Superficial ideals for monomial ideals

2018 ◽  
Vol 17 (06) ◽  
pp. 1850102 ◽  
Author(s):  
Saeed Rajaee ◽  
Mehrdad Nasernejad ◽  
Ibrahim Al-Ayyoub
Keyword(s):  
Noetherian Ring ◽  
Monomial Ideals ◽  
Minimal Set ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Torsion Freeness ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] is a superficial ideal for [Formula: see text] if the following conditions are satisfied: (i) [Formula: see text], where [Formula: see text] denotes a minimal set of generators of an ideal [Formula: see text]. (ii) [Formula: see text] for all positive integers [Formula: see text]. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

scholarly journals On asymptotic values of certain sets of attached prime ideals

1988 ◽  
Vol 30 (3) ◽  
pp. 293-300 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Prime Divisors ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Asymptotic Values ◽  
Attached Prime ◽  
Positive Integers

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

A note on the ascending chain condition of ideals

2019 ◽  
Vol 19 (07) ◽  
pp. 2050135 ◽  
Author(s):  
Ibrahim Al-Ayyoub ◽  
Malik Jaradat ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Regular Ideal ◽  
Reduction Number ◽  
Number Formula ◽  
A Chain ◽  
Ascending Chain Condition

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

Family of quotients of some special rings

2018 ◽  
Vol 17 (12) ◽  
pp. 1850233 ◽  
Author(s):  
Maryam Salimi
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let [Formula: see text] be a commutative Noetherian ring and let [Formula: see text] be a proper ideal of [Formula: see text]. We study some properties of a family of rings [Formula: see text] that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. We deal with the strongly cotorsion property of local cohomology modules of [Formula: see text], when [Formula: see text] is a local ring. Also, we investigate generically Cohen–Macaulay, generically Gorenstein, and generically quasi-Gorenstein properties of [Formula: see text]. Finally, we show that [Formula: see text] is approximately Cohen–Macaulay if and only if [Formula: see text] is approximately Cohen–Macaulay, provided some special conditions.

scholarly journals A family of monomial ideals with the persistence property

2019 ◽  
Vol 18 (05) ◽  
pp. 1950093
Author(s):  
Somayeh Moradi ◽  
Masoomeh Rahimbeigi ◽  
Fahimeh Khosh-Ahang ◽  
Ali Soleyman Jahan
Keyword(s):  
Monomial Ideal ◽  
Independent Set ◽  
Monomial Ideals ◽  
Prime Ideals ◽  
Stable Set ◽  
Ideal Formula ◽  
Path Graph ◽  
Associated Prime Ideals ◽  
Persistence Property ◽  
Positive Integers

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers [Formula: see text] and [Formula: see text], we consider the monomial ideal [Formula: see text] generated by all monomials [Formula: see text], where [Formula: see text] is an independent set of vertices of the path graph [Formula: see text] of size [Formula: see text], which is indeed the facet ideal of the [Formula: see text]th skeleton of the independence complex of [Formula: see text]. We describe the set of associated primes of all powers of [Formula: see text] explicitly. It turns out that any such ideal [Formula: see text] has the persistence property. Moreover, the index of stability of [Formula: see text] and the stable set of associated prime ideals of [Formula: see text] are determined.

scholarly journals Computing the invariants of intersection algebras of principal monomial ideals

2019 ◽  
Vol 29 (02) ◽  
pp. 309-332 ◽  
Author(s):  
Florian Enescu ◽  
Sandra Spiroff
Keyword(s):  
Polynomial Ring ◽  
Noetherian Ring ◽  
Class Group ◽  
Monomial Ideals ◽  
Semigroup Ring ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Commutative Noetherian Ring ◽  
New Class ◽  
Explicit Formulae

We continue the study of intersection algebras [Formula: see text] of two ideals [Formula: see text] in a commutative Noetherian ring [Formula: see text]. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when [Formula: see text] is a polynomial ring over a field and [Formula: see text] are principal monomial ideals. Specifically, we calculate the [Formula: see text]-signature, divisor class group, and Hilbert–Samuel and Hilbert–Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formulæ for the [Formula: see text]-signature and Hilbert–Kunz multiplicity, dependent on families of parameters, are provided.

scholarly journals Cohomological dimension with respect to the linked ideals

2020 ◽  
pp. 2150104
Author(s):  
Maryam Jahangiri ◽  
Khadijeh Sayyari
Keyword(s):  
Noetherian Ring ◽  
Cohomological Dimension ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
The Ideal

Let [Formula: see text] be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if [Formula: see text] is an ideal of [Formula: see text] which is linked by the ideal [Formula: see text], then [Formula: see text] where [Formula: see text]. Also, it is shown that for every ideal [Formula: see text] which is geometrically linked with [Formula: see text] [Formula: see text] does not depend on [Formula: see text].

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

scholarly journals On a class of dual Rickart modules

2020 ◽  
Vol 72 (7) ◽  
pp. 960-970
Author(s):  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Finite Set ◽  
Rickart Modules

UDC 512.5 Let R be a ring and let Ω R be the set of maximal right ideals of R . An R -module M is called an sd-Rickart module if for every nonzero endomorphism f of M , ℑ f is a fully invariant direct summand of M . We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R -module M , provided R is a commutative noetherian ring and A s s ( M ) ∩ Ω R is a finite set. In addition, we introduce and study ageneralization of sd-Rickart modules.

Noetherian domains in which every ideal is isomorphic to a trace ideal

2019 ◽  
Vol 19 (10) ◽  
pp. 2050200
Author(s):  
A. Mimouni
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Large Family ◽  
One Dimensional ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Trace Ideal

This paper seeks an answer to the following question: Let [Formula: see text] be a Noetherian ring with [Formula: see text]. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain [Formula: see text] with [Formula: see text], every ideal is isomorphic to a trace ideal if and only if either [Formula: see text] is a DVR or [Formula: see text] is one-dimensional divisorial domain, [Formula: see text] is a principal ideal of [Formula: see text] and [Formula: see text] posses the property that every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. Next, we globalize our result by showing that a Noetherian domain [Formula: see text] with [Formula: see text] has every ideal isomorphic to a trace ideal if and only if either [Formula: see text] is a PID or [Formula: see text] is one-dimensional divisorial domain, every invertible ideal of [Formula: see text] is principal and for every non-invertible maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] is a principal ideal of [Formula: see text] and every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

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