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Journal articles on the topic 'Zariski topology'

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Published: 6 September 2023

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1

Watase, Yasushige. "Zariski Topology." Formalized Mathematics 26, no. 4 (December 1, 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 commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h−1(𝔭) where 𝔭 2 Spec B.
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2

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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3

Rinaldi, Davide, Giovanni Sambin, and Peter Schuster. "The Basic Zariski Topology." Confluentes Mathematici 7, no. 1 (February 3, 2016): 55–81. http://dx.doi.org/10.5802/cml.18.

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4

Mustafa, Hadi J., and Ameer Mohammad-Husain Hassan. "Near Prime Spectrum." Journal of Kufa for Mathematics and Computer 1, no. 8 (December 30, 2013): 58–70. http://dx.doi.org/10.31642/jokmc/2018/010808.

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Let  be a commutative ring with identity . It is well known that a topology was defined for  called the Zariski topology (prime spectrum) . In this paper we will generalize this idea for near prime ideal . If  be a commutative near-ring with identity ,  be a near prime ideal of  and define  . Then  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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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 (May 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

Ballal, Sachin, and Vilas Kharat. "Zariski topology on lattice modules." Asian-European Journal of Mathematics 08, no. 04 (November 17, 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 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].
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8

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

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9

GOODEARL, K. R., and E. S. LETZTER. "THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS." Journal of Algebra and Its Applications 05, no. 06 (December 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 whose (Gabriel-Rentschler) Krull dimension is a countable ordinal, then each closed set of the refined Zariski topology is the union of a finite set with a Zariski closed set. The approach requires certain auxiliary results guaranteeing embeddings of factor rings into direct products of simple modules. Analysis of these embeddings mimics earlier work of the first author and Zimmermann-Huisgen on products of torsion modules.
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10

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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11

Abbasi, A., and D. Hassanzadeh-lelekaami. "Quasi-prime Submodules and Developed Zariski Topology." Algebra Colloquium 19, spec01 (October 31, 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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12

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 conditions on u.Specs(M) with the dual Zarsiki topology. Also, we topologize u.Specs(M) with the patch topology and the finer patch topology. We show that for every left R-module M, u.Specs(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u.Specs(M) is a compact space with the patch and finer patch topology. Finally, by applying Hochster?s characterization of a spectral space, we show that if M is an Artinian left R-module, then u.Specs(M) with the dual Zariski topology is a spectral space.
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13

Kulak, Ö., and B. N. Türkmen. "Zariski topology over multiplication Krasner hypermodules." Ukrains’kyi Matematychnyi Zhurnal 74, no. 4 (May 20, 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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14

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

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15

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

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16

Aqalmoun, Mohamed. "The Zariski topology graph on scheme." Asian-European Journal of Mathematics 13, no. 04 (December 31, 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: see text] to [Formula: see text], denoted by [Formula: see text], whenever [Formula: see text]. In this paper, we study the connectivity properties of the graph [Formula: see text], we establish the relationship between the connectivity of the graph [Formula: see text] and the structure of irreducible components of the scheme [Formula: see text]. Also, we characterize when the complement graph of the Zariski topology graph [Formula: see text] is a complete multipartite graph.
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17

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

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18

Abad, Manuel, Diego Castaño, and José P. Díaz Varela. "Zariski-type topology for implication algebras." Mathematical Logic Quarterly 56, no. 3 (May 19, 2010): 299–309. http://dx.doi.org/10.1002/malq.200910012.

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19

AZIZI, A. "STRONGLY IRREDUCIBLE IDEALS." Journal of the Australian Mathematical Society 84, no. 2 (April 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 topologies.
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20

Bataineh, Malik, Azzh Saad Alshehry, and Rashid Abu-Dawwas. "Zariski Topologies on Graded Ideals." Tatra Mountains Mathematical Publications 78, no. 1 (October 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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21

Ansari-Toroghy, H., and F. Farshadifar. "The Zariski Topology on the Second Spectrum of a Module." Algebra Colloquium 21, no. 04 (October 6, 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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22

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 (December 1, 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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23

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

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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 (November 6, 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 in fact the restriction of Zariski topology to the lattice of ideals, but we use it for simplicity. Finally, based on pure ideals, we define the normal De Morgan residuated lattice (L is normal iff every proper ideal of L is a pure ideal) and we offer some characterizations.
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25

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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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

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

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28

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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29

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

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30

Türkmen, Ergül, Burcu Nişancı Türkmen, and Öznur Kulak. "Spectrum of Zariski Topology in Multiplication Krasner Hypermodules." Mathematics 11, no. 7 (April 6, 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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31

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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32

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

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33

Marty, Florian. "Relative Zariski Open Objects." Journal of K-Theory 10, no. 1 (January 31, 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 whose points are prime ideals of A and whose open subsets are defined by the same formula as in rings. As a consequence, we can compare the notions of scheme over in [D] and in [TV].
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34

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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35

Bannai, Shinzo, and Momoko Ohno. "Two-graphs and the Embedded Topology of Smooth Quartics and its Bitangent Lines." Canadian Mathematical Bulletin 63, no. 4 (January 24, 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 arrangements consisting of a smooth quartic and five of its bitangent lines and six of its bitangent lines, respectively.
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36

Panpho, Phakakorn, and Pairote Yiarayong. "Zariski topology on the spectrum of fuzzy classical primary submodules." Applied General Topology 23, no. 2 (October 3, 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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37

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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38

Kaczynski, Tomasz, Marian Mrozek, and Anik Trahan. "Ideas from Zariski Topology in the Study of Cubical Homology." Canadian Journal of Mathematics 59, no. 5 (October 1, 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 algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.
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39

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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40

Ansari-Toroghy, H., and Sh Habibi. "The Zariski Topology-Graph of Modules Over Commutative Rings." Communications in Algebra 42, no. 8 (April 4, 2014): 3283–96. http://dx.doi.org/10.1080/00927872.2013.780065.

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41

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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42

VARADARAJAN, K. "CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS." Journal of Algebra and Its Applications 06, no. 04 (August 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 an almost clean ring and proved that C(X) is almost clean if and only if it is clean. We prove a similar result for all Gelfand rings R with J(R) = 0.
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43

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

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44

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 (October 3, 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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46

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><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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47

Eyral, Christophe, and Mutsuo Oka. "Alexander-equivalent Zariski pairs of irreducible sextics." Journal of Topology 2, no. 3 (2009): 423–41. http://dx.doi.org/10.1112/jtopol/jtp017.

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Al-Zoubi, Khaldoun, and Malik Jaradat. "The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings." Tatra Mountains Mathematical Publications 74, no. 1 (December 1, 2019): 7–16. http://dx.doi.org/10.2478/tmmp-2019-0015.

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Abstract Let G be a group with identity e and let R be a G-graded ring. A proper graded ideal P of R is called a graded primary ideal if whenever rgsh∈P, we have rg∈ P or sh∈ Gr(P), where rg,sg∈ h(R). The graded primary spectrum p.Specg(R) is defined to be the set of all graded primary ideals of R.In this paper, we define a topology on p.Specg(R), called Zariski topology, which is analogous to that for Specg(R), and investigate several properties of the topology.
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Çeken, Seçil, and Mustafa Alkan. "On the second spectrum and the second classical Zariski topology of a module." Journal of Algebra and Its Applications 14, no. 10 (September 2015): 1550150. http://dx.doi.org/10.1142/s0219498815501509.

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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.
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Abu-Dawwas, Rashid. "Zariski topology on the spectrum of graded pseudo prime submodules." Boletim da Sociedade Paranaense de Matemática 39, no. 3 (January 1, 2021): 17–26. http://dx.doi.org/10.5269/bspm.39909.

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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.
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