On a question of D. Rees on classical integral closure and integral closure relative to an Artinian module

2019 ◽  
Vol 19 (02) ◽  
pp. 2050033
Author(s):  
V. H. Jorge Pérez ◽  
L. C. Merighe
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Local Cohomology ◽  
Integral Closure ◽  
Minimal Prime Ideal ◽  
Local Cohomology Module ◽  
Artinian Module ◽  
Classical Integral ◽  
Minimal Prime ◽  
The Relationship

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] and [Formula: see text] ideals of [Formula: see text]. Motivated by a question of Rees, we study the relationship between [Formula: see text], the classical Northcott–Rees integral closure of [Formula: see text], and [Formula: see text], the integral closure of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text] (also called here ST-closure of [Formula: see text] on [Formula: see text]), in order to study a relation between [Formula: see text], the multiplicity of [Formula: see text], and [Formula: see text], the multiplicity of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text]. We conclude [Formula: see text] when every minimal prime ideal of [Formula: see text] belongs to the set of attached primes of [Formula: see text]. As an application, we show what happens when [Formula: see text] is a generalized local cohomology module.

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

2019 ◽  
Vol 30 (02) ◽  
pp. 379-396
Author(s):  
V. H. Jorge Pérez ◽  
T. H. Freitas
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Artinian Modules ◽  
Dimension Formula ◽  
Hilbert Coefficients ◽  
Artinian Module ◽  
Samuel Multiplicity

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

scholarly journals On the vanishing and the positivity of intersection multiplicities over local rings with small non complete intersection loci

1994 ◽  
Vol 136 ◽  
pp. 133-155 ◽  
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Prime Ideal ◽  
Local Rings ◽  
Regular Local Ring ◽  
Commutative Noetherian Ring ◽  
Intersection Multiplicities ◽  
Minimal Prime

Throughout this paperAis a commutative Noetherian ring of dimensiondwith the maximal ideal m and we assume that there exists a regular local ringSsuch thatAis a homomorphic image ofS, i.e.,A=S/Ifor some idealIofS. Furthermore we assume thatAis equi-dimensional, i.e., dimA= dimA/for any minimal prime idealofA. We put.

Ordinary Singularities with Decreasing Hilbert Function

1982 ◽  
Vol 34 (1) ◽  
pp. 169-180 ◽  
Author(s):  
Leslie G. Roberts
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Hilbert Function ◽  
Integral Closure ◽  
Minimal Prime Ideal ◽  
Quotient Field ◽  
Graded Ring ◽  
Image Position ◽  
Minimal Prime ◽  
Dimension One

Let A be the co-ordinate ring of a reduced curve over a field k. This means that A is an algebra of finite type over k, A has no nilpotent elements, and that if P is a minimal prime ideal of A, then A/P is an integral domain of Krull dimension one. Let M be a maximal ideal of A. Then G(A) (the graded ring of A relative to M) is defined to be . We get the same graded ring if we first localize at M, and then form the graded ring of AM relative to the maximal ideal MAM. That isLet Ā be the integral closure of A. If P1, P2, …, Ps are the minimal primes of A thenwhere A/Pi is a domain and is the integral closure of A/Pi in its quotient field.

Some properties of Artinian modules and applications

2018 ◽  
Vol 17 (02) ◽  
pp. 1850019
Author(s):  
Tran Nguyen An
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Vanishing Theorem ◽  
Strong Connection ◽  
Artinian Modules ◽  
Local Cohomology Modules ◽  
Artinian Module ◽  
Base Ring ◽  
System Of Parameters ◽  
Cohomology Modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] be an Artinian [Formula: see text]-module. Consider the following property for [Formula: see text] : [Formula: see text] In this paper, we study the property (∗) of [Formula: see text] in order to investigate the relation of system of parameters between [Formula: see text] and the ring [Formula: see text]. We also show that the property (∗) of [Formula: see text] has strong connection with the structure of base ring. Some applications to cofinite Artinian module are given. These are generalizations of [N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra. 42 (2014) 1270–1275; G. Ghasemi, K. Bahmanpour and J. Azami, On the cofiniteness of Artinian local cohomology modules, J. Algebra Appl. 15(4) (2016), Article ID: 1650070, 8 pp.] A generalization of Lichtenbaum–Hartshorne Vanishing Theorem is also given in this paper.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 95-101
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Injective Resolution ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Gorenstein Injective ◽  
Cousin Complexes ◽  
Cohomology Modules ◽  
Cousin Complex

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

scholarly journals Primary Decomposition of Ideals of Lattice Homomorphisms

10.37236/7694 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Leila Sharifan ◽  
Ali Akbar Estaji ◽  
Ghazaleh Malekbala
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Minimal Prime Ideal ◽  
Primary Decomposition ◽  
Planar Lattice ◽  
Combinatorial Properties ◽  
Finite Lattices ◽  
Minimal Prime ◽  
Lattice Homomorphisms ◽  
The Ideal

For two given finite lattices $L$ and $M$, we introduce the ideal of lattice homomorphism $J(L,M)$, whose minimal monomial generators correspond to lattice homomorphisms $\phi : L\to M$. We show that $L$ is a distributive lattice if and only if the equidimensinal part of $J(L,M)$ is the same as the equidimensional part of the ideal of poset homomorphisms $I(L,M)$. Next, we study the minimal primary decomposition of $J(L,M)$ when $L$ is a distributive lattice and $M=[2]$. We present some methods to check if a monomial prime ideal belongs to $\mathrm{ass}(J(L,[2]))$, and we give an upper bound in terms of combinatorial properties of $L$ for the height of the minimal primes. We also show that if each minimal prime ideal of $J(L,[2])$ has height at most three, then $L$ is a planar lattice and $\mathrm{width}(L)\leq 2$. Finally, we compute the minimal primary decomposition when $L=[m]\times [n]$ and $M=[2]$.

On Weakly Laskerian and Weakly Cofinite Modules

2012 ◽  
Vol 19 (04) ◽  
pp. 693-698
Author(s):  
Kazem Khashyarmanesh ◽  
M. Tamer Koşan ◽  
Serap Şahinkaya
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cofinite Modules ◽  
Gorenstein Local Ring

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

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