde I$ does not follow from the paper’s argument.

Editorial extensions

If this is right

The Ratliff–Rush closure of any good ideal can be computed by stabilizing $n$ separate one-dimensional chains; by Remark 7.2 each chain stops as soon as two consecutive ideals coincide, so the computation is finite and explicit.
The same computation works in both $\mathbb{K}[x_1,\dots,x_n]$ and the formal power series ring $\mathbb{K}[[x_1,\dots,x_n]]$, since all arguments are monomial and finitary.
For a good ideal with coloring bound $L$, every power $I^k$ with $k\ge L+1$ is Ratliff–Rush (Proposition 10.9), giving a stabilization threshold in terms of the box coloring.
An equigenerated $\mathfrak{m}$-primary monomial ideal is Freiman (its square has the minimal possible number of generators) exactly when it is very good, i.e. exactly when $I^2=I\langle\mu_1,\dots,\mu_n\rangle$; this is Theorem 11.2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same axis-intersection formula could plausibly be tested on the larger class of ideals that satisfy only the necessary condition from Theorem 3.6; Section 9 supplies a finite test for goodness, so a computer search could locate where the formula first fails.
Interpreting $I_{a_1,\dots,a_n}$ as a directional Newton-polyhedron slice suggests an extension beyond monomial ideals: replace boxes by filtrations and cones by normal fans, and the theorem would become a statement about asymptotic intersections of initial ideals.
The equivalence between very good ideals and $I^2=I\langle\mu_1,\dots,\mu_n\rangle$ offers a fast algebraic certificate for the Freiman property that could be used to build new infinite families of ideals all of whose powers are Ratliff–Rush.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read this paper through, and the main result is real. Theorem 6.2 gives a formula for the Ratliff-Rush closure of an m-primary monomial ideal satisfying the box decomposition principle as an intersection of n one-dimensional stabilizations, and the algorithm in Section 7 is a concrete way to compute those stabilizations. I checked the delicate step, Lemma 6.1 with the cone coloring from Theorem 5.8, and it holds. The two inclusions in Theorem 6.2 follow cleanly once you accept the factorization of large minimal generators by a corner power. The proof of Lemma 6.1 uses the vertex-sum inequality exactly as stated, and the stress-test note confirms that the shifted monomial is indeed a minimal generator because equality of ideals on the cone gives equality of minimal generating sets.

The paper also does some things well beyond the main theorem. The box decomposition framework is a useful way to organize the asymptotics of powers. The algorithm is illustrated on several examples and the Singular computations make the claims plausible. The connection to Freiman ideals is a nice bonus: for equigenerated m-primary monomial ideals, very good equals Freiman, and the alternative proof of Theorem 11.2 is clean and shorter than the original.

The soft spots are peripheral but real. The 'conversely' assertion in Section 9 overclaims: the condition described there is not sufficient for goodness, because monomials of the form \mu^{k1}... m^{k} may fail to be minimal generators. That section needs rewriting or a careful statement of what is actually detected. The proof of Proposition 10.6 also says the old box containing m1 is 'unique' when it is not; the argument only needs 'largest', and the wording should be fixed. Neither issue touches Theorem 6.2 or the main algorithm.

The scope is limited to m-primary monomial ideals satisfying the box decomposition principle, and the sufficient condition in Theorem 3.7 is restrictive, but the class is large enough to include interesting examples and the paper explicitly discusses how to test goodness near the boundary. The presentation is a bit rough in spots—the abstract could be rewritten—but the mathematics is readable.

This paper deserves a serious referee. I would send it to peer review rather than desk reject. If I worked in commutative algebra, I would cite it for the algorithm and the Freiman connection. I would bring it to a reading group on monomial ideals, since the cone decomposition technique is a genuine new tool.

Referee Report

0 major / 5 minor

Summary. The paper studies m-primary monomial ideals in K[x1,...,xn] that satisfy a 'box decomposition principle' (called good ideals). For such an ideal I, with minimal corner generators µ_i = x_i^{d_i}, every minimal generator of I^l lies in a box B_a with |a| = l − 1, and the paper defines ideals I_a by dividing the monomials in B_a by µ^a. It proves a finite cone-decomposition theorem for the assignment a ↦ I_a (Theorem 5.8), then states the main result (Theorem 6.2): the Ratliff–Rush closure of I is the intersection of the one-dimensional stabilizations along the coordinate axes, I_{q1,0,...,0} ∩ ... ∩ I_{0,...,0,qn}. Section 7 turns this into an explicit algorithm for the axis stabilizations, with worked examples in Section 8. Section 9 discusses detection of good ideals, Section 10 studies powers of good ideals and introduces very good ideals, and Section 11 connects very good equigenerated ideals with Freiman ideals.

Significance. The main theorem is a clean structural reduction: instead of computing the full family of colon ideals (I^{k+1}:I^k), one computes n one-dimensional stabilizing ideals. If correct, this is a substantial practical and theoretical contribution to monomial ideal theory and to computational Ratliff–Rush closure. The proof is elementary and self-contained, with the cone-coloring/Lemma 6.1 machinery as the key step; I checked this step and the two inclusions of Theorem 6.2 and found them sound. The examples are helpful, and the connections to very good ideals and Freiman ideals are natural. The paper is generally well organized and readable.

minor comments (5)

[Section 3, Theorem 3.7] The proof of the sufficient condition considers two factors that both satisfy the exponent lower bound, but corner generators have exponent sum 1, which is smaller than n/2 for n ≥ 3. The argument should first remove all corner factors (each corner contributes one unit to the box sum) and then apply the two-factor reduction to the remaining non-corner factors. The statement itself is correct, but the proof as written is incomplete.
[Section 10, Propositions 10.5 and 10.6] The claim that the old box containing m1 is the unique old box is not always true when the non-corner generator has a zero exponent, because the point then lies on a box boundary. What the proof needs is the largest old box, which exists by the second bullet of Proposition 10.4; please replace 'unique' by 'largest' and adjust the reference accordingly.
[Section 6, proof of Theorem 6.2] The containment I^{l+1}:I^l ⊆ I^{l+1}:⟨µ_1^l⟩ is correct because ⟨µ_1^l⟩ ⊆ I^l, but the proof should explicitly note that the quotients (I^{k+1}:I^k) form an increasing sequence, so that proving the containment for all l ≥ q controls the union defining the Ratliff–Rush closure. This is standard but currently implicit.
[Section 9, one-extra-generator paragraph] The converse assertion that every monomial of the form µ^a m^k with k < K is a minimal generator is true, but it deserves a proof. If such a monomial were divisible by a product with fewer copies of m, then m^r would be divisible by r corner monomials for some r < K, contradicting the definition of K. As written, the claim is stated without justification.
[Section 5, Theorem 5.8] The phrase 'for each color the set of points of this color forms a cone' should be clarified, since the recursive construction can in principle assign the same ideal value to several disjoint cones. One can always refine the coloring so that each cone is monochromatic; please state this explicitly or rephrase the condition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 6.2 derives the Ratliff–Rush closure from an independent box/cone structure, with no fitted inputs and no load-bearing self-citation.

full rationale

The paper's central claim, Theorem 6.2, is not circular. The ideals I_{a_1,...,a_n} are defined directly from minimal generators of powers in boxes (Definition 4.2), and Proposition 4.4 identifies them with the principal colons I^{l+1}:⟨μ_1^{a_1}⋯μ_n^{a_n}⟩. The stabilization indices q_i are defined by the one-dimensional chains I_{t,0,...,0}, whose stabilization follows from Corollary 4.5 and Noetherianity, not from any property of the Ratliff–Rush closure itself. The forward inclusion in Theorem 6.2 uses the elementary fact that I^{l+1}:I^l ⊆ I^{l+1}:⟨μ_1^l⟩ together with the definition of q_1; the reverse inclusion uses Lemma 6.1, which is proved from the finite cone coloring of Theorem 5.8. That coloring is constructed by Noetherian stabilization of the ideals I_a and does not presuppose the theorem or the Ratliff–Rush closure. There are no fitted parameters later called predictions, no quantities defined in terms of the target result, and no self-citations at all. The cited characterizations of the Ratliff–Rush closure from [13] and [7] are motivational background, not inputs to the proof. Section 11 gives an independent proof of the Freiman equivalence rather than importing it as a black box. The peripheral unproven 'conversely' remark in Section 9 is a correctness concern, not a circularity, and it does not affect Theorem 6.2. Overall, the derivation is self-contained with respect to the claimed main result.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces new mathematical definitions (good ideals, very good ideals, the ideals I_{a_1,...,a_n}), but these are definitions within a proof system, not postulated entities with independent empirical content. There are no free parameters fitted to data. All listed axioms are standard background or explicit standing hypotheses.

assumptions (5)

standard math R = K[x_1,...,x_n] is Noetherian and every monomial ideal has a unique minimal monomial generating set.
Used throughout, especially in Theorem 5.8, to guarantee that increasing chains of ideals I_{a_1,...,a_n} stabilize.
domain assumption I is an m-primary monomial ideal with \mu_i = x_i^{d_i} among its minimal generators for positive integers d_1,...,d_n.
This standing hypothesis from Section 2 defines the boxes and restricts the theorem to the class of ideals where the construction makes sense.
domain assumption The known characterization of the Ratliff-Rush closure from [13] and [7] (unique largest ideal with the same powers, or positive depth of the associated graded ring) is correct.
Cited in the introduction as motivation and to frame the usefulness of \tilde I; not used as a step in the proof of Theorem 6.2.
domain assumption Results proved in K[x_1,...,x_n] transfer to the power series ring K[[x_1,...,x_n]] in the claimed way.
The paper states in Section 2 that all results also hold in the local ring without proof; this is standard for monomial ideals, and the local statement is part of the announced scope.
domain assumption For an m-primary ideal in an n-dimensional regular local ring, the analytic spread is n.
Used in Theorem 11.2 to equate l(I) with n in the Freiman count; this is a standard fact but is not proved in the paper.

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Cite this review

Pith. "Pith review of Powers of monomial ideals and the Ratliff-Rush operation." pith.science (2026). https://pith.science/paper/YIUMC7D7

@misc{pith2026190810185,
  author       = {Pith},
  title        = {Pith review of: Powers of monomial ideals and the Ratliff-Rush operation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YIUMC7D7}},
  note         = {Machine review of arXiv:1908.10185}
}

abstract

Powers of (monomial) ideals is a subject that still calls attraction in various ways. In this paper we present a nice presentation of high powers of ideals in a certain class in $\mathbb K[x_1, \ldots, x_n]$ and $\mathbb K[[x_1, \ldots, x_n]]$. As an interesting application it leads to an algorithm for computation of the Ratliff--Rush operation on ideals in that class. The Ratliff--Rush operation itself has several applications, for instance, if $I$ is a regular $\mathfrak m$-primary ideal in a local ring $(R,m)$, then the Ratliff--Rush associated ideal $\tilde I$ is the unique largest ideal containing $I$ and having the same Hilbert polynomial as $I$.

Figures

Figures reproduced from arXiv: 1908.10185 by the authors.

**Figure 1.** powers of I: I, I 2 , I 3 and I 4 This gives rise to a more general definition. Definition 4.2. Let I be a good ideal and a1, . . . , an nonnegative integers. We define Ia1,...,an := m µ a1 1 · · · µan n | m ∈ Ba1,...,an ∩ G(I l ) [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** cones of A2,1 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** two examples of possible colorings of N 2 , associated to I Given a good ideal I, any coloring as in Theorem 5.8 represents a finite disjoint union of cones. Each cone has a vertex. Let L denote the maximum of sums of coordinates of these vertices. This number depends on I and on the coloring we choose, but we will not put any additional indices: as soon as we found some coloring (which exists according to Theorem 5… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

Al-Ayyoub , An algorithm for computing the Ratliﬀ–Rush closure , Journal of Algebra and its Applications, 8 (2009), pp

I. Al-Ayyoub , An algorithm for computing the Ratliﬀ–Rush closure , Journal of Algebra and its Applications, 8 (2009), pp. 521–532

work page 2009
[2]

Al-Ayyoub and O

I. Al-Ayyoub and O. Solaiman , Inﬁnite families of Ratliﬀ–Rush ideals , Interna- tional Journal of Algebra, 6 (2012), pp. 815–824

work page 2012
[3]

Decker, G.-M

W. Decker, G.-M. Greuel, G. Pfister, and H. Sch ¨onemann, Sin- gular 4-1-2 — A computer algebra system for polynomial computatio ns. http://www.singular.uni-kl.de, 2019

work page 2019
[4]

Eliahou, J

S. Eliahou, J. Herzog, and M. M. Saem , Monomial ideals with tiny squares , Journal of Algebra, 514 (2018), pp. 99–112

2018
[5]

Elias , On the computation of the Ratliﬀ–Rush closure , Journal of Symbolic Com- putation, 37 (2004), pp

J. Elias , On the computation of the Ratliﬀ–Rush closure , Journal of Symbolic Com- putation, 37 (2004), pp. 717–725. 24

work page 2004
[6]

Heinzer, B

W. Heinzer, B. Johnston, D. Lantz, and K. Shah , Coeﬃcient ideals in and blowups of a commutative Noetherian domain , Journal of Algebra, 162 (1993), pp. 355–391

work page 1993
[7]

Heinzer, D

W. Heinzer, D. Lantz, and K. Shah , The Ratliﬀ–Rush ideals in a Noetherian ring, Communications in algebra, 20 (1992), pp. 591–622

work page 1992
[8]

On the number of generators of powers of an ideal

J. Herzog, M. M. Saem, and N. Zamani , On the number of generators of powers of an ideal , arXiv preprint arXiv:1707.07302, (2017)

work page Pith review arXiv 2017

Show all 16 references

[9]

Herzog and G

J. Herzog and G. Zhu , Freiman ideals, Communications in Algebra, (2019), pp. 1– 17

2019
[10]

K ¨allstr¨om, Liftable derivations for generically separably algebraic morphisms of schemes, Transactions of the American Mathematical Society, 361 (2009) , pp

R. K ¨allstr¨om, Liftable derivations for generically separably algebraic morphisms of schemes, Transactions of the American Mathematical Society, 361 (2009) , pp. 495– 523

2009
[11]

T. J. Puthenpurakal , Bockstein cohomology of associated graded rings , Acta Mathematica Vietnamica, (2013), pp. 1–22

2013
[12]

V. C. Qui ˜nonez, Ratliﬀ–Rush monomial ideals, algebraic and geometric comb ina- torics, Contemp. Math., 423 (2007), pp. 43–50

2007
[13]

L. J. Ratliff and D. E. Rush , Two notes on reductions of ideals , Indiana Univ. Math. J, 27 (1978), pp. 929–934

1978
[14]

M. E. Rossi and I. Swanson , Notes on the behavior of the Ratliﬀ–Rush ﬁltration , Comm. Algebra (Grenoble/Lyon, 2001), Contemp. Math., 331 (200 3), pp. 313–328

2001
[15]

M. E. Rossi, D. T. Trung, and N. V. Trung , Castelnuovo–Mumford regularity and Ratliﬀ–Rush closure , Journal of Algebra, 504 (2018), pp. 568–586

2018
[16]

D. T. Trung , On the computation of Castelnuovo–Mumford regularity of th e Rees algebra and of the ﬁber ring , arXiv preprint arXiv:1904.07484v1, (2019). 25

2019 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Al-Ayyoub , An algorithm for computing the Ratliﬀ–Rush closure , Journal of Algebra and its Applications, 8 (2009), pp

I. Al-Ayyoub , An algorithm for computing the Ratliﬀ–Rush closure , Journal of Algebra and its Applications, 8 (2009), pp. 521–532

work page 2009

[2] [2]

Al-Ayyoub and O

I. Al-Ayyoub and O. Solaiman , Inﬁnite families of Ratliﬀ–Rush ideals , Interna- tional Journal of Algebra, 6 (2012), pp. 815–824

work page 2012

[3] [3]

Decker, G.-M

W. Decker, G.-M. Greuel, G. Pfister, and H. Sch ¨onemann, Sin- gular 4-1-2 — A computer algebra system for polynomial computatio ns. http://www.singular.uni-kl.de, 2019

work page 2019

[4] [4]

Eliahou, J

S. Eliahou, J. Herzog, and M. M. Saem , Monomial ideals with tiny squares , Journal of Algebra, 514 (2018), pp. 99–112

2018

[5] [5]

Elias , On the computation of the Ratliﬀ–Rush closure , Journal of Symbolic Com- putation, 37 (2004), pp

J. Elias , On the computation of the Ratliﬀ–Rush closure , Journal of Symbolic Com- putation, 37 (2004), pp. 717–725. 24

work page 2004

[6] [6]

Heinzer, B

W. Heinzer, B. Johnston, D. Lantz, and K. Shah , Coeﬃcient ideals in and blowups of a commutative Noetherian domain , Journal of Algebra, 162 (1993), pp. 355–391

work page 1993

[7] [7]

Heinzer, D

W. Heinzer, D. Lantz, and K. Shah , The Ratliﬀ–Rush ideals in a Noetherian ring, Communications in algebra, 20 (1992), pp. 591–622

work page 1992

[8] [8]

On the number of generators of powers of an ideal

J. Herzog, M. M. Saem, and N. Zamani , On the number of generators of powers of an ideal , arXiv preprint arXiv:1707.07302, (2017)

work page Pith review arXiv 2017