首页/论文/2003.03953

arXiv 2003.039532020-03-09math.AC

Reducibility index and sum-reducibility index

Tran Nguyen An, Tran Duc Dung, Shinya Kumashiro, Le Thanh Nhan

备注：: 12 pages

英文摘要：

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, Northcott introduced the reducibility index of $M$, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ in $M$. On the other hand, for an Artinian $R$-module $A$, Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of $A$ does not depend on the choice of the representation. This number is called the sum-reducibility index of $A$. In the former part of this paper, we compute the reducibility index of $S\otimes_R M$, where $R\to S$ is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of $R$ are studied. For the latter part of this paper, we clarify the relation among the reducibility index of $M$, that of the completion of $M$, and the sum-reducibility index of the Matlis dual of $M$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Eventually index of reducibility on sequentially Cohen-Macaulay modules

It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the Gorensteinness, Cohen-Macaulayness of local rings in term of eventually index of reducibility for distinguished parameter ideals.

2015-04-23math.AC

On the index of reducibility of powers of a standard parameter ideal

In this paper we study the index of reducibility of powers of a standard parameter ideal. An explicit formula is proved for the extremely case. We apply the main result to compute Hilbert polynomials of socle ideals of standard parameter ideals.

2016-04-24math.AC

G-graded irreducibility and the index of reducibility

Let $R$ be a commutative Noetherian ring graded by a torsionfree abelian group $G$. We introduce the notion of $G$-graded irreducibility and prove that $G$-graded irreducibility is equivalent to irreducibility in the usual sense. This is a generalization of Chen and Kim's result in the $\mathbb{Z}$-graded case. We also discuss the concept of the index of reducibility and give an inequality for the indices of reducibility between any radical non-graded ideal and its largest graded subideal.

2018-12-16math.AC

On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomial type is at most one

Let $(R, \frak m)$ be a Noetherian local ring, $M$ a finitely generated $R$-module. The aim of this paper is to prove a uniform formula for the index of reducibility of paprameter ideals of $M$ provided the polynomial type of $M$ is at most one.

2013-11-05math.AC

On the index of reducibility of parameter ideals: The stable and limit values

Let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. A famous result of Northcott says that if $M$ is Cohen-Macaulay, then the index of reducibility of parameter ideals on $M$ is an invariant of the module. The aim of this paper is to extend Northcott's theorem for any finitely generated $R$-module. We call this invariant the stable value of the indices of reducibility of parameter ideals of $M$. We also introduce the limit value of the indices of reducibility of parameter ideals of $M$.

2019-03-04math.AC

The index of reducibility of parameter ideals in low dimension

Results are presented concerning the following question: If M is a finitely-generated module with finite local cohomologies over a Noetherian local ring (A, m), does there exist an integer n such that every parameter ideal for M contained in m^n has the same index of reducibility? We show that the answer is yes if M has dimension 1 or if M has dimension 2 and positive depth. This research is closely related to work of Goto-Suzuki and Goto-Sakurai; Goto-Sakurai have supplied an answer of yes in case M is Buchsbaum.

2004-08-11math.AC

更多相关论文收起更多论文

The Index of Reducibility of Parameter Ideals and Mostly Zero Finite Local Cohomology

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0 and d, then there is an integer l such that every parameter ideal for M contained in the lth power of the maximal ideal has the same index of reducibility on M. This theorem generalizes work of the second author (math.AC/0408143) and is closely related to work of S. Goto, H. Sakurai (math.AC/0306148), and N. Suzuki.

2005-10-15math.AC

Graded-irreducible modules are irreducible

We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural extension of the index of reducibility to the graded setting coincides with the ordinary index of reducibility. We also investigate the question of uniqueness of the components in a graded-irreducible decomposition, as well as the relation between the index of reducibility of a non-graded ideal and that of its largest graded subideal.

2016-09-29math.AC

On the index of reducibility in Noetherian modules

Let $M$ be a finitely generated module over a Noetherian ring $R$ and $N$ a submodule. The index of reducibility ir$_M(N)$ is the number of irreducible submodules that appear in an irredundant irreducible decomposition of $N$ (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are: (1) $\mathrm{ir}_M(N) = \sum_{{\frak p} \in \mathrm{Ass}_R(M/N)} \dim_{k(\frak p)} \mathrm{Soc}(M/N)_{\frak p} $; (2) For an irredundant primary decomposition of $N = Q_1 \cap \cdots \cap Q_n$, where $Q_i$ is $\frak p_i$-primary, then $\mathrm{ir}_M(N) = \mathrm{i…

2015-04-10math.AC

The Chern Coefficient and Cohen-Macaulay rings

The purpose of this paper is to investigate a relationship between the index of reducibility and the Chern coefficient for primary ideals. Therefore, the main result of this paper gives a characterization of a Cohen-Macaulay ring in terms of its the index of reducibility, its Cohen-Macaulay type, and the Chern coefficient for parameter ideals. As corollaries to the main theorem we obtained the characterizations of a Gorenstein ring in term of its Chern coefficient for parameter ideals.

2015-04-23math.AC

A note on Chern coefficients and Cohen-Macaulay rings

In this paper, we investigate the relationship between the index of reducibility and Chern coefficients for primary ideals. As an application, we give characterizations of a Cohen-Macaulay ring in terms of its type, irreducible multiplicity, and Chern coefficients with respect to certain parameter ideals in Noetherian local rings.

2021-08-25math.AC

理想化的不可约重数

Irreducible multiplicity of idealizations

本文研究诺特局部环及其理想化中准素理想的不可约指数与不可约重数之间的关系，推广了Goto等人的一个主要结果。

2025-09-22math.AC

Burch index, summands of syzygies and linearity in resolutions

The Burch index is a new invariant of a local ring $R$ whose positivity implies a kind of linearity in resolutions of $R$-modules. For example, if $R$ has depth zero and Burch index at least $2$, then any non-free 7th $R$-syzygy contains the residue field as a direct summand. We also compute the Burch index in various cases of interest.

2023-01-15math.AC

$k$ 个正 Burch 指数环上的合冲直和项：通过典范分解

$k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions

本文通过杠分解和 dg 或 $A_\infty$ 分解，对正 Burch 指数环上的模合冲中剩余域直和项的存在性给出新证明，并明确识别 Golod 模情形下的 $k$ 个直和项，其数量随同调度指数增长。

2023-12-30math.AC