Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

scholarly journals On the gamma spectrum of multiplication gamma acts

2021 ◽  
Vol 48 (2) ◽  
Author(s):  
Mehdi S. Abbas ◽  
◽  
Samer A. Gubeir ◽  
Keyword(s):  
Topological Space ◽  
Gamma Spectrum ◽  
Zariski Topology ◽  
Topological Properties ◽  
Separation Axioms ◽  
Algebraic Properties

In this paper, we introduce the concept of topological gamma acts as a generalization of Zariski topology. Some topological properties of this topology are studied. Various algebraic properties of topological gamma acts have been discussed. We clarify the interplay between this topological space's properties and the algebraic properties of the gamma acts under consideration. Also, the relation between this topological space and (multiplication, cyclic) gamma act was discussed. We also study some separation axioms and the compactness of this topological space.

On the second spectrum and the second classical Zariski topology of a module

2015 ◽  
Vol 14 (10) ◽  
pp. 1550150 ◽  
Author(s):  
Seçil Çeken ◽  
Mustafa Alkan
Keyword(s):  
Finite Number ◽  
Associative Ring ◽  
Finite Union ◽  
Zariski Topology ◽  
Topological Properties ◽  
Noetherian Module ◽  
Algebraic Properties ◽  
Second Spectrum

Let R be an associative ring with identity and Specs(M) denote the set of all second submodules of a right R-module M. In this paper, we investigate some interrelations between algebraic properties of a module M and topological properties of the second classical Zariski topology on Specs(M). We prove that a right R-module M has only a finite number of maximal second submodules if and only if Specs(M) is a finite union of irreducible closed subsets. We obtain some interrelations between compactness of the second classical Zariski topology of a module M and finiteness of the set of minimal submodules of M. We give a connection between connectedness of Specs(M) and decomposition of M for a right R-module M. We give several characterizations of a noetherian module M over a ring R such that every right primitive factor of R is artinian for which Specs(M) is connected.

scholarly journals A characterization of left semiartinian rings

1974 ◽  
Vol 11 (3) ◽  
pp. 425-428 ◽  
Author(s):  
Jonathan S. Golan
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Krull Dimension ◽  
Semiartinian Ring ◽  
Torsion Theories ◽  
Necessary And Sufficient

In defining the torsion-theoretic Krull dimension of an associative ring R we make use of a function δ from the complete lattice of all subsets of the torsion-theoretic spectrum of R to the complete lattice of all hereditary torsion theories on R-mod. In this note we give necessary and sufficient conditions for δ to be injective, surjective, and bijective. In particular, δ is bijective if and only if R is a left semiartinian ring.

SIMPLICITY OF CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS

2019 ◽  
Vol 108 (2) ◽  
pp. 202-225
Author(s):  
ALEXANDRE BARAVIERA ◽  
WAGNER CORTES ◽  
MARLON SOARES
Keyword(s):  
Hausdorff Space ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Products ◽  
Topological Properties ◽  
Partial Action ◽  
Necessary And Sufficient ◽  
Skew Group Ring

In this article, we consider a twisted partial action $\unicode[STIX]{x1D6FC}$ of a group $G$ on an associative ring $R$ and its associated partial crossed product $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$. We provide necessary and sufficient conditions for the commutativity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ when the twisted partial action $\unicode[STIX]{x1D6FC}$ is unital. Moreover, we study necessary and sufficient conditions for the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ in the following cases: (i) $G$ is abelian; (ii) $R$ is maximal commutative in $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$; (iii) $C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$ is simple; (iv) $G$ is hypercentral. When $R=C_{0}(X)$ is the algebra of continuous functions defined on a locally compact and Hausdorff space $X$, with complex values that vanish at infinity, and $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ is the associated partial skew group ring of a partial action $\unicode[STIX]{x1D6FC}$ of a topological group $G$ on $C_{0}(X)$, we study the simplicity of $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ by using topological properties of $X$ and the results about the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$.

scholarly journals Quasi-prime Submodules and Developed Zariski Topology

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1089-1108 ◽  
Author(s):  
A. Abbasi ◽  
D. Hassanzadeh-lelekaami
Keyword(s):  
Commutative Ring ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
The Relationship ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity and M be an R-module. Quasi-prime submodules of M and the developed Zariski topology on q Spec (M) are introduced. We also investigate the relationship between algebraic properties of M and topological properties of q Spec (M). Modules whose developed Zariski topology is T0, irreducible or Noetherian are studied, and several characterizations of such modules are given.

Zariski Topologies on Graded Ideals

2021 ◽  
Vol 78 (1) ◽  
pp. 215-224
Author(s):  
Malik Bataineh ◽  
Azzh Saad Alshehry ◽  
Rashid Abu-Dawwas
Keyword(s):  
Commutative Ring ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Algebraic Properties

Abstract In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.

scholarly journals Formal Analysis of SET and NSL Protocols Using the Interpretation Functions-Based Method

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Hanane Houmani ◽  
Mohamed Mejri
Keyword(s):  
Communication Channel ◽  
Formal Analysis ◽  
Sufficient Conditions ◽  
Main Idea ◽  
The Internet ◽  
Cryptographic Protocol ◽  
Cryptographic Primitives ◽  
Algebraic Properties ◽  
Security Properties ◽  
Set Protocol

Most applications in the Internet such as e-banking and e-commerce use the SET and the NSL protocols to protect the communication channel between the client and the server. Then, it is crucial to ensure that these protocols respect some security properties such as confidentiality, authentication, and integrity. In this paper, we analyze the SET and the NSL protocols with respect to the confidentiality (secrecy) property. To perform this analysis, we use the interpretation functions-based method. The main idea behind the interpretation functions-based technique is to give sufficient conditions that allow to guarantee that a cryptographic protocol respects the secrecy property. The flexibility of the proposed conditions allows the verification of daily-life protocols such as SET and NSL. Also, this method could be used under different assumptions such as a variety of intruder abilities including algebraic properties of cryptographic primitives. The NSL protocol, for instance, is analyzed with and without the homomorphism property. We show also, using the SET protocol, the usefulness of this approach to correct weaknesses and problems discovered during the analysis.

SOME TOPOLOGICAL RESULTS ON LEVEL SETS AND MULTIFRACTAL SETS

Fractals ◽  
2019 ◽  
Vol 28 (01) ◽  
pp. 2050009
Author(s):  
CHUNTAI LIU
Keyword(s):  
Level Sets ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Properties ◽  
Symbolic Space ◽  
Necessary And Sufficient

In this paper, we study topological properties of some level sets and some multifractal sets induced by Rademacher’s series and Takagi’s series, respectively. By using symbolic space, we obtain necessary and sufficient conditions for them to be residual.

scholarly journals On generalizations of C*-embedding for wallman rings

1978 ◽  
Vol 25 (2) ◽  
pp. 215-229 ◽  
Author(s):  
H. L. Bentley ◽  
B. J. Taylor
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Properties ◽  
Zero Sets ◽  
Algebraic Properties

AbstractBiles (1970) has called a subring A of the ring C(X), of all real valued continuous functions on a topological space X, a Wallman ring on X whenever Z(A), the zero sets of functions belonging to A, forms a normal base on X in the sense of Frink (1964). Previously, we have related algebraic properties of a Wallman ring A to topological properties of the Wallman compactification w(Z(A)) of X determined by the normal base Z(A). Here we introduce two different generalizations of the concept of “a C*-embedded subset” and study relationships between these and topological (respectively, algebraic) properties of w(Z(A)) (respectively, A).

NORMALITY OF f-UNITARY UNITS IN AN ALTERNATIVE LOOP RING

2006 ◽  
Vol 05 (04) ◽  
pp. 537-548
Author(s):  
EDGAR G. GOODAIRE ◽  
CÉSAR POLCINO MILIES
Keyword(s):  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Loop Ring ◽  
Necessary And Sufficient

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let f : L → {±1} be a homomorphism and for α = ∑αℓℓ in the integral loop ring Z L, define αf = ∑αℓf(ℓ)ℓ-1. A unit u ∈ Z L is said to be f-unitary if uf = ±u-1. The set [Formula: see text] of all f-unitary units is a subloop of [Formula: see text], the loop of all units in Z L. In this paper, we find necessary and sufficient conditions for [Formula: see text] to be normal in [Formula: see text].

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