scholarly journals Lower and upper bounds for Cohen-Macaulay dimension

2005 ◽  
Vol 71 (2) ◽  
pp. 337-346
Author(s):  
J. Asadollahi ◽  
Sh. Salarain
Keyword(s):  
Noetherian Ring ◽  
Upper Bounds ◽  
Finitely Generated ◽  
Lower And Upper Bounds ◽  
Finitely Generated Module ◽  
Basic Properties ◽  
Homological Dimensions

Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen–Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula.

Artinianness and Attached Primes of Formal Local Cohomology Modules

2014 ◽  
Vol 21 (02) ◽  
pp. 307-316 ◽  
Author(s):  
Mohammad Hasan Bijan-Zadeh ◽  
Shahram Rezaei
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Upper Bounds ◽  
Finitely Generated ◽  
Lower And Upper Bounds ◽  
Local Cohomology Modules ◽  
Formal Local Cohomology ◽  
Cohomology Modules

Let 𝔞 be an ideal of a local ring (R, 𝔪) and M a finitely generated R-module. In this paper we study the Artinianness properties of formal local cohomology modules and we obtain the lower and upper bounds for Artinianness of formal local cohomology modules. Additionally, we determine the set [Formula: see text] and we show that the set of all non-isomorphic formal local cohomology modules [Formula: see text] is finite.

scholarly journals REFLEXIVITY AND CONNECTEDNESS

2014 ◽  
Vol 57 (1) ◽  
pp. 231-240 ◽  
Author(s):  
SEAN SATHER-WAGSTAFF
Keyword(s):  
Noetherian Ring ◽  
Prime Spectrum ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring

AbstractGiven a finitely generated module over a commutative noetherian ring that satisfies certain reflexivity conditions, we show how failure of the semidualizing property for the module manifests in a disconnection of the prime spectrum of the ring.

scholarly journals On the Diameter of Dual Graphs of Stanley-Reisner Rings and Hirsch Type Bounds on Abstractions of Polytopes

10.37236/6831 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Brent Holmes
Keyword(s):  
Noetherian Ring ◽  
Dual Graph ◽  
Upper Bounds ◽  
Maximum Diameter ◽  
Prime Ideals ◽  
Lower And Upper Bounds ◽  
Commutative Noetherian Ring ◽  
Positive Dimension ◽  
Dual Graphs ◽  
Minimal Prime

Let $R$ be an equidimensional commutative Noetherian ring of positive dimension. The dual graph $\mathcal{G} (R)$ of $R$ is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height$(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of dual graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of variables and the dimension. Dual graphs of $(S_2)$ Stanley-Reisner rings are a natural abstraction of the $1$-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type bounds on $1$-skeletons of polyhedra.

scholarly journals A note on the countable union of prime submodules

2001 ◽  
Vol 27 (10) ◽  
pp. 641-643 ◽  
Author(s):  
M. R. Pournaki ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Countable Family ◽  
Prime Ideals ◽  
Countable Union ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Prime Submodules

LetMbe a finitely-generated module over a Noetherian ringR. Suppose𝔞is an ideal ofRand letN=𝔞Mand𝔟=Ann(M/N). If𝔟⫅J(R),Mis complete with respect to the𝔟-adic topology,{Pi}i≥1is a countable family of prime submodules ofM, andx∈M, thenx+N⫅∪i≥1Piimplies thatx+N⫅Pjfor somei≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.

scholarly journals s-Sequences and Monomial Modules

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2659
Author(s):  
Gioia Failla ◽  
Paola Lea Staglianó
Keyword(s):  
Noetherian Ring ◽  
Symmetric Algebra ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Syzygy Module ◽  
Homological Invariants

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noetherian ring with unit, has a specific initial module with respect to an admissible order, provided M is generated by an s-sequence. Significant examples complement the results.

scholarly journals On the Aα-Eigenvalues of Signed Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1990
Author(s):  
Germain Pastén ◽  
Oscar Rojo ◽  
Luis Medina
Keyword(s):  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Signed Graph ◽  
Lower And Upper Bounds ◽  
Signed Graphs ◽  
Basic Properties ◽  
Underlying Graph ◽  
Positive Semidefiniteness ◽  
Vertex Degrees

For α∈[0,1], let Aα(Gσ)=αD(G)+(1−α)A(Gσ), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(Gσ) is the adjacency matrix of the signed graph Gσ whose underlying graph is G. In this paper, basic properties of Aα(Gσ) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained.

scholarly journals Gorenstein dimensions over some rings of the form R ⊕ C

2016 ◽  
Vol 15 (03) ◽  
pp. 1650043 ◽  
Author(s):  
Pye Phyo Aung
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Positive Characteristic ◽  
Finiteness Condition ◽  
Trivial Extension ◽  
Commutative Noetherian Ring ◽  
Basic Properties ◽  
Semidualizing Module ◽  
Homological Dimensions ◽  
Amalgamated Duplication

Given a semidualizing module [Formula: see text] over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205(2) (2006) 423–445] investigate some connections between [Formula: see text]-Gorenstein dimensions of an [Formula: see text]-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” [Formula: see text]. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra 216(1) (2012) 115–118].

scholarly journals Some basic properties of Sombor indices

2021 ◽  
Vol 4 (1) ◽  
pp. 1-3
Author(s):  
Ivan Gutman ◽  
Keyword(s):  
Molecular Structure ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Lower And Upper Bounds ◽  
Basic Properties ◽  
Special Case

The recently introduced class of vertex-degree-based molecular structure descriptors, called Sombor indices (\(SO\)), are examined and a few of their basic properties established. Simple lower and upper bounds for \(SO\) are determined. It is shown that any vertex-degree-based descriptor can be viewed as a special case of a Sombor-type index.

scholarly journals An intrinsic definition of the Rees algebra of a module

2017 ◽  
Vol 61 (1) ◽  
pp. 13-30
Author(s):  
Gustav Sædén Ståhl
Keyword(s):  
Noetherian Ring ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Definition Of ◽  
Free Modules ◽  
Divided Powers ◽  
Intrinsic Definition

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

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