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.

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.

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 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).

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 Zariski topology on the spectrum of graded pseudo prime submodules

2021 ◽  
Vol 39 (3) ◽  
pp. 17-26
Author(s):  
Rashid Abu-Dawwas
Keyword(s):  
Topological Space ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Graded Modules ◽  
Prime Submodules ◽  
Algebraic Properties

In this article, we introduce the concept of graded pseudo prime submodules of graded modules that is a generalization of the graded prime ideals over commutative rings. We study the Zariski topology on the graded spectrum of graded pseudo prime submodules. We clarify the relation between the properties of this topological space and the algebraic properties of the graded modules under consideration.

Zariski topology on lattice modules

2015 ◽  
Vol 08 (04) ◽  
pp. 1550066 ◽  
Author(s):  
Sachin Ballal ◽  
Vilas Kharat
Keyword(s):  
New York ◽  
Spectral Theory ◽  
Zariski Topology ◽  
Topological Properties ◽  
Algebraic Properties ◽  
Prime Elements ◽  
Multiplicative Lattices

Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

γ-Operation and Some Types of Soft Sets and Soft Continuity of (Supra) Soft Topological Spaces

2016 ◽  
pp. 127-171
Author(s):  
Ali Kandil ◽  
Osama A. El-Tantawy ◽  
Sobhy A. El-Sheikh ◽  
A. M. Abd El-latif
Keyword(s):  
Topological Space ◽  
Soft Set ◽  
Topological Spaces ◽  
Topological Properties ◽  
Soft Sets ◽  
Soft Mapping ◽  
Separation Axioms ◽  
Different Types

The main purpose of this chapter is to introduce the notions of ?-operation, pre-open soft set a-open sets, semi open soft set and ß-open soft sets to soft topological spaces. We study the relations between these different types of subsets of soft topological spaces. We introduce new soft separation axioms based on the semi open soft sets which are more general than of the open soft sets. We show that the properties of soft semi Ti-spaces (i=1,2) are soft topological properties under the bijection and irresolute open soft mapping. Also, we introduce the notion of supra soft topological spaces. Moreover, we introduce the concept of supra generalized closed soft sets (supra g-closed soft for short) in a supra topological space (X,µ,E) and study their properties in detail.

The flat topology on the minimal and maximal prime spectrum of a commutative ring

2019 ◽  
Vol 18 (06) ◽  
pp. 1950110
Author(s):  
Esmaeil Rostami ◽  
Masoumeh Hedayati ◽  
Nosratollah Shajareh Poursalavati
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Compact Topological Space ◽  
Algebraic Properties ◽  
Ascending Chain Condition

In this paper, we investigate connections between some algebraic properties of commutative rings and topological properties of their minimal and maximal prime spectrum with respect to the flat topology. We show that for a commutative ring [Formula: see text], the ascending chain condition on principal annihilator ideals of [Formula: see text] holds if and only if [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology and we give a characterization for a topological space [Formula: see text] for which [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology. Also, we give a characterization for rings whose maximal prime spectrum is a compact topological space with respect to the flat topology. Some other results are obtained too.

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.

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