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Journal articles on the topic 'Zariski topology'

Author: Grafiati

Published: 6 September 2023

Last updated: 25 July 2025

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1

Watase, Yasushige. "Zariski Topology." Formalized Mathematics 26, no. 4 (2018): 277–83. http://dx.doi.org/10.2478/forma-2018-0024.

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Summary We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutat

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2

Mustafa, Hadi J., and Ameer Mohammad-Husain Hassan. "Near Prime Spectrum." Journal of Kufa for Mathematics and Computer 1, no. 8 (2013): 58–70. http://dx.doi.org/10.31642/jokmc/2018/010808.

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Let Â be a commutative ring with identity . It is well known that a topology was defined for Â called the Zariski topology (prime spectrum) . In this paper we will generalize this idea for near prime ideal . If Â be a commutative near-ring with identity , Â be a near prime ideal of Â and define Â . Then Â can be endowed with a topology similar to the Zariski topology which is called near Zariski topology (near prime spectrum) . we studies and discuss some of properties of such topology .

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3

Hassanzadeh-Lelekaami, Dawood, and Maryam Karimi. "Developed Zariski topology-graph." Discussiones Mathematicae - General Algebra and Applications 37, no. 2 (2017): 233. http://dx.doi.org/10.7151/dmgaa.1272.

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4

Rinaldi, Davide, Giovanni Sambin, and Peter Schuster. "The Basic Zariski Topology." Confluentes Mathematici 7, no. 1 (2016): 55–81. http://dx.doi.org/10.5802/cml.18.

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5

Dikranjan, Dikran, and Daniele Toller. "Zariski topology and Markov topology on groups." Topology and its Applications 241 (June 2018): 115–44. http://dx.doi.org/10.1016/j.topol.2018.03.025.

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6

JUNKER, MARKUS, and DANIEL LASCAR. "THE INDISCERNIBLE TOPOLOGY: A MOCK ZARISKI TOPOLOGY." Journal of Mathematical Logic 01, no. 01 (2001): 99–124. http://dx.doi.org/10.1142/s0219061301000041.

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We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies (one topology on each Mn). The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.

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7

GOODEARL, K. R., and E. S. LETZTER. "THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS." Journal of Algebra and Its Applications 05, no. 06 (2006): 719–30. http://dx.doi.org/10.1142/s0219498806001922.

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In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simple left modules over a ring R. Namely, if R is left noetherian (or satisfies the ascending chain condition for semiprimitive ideals), and if R is either a countable dimensional algebra (over a field) or a ring who

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8

Ballal, Sachin, and Vilas Kharat. "Zariski topology on lattice modules." Asian-European Journal of Mathematics 08, no. 04 (2015): 1550066. http://dx.doi.org/10.1142/s1793557115500667.

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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 uni

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9

Abuhlail, Jawad. "A Zariski Topology for Modules." Communications in Algebra 39, no. 11 (2011): 4163–82. http://dx.doi.org/10.1080/00927872.2010.519748.

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10

Xie, Junyi. "The existence of Zariski dense orbits for endomorphisms of projective surfaces." Journal of the American Mathematical Society 38, no. 1 (2022): 1–62. http://dx.doi.org/10.1090/jams/1004.

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Let f f be a dominant endomorphism of a smooth projective surface X X over an algebraically closed field k \mathbf {k} of characteristic 0 0 . We prove that if there is no rational function H ∈ k ( X ) H \in \mathbf {k}(X) such that H ∘ f = H H \circ f = H , then there exists a point x ∈ X ( k ) x \in X(\mathbf {k}) such that the forward orbit of x x under f f is Zariski dense in X X . This result gives us a positive answer to the Zariski dense orbit conjecture for endomorphisms of smooth projective surfaces. We also define a new topology on varieties over algebraically closed fields with fini

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11

Mouadi, Hassan, and Driss Karim. "Some topology on zero-dimensional subrings of product of rings." Filomat 34, no. 14 (2020): 4589–95. http://dx.doi.org/10.2298/fil2014589m.

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Let R be a ring and {Ri}i?I a family of zero-dimensional rings. We define the Zariski topology on Z(R,?Ri) and study their basic properties. Moreover, we define a topology on Z(R,?Ri) by using ultrafilters; it is called the ultrafilter topology and we demonstrate that this topology is finer than the Zariski topology. We show that the ultrafilter limit point of a collections of subrings of Z(R,?Ri) is a zero-dimensional ring. Its relationship with F-lim and the direct limit of a family of rings are studied.

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12

AZIZI, A. "STRONGLY IRREDUCIBLE IDEALS." Journal of the Australian Mathematical Society 84, no. 2 (2008): 145–54. http://dx.doi.org/10.1017/s1446788708000062.

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AbstractA proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, $A\cap B \subseteq I$ implies that either $A \subseteq I$ or $B \subseteq I$. In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace top

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13

Abbasi, A., and D. Hassanzadeh-lelekaami. "Quasi-prime Submodules and Developed Zariski Topology." Algebra Colloquium 19, spec01 (2012): 1089–108. http://dx.doi.org/10.1142/s1005386712000879.

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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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14

Bataineh, Malik, Azzh Saad Alshehry, and Rashid Abu-Dawwas. "Zariski Topologies on Graded Ideals." Tatra Mountains Mathematical Publications 78, no. 1 (2021): 215–24. http://dx.doi.org/10.2478/tmmp-2021-0015.

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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.

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15

Çeken, Seçil. "On the upper dual Zariski topology." Filomat 34, no. 2 (2020): 483–89. http://dx.doi.org/10.2298/fil2002483c.

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Let R be a ring with identity and M be a left R-module. The set of all second submodules of M is called the second spectrum of M and denoted by Specs(M). For each prime ideal p of R we define Specsp(M) := {S? Specs(M) : annR(S) = p}. A second submodule Q of M is called an upper second submodule if there exists a prime ideal p of R such that Specs p(M)? 0 and Q = ? S2Specsp(M)S. The set of all upper second submodules of M is called upper second spectrum of M and denoted by u.Specs(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological condi

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16

Kulak, Ö., and B. N. Türkmen. "Zariski topology over multiplication Krasner hypermodules." Ukrains’kyi Matematychnyi Zhurnal 74, no. 4 (2022): 525–33. http://dx.doi.org/10.37863/umzh.v74i4.6626.

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UDC 512.5 In this paper, we introduce the notion of multiplication Krasner hypermodules over commutative hyperrings and topologize the collection of all multiplication Кrasner hypermodules. In addition, we investigate some properties of this topological space.

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17

Gierz, Gerhard, and Albert Stralka. "The Zariski Topology for distributive lattices." Rocky Mountain Journal of Mathematics 17, no. 2 (1987): 195–218. http://dx.doi.org/10.1216/rmj-1987-17-2-195.

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18

Abuhlail, Jawad. "A dual Zariski topology for modules." Topology and its Applications 158, no. 3 (2011): 457–67. http://dx.doi.org/10.1016/j.topol.2010.11.021.

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19

Aqalmoun, Mohamed. "The Zariski topology graph on scheme." Asian-European Journal of Mathematics 13, no. 04 (2018): 2050075. http://dx.doi.org/10.1142/s1793557120500758.

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Let [Formula: see text] be a quasi-compact scheme and [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the set of closed points of [Formula: see text] and the closure of the subset [Formula: see text]. Let [Formula: see text] be a nonempty subset of [Formula: see text]. We define the [Formula: see text]-Zariski topology graph on the scheme [Formula: see text], denoted by [Formula: see text], as an undirected graph whose vertex set is the set [Formula: see text], for two distinct vertices [Formula: see text] and [Formula: see text], there is an arc from [Formula: s

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20

Schuster, Peter. "Formal Zariski topology: Positivity and points." Annals of Pure and Applied Logic 137, no. 1-3 (2006): 317–59. http://dx.doi.org/10.1016/j.apal.2005.05.026.

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21

Abad, Manuel, Diego Castaño, and José P. Díaz Varela. "Zariski-type topology for implication algebras." Mathematical Logic Quarterly 56, no. 3 (2010): 299–309. http://dx.doi.org/10.1002/malq.200910012.

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22

Ansari-Toroghy, H., and F. Farshadifar. "The Zariski Topology on the Second Spectrum of a Module." Algebra Colloquium 21, no. 04 (2014): 671–88. http://dx.doi.org/10.1142/s1005386714000625.

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Let R be a commutative ring and M be an R-module. The second spectrum Spec s(M) of M is the collection of all second submodules of M. We topologize Spec s(M) with Zariski topology, which is analogous to that for Spec (M), and investigate this topological space. For various types of modules M, we obtain conditions under which Spec s(M) is a spectral space. We also investigate Spec s(M) with quasi-Zariski topology.

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23

Ansari-Toroghy, H., and Sh Habibi. "The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 3 (2018): 41–56. http://dx.doi.org/10.2478/auom-2018-0032.

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AbstractLet M be a module over a commutative ring and let Max(M) be the collection of all maximal submodules of M. We topologize Max(M) with quasi-Zariski topology, where M is a Max-top module. For a subset T of Max(M), we introduce a new graph $G(\tau_T^{*m})$, called the quasi-Zariski topology-graph on the maximal spectrum of M. It helps us to study algebraic (resp. topological) properties of M (resp. Max(M)) by using the graphs theoretical tools.

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24

Holdon, Liviu-Constantin. "The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices." Open Mathematics 18, no. 1 (2020): 1206–26. http://dx.doi.org/10.1515/math-2020-0061.

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Abstract In this paper, by using the ideal theory in residuated lattices, we construct the prime and maximal spectra (Zariski topology), proving that the prime and maximal spectra are compact topological spaces, and in the case of De Morgan residuated lattices they become compact {T}_{0} topological spaces. At the same time, we define and study the reticulation functor between De Morgan residuated lattices and bounded distributive lattices. Moreover, we study the I-topology (I comes from ideal) and the stable topology and we define the concept of pure ideal. We conclude that the I-topology is

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25

Abuhlail, Jawad Y. "A Zariski Topology for Bicomodules and Corings." Applied Categorical Structures 16, no. 1-2 (2007): 13–28. http://dx.doi.org/10.1007/s10485-007-9088-1.

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26

Ansari-Toroghy, Habibollah, Shokoufeh Habibi, and Masoomeh Hezarjaribi. "On the graph of modules over commutative rings II." Filomat 32, no. 10 (2018): 3657–65. http://dx.doi.org/10.2298/fil1810657a.

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Let M be a module over a commutative ring R. In this paper, we continue our study about the quasi-Zariski topology-graph G(?*T) which was introduced in (On the graph of modules over commutative rings, Rocky Mountain J. Math. 46(3) (2016), 1-19). For a non-empty subset T of Spec(M), we obtain useful characterizations for those modules M for which G(?*T) is a bipartite graph. Also, we prove that if G(?*T) is a tree, then G(?*T) is a star graph. Moreover, we study coloring of quasi-Zariski topology-graphs and investigate the interplay between ?(G(?+T)) and ?(G(?+T)).

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27

Degtyarev, Alex. "Classical Zariski pairs." Journal of Singularities 2 (2010): 51–55. http://dx.doi.org/10.5427/jsing.2010.2c.

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28

Hussein, Salah El Din S. "Control subgroups and birational extensions of graded rings." International Journal of Mathematics and Mathematical Sciences 22, no. 2 (1999): 411–15. http://dx.doi.org/10.1155/s0161171299224118.

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In this paper, we establish the relation between the concept of control subgroups and the class of graded birational algebras. Actually, we prove that ifR=⊕σ∈GRσis a stronglyG-graded ring andH⊲G, then the embeddingi:R(H)↪R, whereR(H)=⊕σ∈HRσ, is a Zariski extension if and only ifHcontrols the filterℒ(R−P)for every prime idealPin an open set of the Zariski topology onR. This enables us to relate certain ideals ofRandR(H)up to radical.

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Bannai, Shinzo, and Momoko Ohno. "Two-graphs and the Embedded Topology of Smooth Quartics and its Bitangent Lines." Canadian Mathematical Bulletin 63, no. 4 (2020): 802–12. http://dx.doi.org/10.4153/s0008439520000053.

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AbstractIn this paper, we study how to distinguish the embedded topology of a smooth quartic and its bitangent lines. In order to do this, we introduce the concept of two-graphs and switching classes from graph theory. This new method improves previous results about a quartic and three bitangent lines considered by E. Artal Bartolo and J. Vallès, four bitangent lines considered by the authors and H. Tokunaga, and enables us to distinguish the embedded topology of a smooth quartic and five or more bitangent lines. As an application, we obtain a new Zariski 5-tuple and a Zariski 9-tuple for arra

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Marty, Florian. "Relative Zariski Open Objects." Journal of K-Theory 10, no. 1 (2012): 9–39. http://dx.doi.org/10.1017/is011012004jkt176.

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AbstractIn [TV], Bertrand Toën and Michel Vaquié define a scheme theory for a closed monoidal category (,⊗, 1) One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative monoidal objects in . The purpose of this article is to prove that under some hypotheses, Zariski open subobjects of affine schemes can be classified almost as in the usual case of rings (ℤ-mod,⊗,ℤ). The main result states that for any commutative monoidal object A in , the locale of Zariski open subobjects of the affine scheme Spec(A) is associated to a topological space w

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31

Gierz, Gerhard, and Albert Stralka. "The Zariski topology and essential extensions of semilattices." Journal of Pure and Applied Algebra 68, no. 1-2 (1990): 135–48. http://dx.doi.org/10.1016/0022-4049(90)90139-9.

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32

Türkmen, Ergül, Burcu Nişancı Türkmen, and Öznur Kulak. "Spectrum of Zariski Topology in Multiplication Krasner Hypermodules." Mathematics 11, no. 7 (2023): 1754. http://dx.doi.org/10.3390/math11071754.

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In this paper, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings. In particular, we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hypermodules. Moreover, we provide some relevant properties of the hypermodule in this topological hyperspace.

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Yildiz, Eda, Bayram Ersoy, and Ünsal Tekir. "S-Zariski topology on S-spectrum of modules." Filomat 36, no. 20 (2022): 7103–12. http://dx.doi.org/10.2298/fil2220103y.

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Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, first we give some relations between S-prime and S-maximal submodules that are generalizations of prime and maximal submodules, respectively. Then we construct a topology on the set of all S-prime submodules of M , which is generalization of prime spectrum of M. We investigate when SpecS(M) is T0 and T1-space. We also study on some continuous maps and irreducibility on SpecS(M). Moreover, we introduce the notion of S-radical of a submodule N of M and use it to show the irreducibility of S-variety VS(N).

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Dikranjan, Dikran, and Dmitri Shakhmatov. "The Markov–Zariski topology of an abelian group." Journal of Algebra 324, no. 6 (2010): 1125–58. http://dx.doi.org/10.1016/j.jalgebra.2010.04.025.

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35

Dumas, David, and Richard P. Kent. "Bers slices are Zariski dense." Journal of Topology 2, no. 2 (2009): 373–79. http://dx.doi.org/10.1112/jtopol/jtp014.

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Giuli, Eraldo. "Zariski closure, completeness and compactness." Topology and its Applications 153, no. 16 (2006): 3158–68. http://dx.doi.org/10.1016/j.topol.2005.04.014.

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Panpho, Phakakorn, and Pairote Yiarayong. "Zariski topology on the spectrum of fuzzy classical primary submodules." Applied General Topology 23, no. 2 (2022): 333–43. http://dx.doi.org/10.4995/agt.2022.17427.

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Let R be a commutative ring with identity and M a unitary R-module. The fuzzy classical primary spectrum F cp.spec(M) is the collection of all fuzzy classical primary submodules A of M, the recent generalization of fuzzy primary ideals and fuzzy classical prime submodules. In this paper, we topologize FM(M) with a topology having the fuzzy primary Zariski topology on the fuzzy classical primary spectrum F cp.spec(M) as a subspace topology, and investigate the properties of this topological space.

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Moghimi, Hosein, and Javad Harehdashti. "The Radical-Zariski topology on the radical spectrum of modules." Filomat 36, no. 9 (2022): 3037–50. http://dx.doi.org/10.2298/fil2209037m.

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For a module M over a commutative ring R with identity, let RSpec(M) denote the collection of all submodules L of Msuch that ?(L:M) is a prime ideal of R and is equal to (rad L:M). In this article, we topologies RSpec(M) with a topology which enjoys analogs of many of the properties of the Zariski topology on the prime spectrum Spec(M) (as a subspace topology). We investigate this topological space from the point of view of spectral spaces by establishing interrelations between RSpec(M) and Spec(R/Ann(M)).

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Kaczynski, Tomasz, Marian Mrozek, and Anik Trahan. "Ideas from Zariski Topology in the Study of Cubical Homology." Canadian Journal of Mathematics 59, no. 5 (2007): 1008–28. http://dx.doi.org/10.4153/cjm-2007-043-3.

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AbstractCubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ℝd in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorit

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VARADARAJAN, K. "CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS." Journal of Algebra and Its Applications 06, no. 04 (2007): 671–85. http://dx.doi.org/10.1142/s0219498807002466.

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We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology. In the class of rings C(X), W. Wm Mc Govern [6] has characterized potent rings as the ones with X admitting a clopen π-base. We prove the analogous result for any commutative ring in terms of the Zariski topology on Max(R). Mc Govern also introduced the concept of

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Tekir, Ünsal. "The Zariski Topology on the Prime Spectrum of a Module over Noncommutative Rings." Algebra Colloquium 16, no. 04 (2009): 691–98. http://dx.doi.org/10.1142/s1005386709000650.

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Salam, Saif, and Khaldoun Al-Zoubi. "The Zariski topology on the graded primary spectrum of a graded module over a graded commutative ring." Applied General Topology 23, no. 2 (2022): 345–61. http://dx.doi.org/10.4995/agt.2022.16332.

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Let R be a G-graded ring and M be a G-graded R-module. We define the graded primary spectrum of M, denoted by PSG(M), to be the set of all graded primary submodules Q of M such that (GrM(Q) :RM) = Gr((Q:RM)). In this paper, we define a topology on PSG(M) having the Zariski topology on the graded prime spectrum SpecG(M) as a subspace topology, and investigate several topological properties of this topological space.

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Hassanzadeh-Lelekaami, Dawood. "Attaching a topological space to a module." Filomat 32, no. 9 (2018): 3171–80. http://dx.doi.org/10.2298/fil1809171h.

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Let R be a commutative ring with identity and let M be an R-module. We investigate when the strongly prime spectrum of M has a Zariski topology analogous to that for R. We provide some examples of such modules.

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Çeken, Seçil. "Dual Zariski Spaces of Modules." Algebra Colloquium 30, no. 04 (2023): 569–84. http://dx.doi.org/10.1142/s1005386723000445.

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Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an [Formula: see text]-module, [Formula: see text] denote the set of all submodules of [Formula: see text] and [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we set [Formula: see text] and [Formula: see text]. Consider [Formula: see text], where [Formula: see text] is the set of all ideals of [Formula: see text]. We set [Formula: see text] and [Formula: see text] for any ideal [Formula: see text] of [Formula: see text]. In this paper, we investigate when, for arbitrary [Formu

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Sharma, Poonam Kumar. "Zariski topology on the spectrum of intuitionistic fuzzy classical primary submodules." Notes on Intuitionistic Fuzzy Sets 30, no. 3 (2024): 203–14. http://dx.doi.org/10.7546/nifs.2024.30.3.203-214.

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In this paper, we define and study the notion of intuitionistic fuzzy classical primary submodules over a unitary $R$-module $M$, where $R$ is a commutative ring with unity. This is a generalisation of intuitionistic fuzzy primary ideals and intuitionistic fuzzy classical prime submodules. We further topologize the collection of all intuitionistic fuzzy submodules on an $R$-module $M$ with a topology having the intuitionistic fuzzy primary Zariski topology on the intuitionistic fuzzy classical primary spectrum $IF_{cp}spec(M)$ as a subspace topology and investigate the properties of this topol

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Salam, Saif, and Khaldoun Al-Zoubi. "Graded modules with Noetherian graded second spectrum." AIMS Mathematics 8, no. 3 (2023): 6626–41. http://dx.doi.org/10.3934/math.2023335.

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Abstract:

<abstract><p>Let $ R $ be a $ G $ graded commutative ring and $ M $ be a $ G $-graded $ R $-module. The set of all graded second submodules of $ M $ is denoted by $ Spec_G^s(M), $ and it is called the graded second spectrum of $ M $. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate $ Spec^s_G(M) $ with the Zariski topology from the viewpoint of being a Noetherian space.</p></abstract>

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Camerlo, Riccardo, and Carla Massaza. "The Wadge hierarchy on Zariski topologies." Topology and its Applications 294 (May 2021): 107661. http://dx.doi.org/10.1016/j.topol.2021.107661.

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Ansari-Toroghy, H., and Sh Habibi. "The Zariski Topology-Graph of Modules Over Commutative Rings." Communications in Algebra 42, no. 8 (2014): 3283–96. http://dx.doi.org/10.1080/00927872.2013.780065.

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ÇALLIALP, Fethi, Gülşen ULUCAK, and Ünsal TEKİR. "On the Zariski topology over an $L$-module $M$." TURKISH JOURNAL OF MATHEMATICS 41 (2017): 326–36. http://dx.doi.org/10.3906/mat-1502-31.

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Yıldız, Eda, Bayram Ersoy, and Ünsal Tekir. "Erratum: S-Zariski topology on s-spectrum of modules." Filomat 37, no. 23 (2023): 8039. http://dx.doi.org/10.2298/fil2323039y.

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