Dancheng Lu, Tongsuo Wu
Let $R$ be a standard graded algebra over an infinite field $\mathbb{K}$ and $M$ a finitely generated $\ZZ$-graded $R$-module. Let $I_1,\ldots I_m$ be graded ideals of $R$. The functions $r(M/I_1^{a_1}\ldots I_m^{a_m}M)$ and $r(I_1^{a_1}\ldots I_m^{a_m}M)$ are investigated and their asymptotical behaviours are given. Here $r(\bullet)$ stands for the reduction number of a finitely graded module $\bullet$.
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The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some special cases the reduction number can be explicitly computed.
Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in $2$-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.
We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.
Reduction numbers and initial ideals
标准分次代数A的约化数r（A）是最小整数k，使得存在A的齐次极大理想m的最小约化J，使得Jm^k=m^{k+1}. Vasconcelos证明了A=R/I的约化数只能通过传递到初始理想r（R/I）\leq r（R/in（I））而增加。本文的目的就是证明这个猜想。
The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring $(A,\mathfrak{m})$. As an application, we present complete descriptions of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small reduction number.
Let R be a commutative ring and I an ideal of R. A sub-ideal J of I is a reduction of I if JI^n = I^n+1 for some positive integer n. The ring R has the (finite) basic ideal property if (finitely generated) ideals of R do not have proper reductions. Hays characterized (one-dimensional) Prufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays' results to Prufer v-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among star-basic properties for certain star operations.
In this paper, we give a way to construct graded filtrations of graded modules. We then apply it to the Sally module, which describes a correction term of the Hilbert function. As a result, we obtain the inequality of the Hilbert coefficients for ideals of reduction number $2$ or $3$.
An upper bound on the reduction number of an ideal
设A是一个交换环，I是A的一个理想，它具有一个约化Q.本文给出了当给定A中的理想族时，I关于Q的约化数的一个上界。作为推论，如果某个包含I的理想J满足J^2 = QJ，则I^{v + 2} = QI^{v + 1}，其中v表示J / I作为A-模的生成元的个数。
Minimal Monomial Reductions and the Reduced Fiber Ring of an Extremal Ideal
设$I$是多项式环$A=K[x_1,...,x_n]$中的单项理想.如果不存在真的单项理想$L \subset J$使得$L$是$I$的一个约化理想，我们称一个单项理想$J$是$I$的一个极小单项约化理想.证明了$I$存在唯一的极小单项式约化理想$J$，并证明了$J$的单项式生成元的最大度决定了$t\gg 0$的线性函数$\reg(I^t)=pt+c$的斜率$p$.我们确定了$J$的约化纤维环$\mathcal{F}(J)_{\red}$的结构，并证明了$\mathcal{F}(J)_{\red}$同构于由$J$的凸几何性质确定的半群环的逆系统的逆极限.
Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1} +...+ a_{n-1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of i…
In this paper, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein. First, we clarify the relation between almost Gorenstein rings and nearly Gorenstein rings by using the invariant in dimension one. We next characterize the idealization of trace ideals over Gorenstein rings in terms of the invariant. It provides better prospects for a result of the almost Gorenstein property of idealization.
In this paper, we present an algorithm for computing the minimal reductions of $\mathfrak{m}$-primary ideals of Cohen--Macaulay local rings. Using this algorithm, we are able to compute the Hilbert--Samuel multiplicities and solve the membership problem for the integral closure of $\mathfrak{m}$-primary ideals.
We study minimal reductions of edge ideals of graphs and determine restrictions on the coefficients of the generators of these minimal reductions. We prove that when $I$ is not basic, then $\core{I}\subset \m I$, where $I$ is an edge ideal in the corresponding localized polynomial ring and $\m$ is the maximal ideal of this ring. We show that the inclusion is an equality for the edge ideal of an even cycle with an arbitrary number of whiskers. Moreover, we show that the core is obtained as a finite intersection of homogeneous minimal reductions in the case of even cycles. The formula for the co…
Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We investigate the existence (or lack thereof) of proper reductions of an ideal of $R$ and the number of generators needed for a reduction in the case $k$ is a finite field. When $R$ is one-dimensional, we give a formula for the smallest integer $n$ for which every ideal has an $n$-generated reduction. It follows that in a one-dimensional local Noetherian ring every ide…
Let $(R,m)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an m-primary ideal and $J$ a minimal reduction of $I$. In this paper, we show that $\widetilde{r_J(I)}\leq r_J(I)$. This answer to a question that made by M.E. Rossi and I. Swanson in [\ref{Rs}, Question 4.6].
We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of Reid-Roberts-Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.
We discuss the problem of determining reduction number of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computation in a polynomial ring with (n-dim(I))dim(I) parameters and n-dim(I) variables. The second one computes via a Grobner system the set of all reduction numbers of the ideal I and thus in particular also its big reduction number. However,it requires computations in a ring with n.dim(I) parameters and n variables.
This paper investigates t-reductions of ideals in pullback constructions. Section 2 examines the correlation between the notions of reduction and t-reduction in pseudo-valuation domains. Section 3 solves an open problem on whether the finite t-basic and v-basic ideal properties are distinct. We prove that these two notions coincide in any arbitrary domain. Section 4 features the main result, which establishes the transfer of the finite t-basic ideal property to pullbacks in line with Fontana-Gabelli's result on PvMDs and Gabelli-Houston's result on v-domains. This allows us to enrich the liter…
Let $R$ be a Noetherian local ring and let $I$ be an ideal in $R$. The ideal $I$ is called balanced if the colon ideal $J:I$ is independent of the choice of the minimal reduction $J$ of $I$. Under suitable assumptions, Ulrich showed that $I$ is balanced if and only if the reduction number, $r(I)$, of $I$ is at most the `expected' one, namely $\ell(I)- \height I+1$, where $\ell(I)$ is the analytic spread of $I$. In this article we propose a generalization of balanced. We prove under suitable assumptions that if either $R$ is one-dimensional or the associated graded ring of $I$ is Cohen-Macaulay…
We study relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal F= \{\mathcal F (\underline n)\}_{\underline n\in\mathbb Z^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalisation of the Kirby-Mehran Complex. An analysis of its homology leads to an analogue of Huneke's Fundamental Lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal F$ and reduction vectors with respect to c…
We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) To develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen-Macaulay rings to more general rings. (ii) To use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen-Macaulayness of the special fiber ring. (iii) To extend some of the results of (i) to two-dimension…
Minimal cellular resolutions of monomial ideals with five generators and their Artinian reductions
本文证明了至多五个生成元的单项式理想及其阿廷约化具有极小广义Barile-Macchia消解，从而推出其具有极小胞腔消解，推广了已有结果并与剪枝消解方法相互印证。
In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{ \overline{I^rJ^s}\}$ of complete ${\mathfrak m}$-primary ideals and vanishing of the second normal Hilbert coefficient $\overline{e}_2(IJ)$ where $R$ is a two-dimensional Cohen-Macaulay analytically unramified local ring with maximal ideal $\mathfrak m.$ This generalization is obtained as a consequence of a formula for the third local cohomology module of the extended Rees algebras of the $\mathbb Z^3$-graded filtration $\{\overline{I^rJ^sK^t}\}$ with su…
In a local Cohen-Macaulay ring $(A, \mathrm{m})$, we study the Hilbert function of an $\mathrm{m}$-primary ideal $I$ whose reduction number is two. It is a continuous work of the papers of Huneke, Ooishi, Sally, and Goto-Nishida-Ozeki. With some conditions, we show the inequality $\mathrm{e}_1(I)\ge \mathrm{e}_0(I) - \ell_A (A/I) + \mathrm{e}_2(I)$ of the Hilbert coefficients, which is the converse inequality of Sally and Itoh. We also study relations between the Hilbert coefficients and the depth of the associated graded ring.
Green's general hyperplane restriction theorem gives a sharp upper bound for the Hilbert function of a standard graded algebra over and infinite field K modulo a general linear form. We strengthen Green's result by showing that the linear forms that do not satisfy such estimate belong to a finite union of proper linear spaces. As an application we give a method to derive variations of the Eakin-Sathaye theorem on reductions. In particular, we recover and extend results by O'Carroll on the Eakin-Sathaye theorem for complete and joint reductions.
In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge \mathrm{e}_0(I) - \mathrm{e}_1(I) + \dfrac{\mathrm{e}_2(I) + \ell_A(I^2/QI)}{2}$, where $\mathrm{e}_i(I)$ denotes the $i$th Hilbert coefficients and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of …
In this article, we define a class of binomial ideals associated to a simplicial complex. This class of ideals appears in the presentation of fiber cones of codimension 2 lattice ideals \cite{hm}, and in the work of Barile and Morales \cite{bm2}, \cite{bm3}, \cite{bm4}. We compute the reduction number of Binomial extensions of Simplicial ideals. This extends all the previous results in this area.
Bound on the a-invariant and reduction numbers of ideals
设$R$是Artin局部环上的$d$维标准分次环.设$M$是$R.$的唯一极大齐性理想，$h^i(R)_n$表示$H^i_M(R)_n$的长度，即R的第i个局部上同调模关于M的第n个分次分支.定义$R$的Eisenbud-Goto不变量为数$$EG(R)= \sum_{q=0}^{d-1} \binom{d-1}{q} h^q(R)_{1-q}.$$证明了$R$的$a$ -不变量满足$$ a(R) \leq e(R)-length(R_1)+(d-1)(length(R_0)-1)+ EG(R).$$利用这一界得到了Cohen-Macaulay局部环$(R,m)$的$m$ -准素理想的约化数的上界，该局部环的分次环$G(m)$具有几乎最大深度.
On the analytic spread and the reduction number of the ideal of maximal minors
令$m$，$n$，$a_1$，.，$a_r$，$b_1$，.，$b_r$是整数，具有$1\leq a_1<...<a_r\leq m$和$1\leq b_1<...<b_r\leq n$。并且令$x$是通用$m\times n$矩阵，其具有第一$a_i-1$行和第一$b_i-1$列的$i$ -子式都为零的性质，对于$i=1$，..，$r+1$（$a_{r+1}=m+1$和$b_{r+1}=n+1$）。对于具有$1\leq u\leq m$的整数$u$，我们用$U$表示由$x$的第一$u$行组成的$u\times n$矩阵。本文研究了PDMATHBNF的极大子式理想的解析展度和约化数
Borel-fixed ideals and reduction number
本文的目的是研究约化数与Borel不动理想在所有特征中的关系。根据定义，Borel不动理想在某些特殊化下是闭的，这类似于强稳定性。我们将估计可以专门化为给定单项式的单项式的数量。因此，我们得到了一个组合版本的著名的Eakin-Sathaye定理的范围内的减少数量的希尔伯特函数。此外，我们证明了Eakin-Sathaye定理的界是由一个lex-segment单项理想的约化数得到的。这一结果肯定地回答了孔卡的一个问题。我们还将证明lex-segment理想的约简数是由给定理想的约简数指数有界的。
We study a notion called $n$-standardness (defined by M. E. Rossi and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is three-standard, first proving results when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic zero case. As an application, we extend a result due to T. Puthenpurakal and show that a certain length associated to a minimal reduction of the max…
Some algebraic invariants of mixed product ideals
我们计算了一类特殊的单项理想，即混合积的理想的一些代数不变式（如深度，Castelnuovo - Mumford正则性）。因此，我们刻画了混合积的Cohen-Macaulay理想。
We consider classes of ideals which generalize the mixed product ideals introduced by Restuccia and Villarreal, and also generalize the expansion construction by Bayati and the first author \cite{BH}. We compute the minimal graded free resolution of generalized mixed product ideals and show that the regularity of a generalized mixed product ideal coincides with regularity of the monomial ideal by which it is induced.
It is well known that in the Noetherian local ring with infinite residue field the reduction of $\mm$-primary ideal may be given in the form of a sufficiently general linear combination of its generators. In the paper we give a condition for the existence of such reduction in terms of the sum of degrees of the ideal fiber cone prime divisors in the case of any Noetherian local ring.
Let $J\subset I$ be ideals in a formally equidimensional local ring with $λ(I/J)<\infty.$ Rees proved that for all $n\gg0$, $λ(I^n/J^n)$ is a polynomial $P(I/J)(X)$ in $n$ of degree at most dim $R$ and $J$ is a reduction of $I$ if and only if deg $P(I/J)(X)\leq$ dim $R-1.$ We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg $P(I/J)$ achieves its maximal degree. On the other hand, for ideals $J\subset…
Betti numbers of mixed product ideals
本文计算了一类特殊的无平方根的单项理想--混合积理想的分解的贝蒂数。此外，当这些理想是Cohen-Macaulay时，我们计算它们的型。
Reduction numbers of links of irreducible varieties
理想$I$的约化为$I$的性质提供了一条自然的途径，具有生成元较少的优点。本文主要研究一类准素理想的环的约化指数的猜想。在Koszul和Keszon-Northcott解上存在一个代数结构是详细说明该猜想已知情况的主要工具。在最后一节中，我们涉及的猜想，以公式涉及的长度的第一Koszul同调模的这些理想。
The structure of the core of ideals
$R$ -理想$I$的核心是$I$的所有约简的交集。该对象是由D. Rees和J. Sally以及后来的C. Huneke和我斯旺森，谁表明，特别是其连接到李普曼的概念伴随的一个理想。作为一个先验的无限理想的交集，核心是很难明确描述。我们证明了在一个广泛的设置：${\rm core}(I)$是一个有限的交集的最小减少; ${\rm core}(I)$是一个有限的交集的一般最小减少; ${\rm core}(I)$是收缩$R$的一个“普遍”的理想; ${\rm core}(I)$表现良好的平坦扩展。证明是基于一般的多重性估计某些模块。
Polymatroidal property of generalized mixed product ideals
本文研究了由单项式理想诱导的广义混合乘积理想的拟阵性，并计算了其部分代数不变量。
Distance Reduction in Bouquet Decompositions and Toric Ideals of Graphs
本文研究图的托ric理想距离缩减性质与花结构的关系，证明完全交义图的最小马尔可夫基距离缩减当且仅当其理想电路距离缩减，并探讨同质托ric理想在花结构下距离缩减性质的保持条件。
Let $R$ be a commutative ring with identity. In this note, we study the property: If $ I \subsetneqq J$ are ideals in $R$, then $ I^n \subsetneqq J^n$ for all $ n\geq 1$. We define the notion of a big ideal (Definition 1.2). It is noted that the property has close relationship with the notions of reduction of an ideal and Ratliff-Rush ideal [7]. Apart from other results, it is proved that a Noetherian domain satifies the property if and only if every ideal in $R$ is a Ratliff-Rush ideal. We also prove that ideals having no proper reduction are big ideals, and maximal ideals in regular rings ar…
In this paper we give versions of Hilbert's syzygy theorem for finitely generated modules over polynomial rings over direct product of principal ideal rings.
The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-co…
Castelnuovo-Mumford regularity of products of ideals
我们讨论了Castelnuovo-Mumford正则性在理想和模上的某些运算下的行为，如乘积或幂。特别地，我们证明了即使I是线性型的理想且M是具有线性分解的模，reg（IM）也可以大于reg（M）+reg（I）.另一方面，我们证明了任何线性型的理想的理想的积都有线性分解。我们还讨论了多拟阵理想的情况下，并表明，任何产品的一般汉克尔矩阵的行列式理想有一个线性决议。
Let $(A, \frak m)$ be a noetherian local ring with maximal ideal $\frak{m}$ and infinite residue field $k = A/\frak{m}.$ Let $J$ be an $\frak m$-primary ideal, $I_1,...,I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this paper, we interpret mixed multiplicities of $(I_1,..., I_s,J)$ with respect to $M$ as multiplicities of joint reductions of them. This generalizes the Rees's theorem on mixed multiplicity (Theorem 2.4). As an application we show that mixed multiplicities are also multiplicities of Rees's superficial sequences.
Reduction of the Finite Generation Problem for Leading Term Ideals under Arbitrary Rational Monomial Orders to the Lexicographic
研究将任意有理单项式序下首项理想的有限生成问题归约到字典序，通过带标签的单项式嵌入等方法证明相关性质等价，还应用此归约于赋值环，给出格罗贝纳环猜想有理单项式序版本的证明。
The core of an ideal is defined as the intersection of all of its reductions. In this paper we provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$ coincides with the largest monomial ideal contained in a general reduction of $I$. We prove that the class of lex-segment ideals satisfies these residual conditions and study the core of lex-segment ideals generated in one degree. For monomial ideals that do not necessarily satisfy the residual conditions and that are generated in one degree, we conjecture an expli…
In this paper, we study the strong Lefschetz property of artinian complete intersection ideals generated by products of linear forms. We prove the strong Lefschetz property for a class of such ideals with binomial generators.
We exhibit an example of a product of two proper monomial ideals such that the residue class ring is not Golod. We also discuss the strongly Golod property for rational powers of monomial ideals, and introduce some sufficient conditions for weak Golodness of monomial ideals. Along the way, we ask some related questions.
We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing…
In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a $3$-dimensional regular local ring (or $3$-variable polynomial ring) $(R , \m)$, the ideal $I \m$ always defines a Golod ring for any proper ideal $I \subset R$. We also show that non-Golod products of ideals are ubiquitous; more precisely, we prove that for any proper ideal with grade $\geq 4$, there exists an ideal $J \subseteq I$ such that $IJ$ is not Golod. We conclude by showing that if $I$ is any proper ideal in a $3$-dimensional regular local ring and $\mfa \subseteq I$ a co…
We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which i…
We give bounds for the Hilbert-Kunz multiplicity of the product of two ideals, and we characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations.
Two ideals $I$ and $J$ are called transverse if $I \cap J = IJ$. We show that the obstructions defined by Avramov for classes of (sequentially) transverse ideals in regular local rings are always $0$. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative mul…
In this paper we use some results related to regularity, Betti numbers and reduction of generic initial ideals, showing their stability in passing from an ideal to its initial ideal if the last has some simple properties.
New title and minor adjustments. To appear in the Journal of Pure and Applied Algebra
Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension $d\geq 3$, for the vanishing of the normal Hilbert coefficients $\overline{e}_k(I)$ for $k\leq d,$ in terms of the normal reduction number.
Notes on the behavior of the Ratliff-Rush filtration
我们建立了新的Ratliff-Rush闭理想类和Ratliff-Rush闭包的一些病态行为。特别是，Ratliff-Rush闭包在通道模表面元素下表现不佳，需要理想，相关素数，首项理想和生成元的最小数量的幂。相比之下，Ratliff-Rush过滤的约简数表现得更好：它保留了理想约简数的一些信息。
Bounds for the reduction number of primary ideal in dimension three
本文研究了Cohen-Macaulay局部环中三维准素理想的约化数界限，在特定深度条件下利用Hilbert系数给出了约化数的上界估计。
This note proves that if S is an unramified regular local ring and I and J are proper ideals of height at least two, then S/IJ is never Gorenstein.
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