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Статті в журналах з теми "An elementary divisor ring"

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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Chen, Huanyin, and Marjan Sheibani Abdolyousefi. "Elementary matrix reduction over Bézout domains." Journal of Algebra and Its Applications 18, no. 08 (July 5, 2019): 1950141. http://dx.doi.org/10.1142/s021949881950141x.

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Анотація:

A ring [Formula: see text] is an elementary divisor ring if every matrix over [Formula: see text] admits a diagonal reduction. If [Formula: see text] is an elementary divisor domain, we prove that [Formula: see text] is a Bézout duo-domain if and only if for any [Formula: see text], [Formula: see text] such that [Formula: see text]. We explore certain stable-like conditions on a Bézout domain under which it is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

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2

Zabavsky, B. V., O. Romaniv, B. Kuznitska, and T. Hlova. "Comaximal factorization in a commutative Bezout ring." Algebra and Discrete Mathematics 30, no. 1 (2020): 150–60. http://dx.doi.org/10.12958/adm1203.

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3

KABBOUR, MOHAMMED, and NAJIB MAHDOU. "AMALGAMATION OF RINGS DEFINED BY BÉZOUT-LIKE CONDITIONS." Journal of Algebra and Its Applications 10, no. 06 (December 2011): 1343–50. http://dx.doi.org/10.1142/s0219498811005683.

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Анотація:

Let f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we investigate the transfer of notions elementary divisor ring, Hermite ring and Bézout ring to the amalgamation A ⋈f J. We provide necessary and sufficient conditions for A ⋈f J to be an elementary divisor ring where A and B are integral domains. In this case it is shown that A ⋈f J is an Hermite ring if and only if it is a Bézout ring. In particular, we study the transfer of the previous notions to the amalgamated duplication of a ring A along an A-submodule E of Q(A) such that E2 ⊆ E.

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4

Gatalevich, A. I., and B. V. Zabavs'kii. "Noncommutative elementary divisor rings." Journal of Mathematical Sciences 96, no. 2 (August 1999): 3013–16. http://dx.doi.org/10.1007/bf02169697.

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5

Zabavskii, B. V. "Noncommutative elementary divisor rings." Ukrainian Mathematical Journal 39, no. 4 (1988): 349–53. http://dx.doi.org/10.1007/bf01060766.

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6

Zabavs’kyi, B. V. "A Sharp Bézout Domain is an Elementary Divisor Ring." Ukrainian Mathematical Journal 66, no. 2 (July 2014): 317–21. http://dx.doi.org/10.1007/s11253-014-0932-9.

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7

Zabavsky, B. V., and O. M. Romaniv. "Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals." Carpathian Mathematical Publications 10, no. 2 (December 31, 2018): 402–7. http://dx.doi.org/10.15330/cmp.10.2.402-407.

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Анотація:

We investigate commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals. In particular, we have described the class of such rings, which are elementary divisor rings. A ring $R$ is called an elementary divisor ring if every matrix over $R$ has a canonical diagonal reduction (we say that a matrix $A$ over $R$ has a canonical diagonal reduction if for the matrix $A$ there exist invertible matrices $P$ and $Q$ of appropriate sizes and a diagonal matrix $D=\mathrm{diag}(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r,0,\dots,0)$ such that $PAQ=D$ and $R\varepsilon_i\subseteq R\varepsilon_{i+1}$ for every $1\le i\le r-1$). We proved that a commutative Bezout domain $R$ in which any nonze\-ro prime ideal is contained in a finite set of maximal ideals and for any nonzero element $a\in R$ the ideal $aR$ a decomposed into a product $aR = Q_1\ldots Q_n$, where $Q_i$ ($i=1,\ldots, n$) are pairwise comaximal ideals and $\mathrm{rad}\,Q_i\in\mathrm{spec}\, R$, is an elementary divisor ring.

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8

Zabavsky, Bohdan. "Rings of dyadic range 1." Journal of Algebra and Its Applications 18, no. 11 (August 19, 2019): 1950206. http://dx.doi.org/10.1142/s0219498819502062.

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Анотація:

In this paper, we introduced the concept of a ring of a right (left) dyadic range 1. We proved that a Bezout ring of right (left) dyadic range 1 is a ring of stable range 2. And we proved that a commutative Bezout ring is an elementary divisor ring if and only if it is a ring of dyadic range 1.

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9

Zabavs’kyi, B. V., and B. M. Kuznits’ka. "A Stable Range of Class Full Matrices over Elementary Divisor Ring." Ukrainian Mathematical Journal 66, no. 5 (October 2014): 792–95. http://dx.doi.org/10.1007/s11253-014-0973-0.

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10

Zabavsky, B. V., O. V. Domsha, and O. M. Romaniv. "Clear rings and clear elements." Matematychni Studii 55, no. 1 (March 3, 2021): 3–9. http://dx.doi.org/10.30970/ms.55.1.3-9.

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Анотація:

An element of a ring $R$ is called clear if it is a sum of a unit-regular element and a unit. An associative ring is clear if each of its elements is clear.In this paper we defined clear rings and extended many results to a wider class. Finally, we proved that a commutative Bezout domain is an elementary divisor ring if and only if every full $2\times 2$ matrix over it is nontrivially clear.

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11

Zheng, Licui, Jinwang Liu, and Weijun Liu. "Generalized Serre Problem over Elementary Divisor Rings." Mathematical Problems in Engineering 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/926046.

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Анотація:

Matrix factorization has been widely investigated in the past years due to its fundamental importance in several areas of engineering. This paper investigates completion and zero prime factorization of matrices over elementary divisor rings (EDR). The Serre problem and Lin-Bose problems are generalized to EDR and are completely solved.

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12

Zabavsky, B. V., and O. V. Pihura. "Gelfand local Bezout domains are elementary divisor rings." Carpathian Mathematical Publications 7, no. 2 (December 19, 2015): 188–90. http://dx.doi.org/10.15330/cmp.7.2.188-190.

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13

Bohdan, Zabavsky. "Conditions for stable range of an elementary divisor rings." Communications in Algebra 45, no. 9 (November 21, 2016): 4062–66. http://dx.doi.org/10.1080/00927872.2016.1259418.

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14

Zabavs’kyi, B. V., and M. Ya Komarnyts’kyi. "Cohen-type theorem for adequacy and elementary divisor rings." Journal of Mathematical Sciences 167, no. 1 (April 30, 2010): 107–11. http://dx.doi.org/10.1007/s10958-010-9906-0.

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15

Dmytruk, A. A., A. I. Gatalevych, and M. I. Kuchma. "Stable range conditions for abelian and duo rings." Matematychni Studii 57, no. 1 (March 31, 2022): 92–97. http://dx.doi.org/10.30970/ms.57.1.92-97.

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Анотація:

The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1. The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).

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16

Sorokin, O. S. "Finite homomorphic images of Bezout duo-domains." Carpathian Mathematical Publications 6, no. 2 (December 29, 2014): 360–66. http://dx.doi.org/10.15330/cmp.6.2.360-366.

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Анотація:

It is proved that for a quasi-duo Bezout ring of stable range 1 the duo-ring condition is equivalent to being an elementary divisor ring. As an application of this result a couple of useful properties are obtained for finite homomorphic images of Bezout duo-domains: they are coherent morphic rings, all injective modules over them are flat, their weak global dimension is either 0 or infinity. Moreover, we introduce the notion of square-free element in noncommutative case and it is shown that they are adequate elements of Bezout duo-domains. In addition, we are going to prove that these elements are elements of almost stable range 1, as well as necessary and sufficient conditions for being square-free element are found in terms of regularity, Jacobson semisimplicity, and boundness of weak global dimension of finite homomorphic images of Bezout duo-domains.

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17

Eljeri, Mosbah. "The EA-Dimension of a Commutative Ring." ISRN Algebra 2013 (December 26, 2013): 1–6. http://dx.doi.org/10.1155/2013/293207.

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Анотація:

An elementary annihilator of a ring A is an annihilator that has the form (0:a)A; a∈R∖(0). We define the elementary annihilator dimension of the ring A, denoted by EAdim(A), to be the upper bound of the set of all integers n such that there is a chain (0:a0)⊂⋯⊂(0:an) of annihilators of A. We use this dimension to characterize some zero-divisors graphs.

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18

Mayer, Daniel C. "Quadratic p-ring spaces for counting dihedral fields." International Journal of Number Theory 10, no. 08 (October 29, 2014): 2205–42. http://dx.doi.org/10.1142/s1793042114500754.

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Анотація:

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field [Formula: see text], p-ring spaces Vp(c) modulo c are introduced by defining a morphism ψ : f ↦ Vp(f) from the divisor lattice ℕ of positive integers to the lattice 𝒮 of subspaces of the direct product Vp of the p-elementary class group 𝒞/𝒞p and unit group U/Up of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group Gal (N | ℚ) and sharing a common discriminant dN and conductor c over K. The number mp(d, c) of these extensions is given by a formula in terms of positions of p-ring spaces in 𝒮, whose complexity increases with the dimension of the vector space Vp over the finite field 𝔽p, called the modified p-class rank σp of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σp ≤ 1 only. Here, the results are extended to σp = 2, underpinned by concrete numerical examples.

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19

COYETTE, CÉCILE. "MAL'CEV–NEUMANN RINGS AND NONCROSSED PRODUCT DIVISION ALGEBRAS." Journal of Algebra and Its Applications 11, no. 03 (May 24, 2012): 1250052. http://dx.doi.org/10.1142/s0219498811005804.

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Анотація:

The first section of this paper yields a sufficient condition for a Mal'cev–Neumann ring of formal series to be a noncrossed product division algebra. This result is used in Sec. 2 to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p2 for p any odd prime). The arguments are based on those of Hanke in [A direct approach to noncrossed product division algebras, thesis dissertation, Postdam (2001), An explicit example of a noncrossed product division algebra, Math. Nachr.251 (2004) 51–68, A twisted Laurent series ring that is a noncrossed product, Israel. J. Math.150 (2005) 199–2003].

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20

Zabavskii, B. V., and N. Ya Komarnitskii. "Distributive elementary divisor domains." Ukrainian Mathematical Journal 42, no. 7 (July 1990): 890–92. http://dx.doi.org/10.1007/bf01062100.

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21

Tuganbaev, A. A. "Rings of elementary divisors and distributive rings." Russian Mathematical Surveys 46, no. 6 (December 31, 1991): 230–32. http://dx.doi.org/10.1070/rm1991v046n06abeh002857.

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22

Zabavskii, B. V. "On noncommutative rings with elementary divisors." Ukrainian Mathematical Journal 42, no. 6 (June 1990): 748–50. http://dx.doi.org/10.1007/bf01058928.

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23

Hirano, Yasuyuki. "On rings all of whose factor rings are integral domains." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 55, no. 3 (December 1993): 325–33. http://dx.doi.org/10.1017/s1446788700034078.

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AbstractA ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

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24

Guralnick, Robert M., Lawrence S. Levy, and Charles Odenthal. "Elementary divisor theorem for noncommutative PIDs." Proceedings of the American Mathematical Society 103, no. 4 (April 1, 1988): 1003. http://dx.doi.org/10.1090/s0002-9939-1988-0954973-3.

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25

Lorenzini, Dino. "Elementary Divisor domains and Bézout domains." Journal of Algebra 371 (December 2012): 609–19. http://dx.doi.org/10.1016/j.jalgebra.2012.08.020.

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26

Brown, W. J., and D. D. Rockey. "Identification of an Antigen Localized to an Apparent Septum within Dividing Chlamydiae." Infection and Immunity 68, no. 2 (February 1, 2000): 708–15. http://dx.doi.org/10.1128/iai.68.2.708-715.2000.

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Анотація:

ABSTRACT The process of chlamydial cell division has not been thoroughly investigated. The lack of detectable peptidoglycan and the absence of an FtsZ homolog within chlamydiae suggest an unusual mechanism for the division process. Our laboratory has identified an antigen (SEP antigen) localized to a ring-like structure at the apparent septum within dividing chlamydial reticulate bodies (RB). Antisera directed against SEP show similar patterns of antigen distribution inChlamydia trachomatis and Chlamydia psittaciRB. In contrast to localization in RB, SEP in elementary bodies appears diffuse and irregular, suggesting that the distribution of the antigen is developmental-stage specific. Treatment of chlamydiae with inhibitors of peptidoglycan synthesis or culture of chlamydiae in medium lacking tryptophan leads to the formation of nondividing, aberrant RB. Staining of aberrant RB with anti-SEP reveals a marked redistribution of the antigen. Within C. trachomatis-infected cells, ampicillin treatment leads to high levels of SEP accumulation at the periphery of aberrant RB, while inC. psittaci, treatment causes SEP to localize to distinct punctate sites within the bacteria. Aberrancy produced via tryptophan depletion results in a different pattern of SEP distribution. In either case, the reversal of aberrant formation results in the production of normal RB and a redistribution of SEP to the apparent plane of bacterial division. Collectively these studies identify a unique chlamydial-genus-common and developmental-stage-specific antigen that may be associated with RB division.

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27

REDMOND, SHANE P. "RECOVERING RINGS FROM ZERO-DIVISOR GRAPHS." Journal of Algebra and Its Applications 12, no. 08 (July 31, 2013): 1350047. http://dx.doi.org/10.1142/s0219498813500473.

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Анотація:

Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R1 × R2 × ⋯ × Rn, where each Ri is local and this representation of R is unique up to factors Ri with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.

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28

Ðurić, Alen, Sara Jevđenić, Polona Oblak, and Nik Stopar. "The total zero-divisor graph of a commutative ring." Journal of Algebra and Its Applications 18, no. 10 (August 6, 2019): 1950190. http://dx.doi.org/10.1142/s0219498819501901.

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Анотація:

In this paper, we initiate the study of the total zero-divisor graph over a commutative ring with unity. This graph is constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring and give a joint insight of the structure of zero-divisors in a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graph of the ring [Formula: see text] of integers modulo [Formula: see text] and prove that the total zero-divisor graphs of [Formula: see text] and [Formula: see text] are isomorphic if and only if [Formula: see text].

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29

Esin, Songül, Müge Kanuni, Ayten Koç, Katherine Radler, and Kulumani M. Rangaswamy. "On Prüfer-like properties of Leavitt path algebras." Journal of Algebra and Its Applications 19, no. 07 (July 5, 2019): 2050122. http://dx.doi.org/10.1142/s0219498820501224.

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Анотація:

Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra [Formula: see text], in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173–199], it was shown that the ideals of [Formula: see text] satisfy the distributive law, a property of Prüfer domains and that [Formula: see text] is a multiplication ring, a property of Dedekind domains. In this paper, we first show that [Formula: see text] satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers [Formula: see text], [Formula: see text] and [Formula: see text]. We also show that [Formula: see text] satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which [Formula: see text] satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals.

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30

Simonetta, Patrick. "Une correspondance entre anneaux partiels et groupes." Journal of Symbolic Logic 62, no. 1 (March 1997): 60–78. http://dx.doi.org/10.2307/2275732.

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AbstractThis work is inspired by the correspondence of Malcev between rings and groups. Let A be a domain with unit, and S a multiplicative group of invertible elements. We define AS as the structure obtained from A by restraining the multiplication to A × S, and σ(AS) as the group of functions from A to A of the form x → xa + b, where (a, b) belongs to S × A. We show that AS and σ(As) are interpretable in each other, and then, that we can transfer some properties between classes (or theories) of “reduced” domains and corresponding groups, such as being elementary, axiomatisability (for classes), decidability, completeness, or, in some cases, existence of a model-completion (for theories).We study the extensions of the additive group of A by the group S, acting by right multiplication, and show that sometimes σ(AS) is the unique extension of this type. We also give conditions allowing us to eliminate parameters appearing in interpretations.We emphasize the case where the domain is a division ring K and S is its multiplicative group K×. Here, the interpretations can always be done without parameters. If the centre of K contains more than two elements, then σ(K) is the only extension of the additive group of K by its multiplicative group acting by right multiplication, and the class of all such σ(K)'s is elementary and finitely axiomatisable. We give, in particular, an axiomatisation for this class and for the class of σ(K)'s where K is an algebraically closed field of characteristic 0. From these results it follows that some classical model-companion results about theories of fields can be translated and restated as results about theories of solvable groups of class 2.

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31

Nazary, Abdul Jamil. "Role of zero-divisor graph in power set ring." International Journal of Innovative Research and Scientific Studies 4, no. 3 (June 30, 2021): 181–85. http://dx.doi.org/10.53894/ijirss.v4i3.75.

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Анотація:

The first article about graph theory was written by Leonhard Euler the famous Swiss mathematician which was published in 1736. Primarily, the idea of graph was not important as point of mathematics because it mostly deals with recreational puzzles. But the recent improvement in mathematics specially, its application brought a strong revolution in graph theory. Therefore, this article is written under the title of (Role of Zero-divisor graph in power set ring). This study clarifies the role of the zero-fraction graph in the strength ring and the library research method was used for the collection of information from the past research. This study first introduces the zero-divisor graph set of alternatives in R ring. Then presents the Role of Zero-divisor graph in power set ring. Whereas the vertex is denoted with K1 and the elements of an optional ring which are not zero-divisors they are the vertices without edges of that ring in zero-divisor graph. Next, we will study when a graph is planar in power set ring. The research found out that if the element numbers of set X  is less than 4, the graph of zero-divisor graph ring of P X( ) is planar and if the element numbers of set X is greater than or equals to 4, the zero-divisor graph ring is not planar.

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32

Nagy, Béla. "Inverse elementary divisor problems for nonnegative matrices." Operators and Matrices, no. 2 (2011): 289–301. http://dx.doi.org/10.7153/oam-05-20.

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33

Hillman, Jonathan A. "Knot modules and the elementary divisor theorem." Journal of Pure and Applied Algebra 40 (1986): 115–24. http://dx.doi.org/10.1016/0022-4049(86)90034-4.

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34

Afkhami, Mojgan. "The cozero-divisor graph of a noncommutative ring." Journal of Algebra and Its Applications 13, no. 08 (June 24, 2014): 1450062. http://dx.doi.org/10.1142/s0219498814500625.

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Анотація:

In this paper, we extend the concept of the cozero-divisor graph to any arbitrary ring with nonzero identity. Also we study some basic properties of this graph and gain some results on the cozero-divisor graphs of matrix rings.

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35

Siddiqui, Hafiz Muahmmad Afzal, Ammar Mujahid, Muhammad Ahsan Binyamin, and Muhammad Faisal Nadeem. "On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings." Mathematical Problems in Engineering 2021 (December 29, 2021): 1–7. http://dx.doi.org/10.1155/2021/5826722.

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Анотація:

Given a finite commutative unital ring S having some non-zero elements x , y such that x . y = 0 , the elements of S that possess such property are called the zero divisors, denoted by Z S . We can associate a graph to S with the help of zero-divisor set Z S , denoted by ζ S (called the zero-divisor graph), to study the algebraic properties of the ring S . In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of S . To do so, we will discuss the zero-divisor graphs for the ring of integers ℤ m modulo m , some quotient polynomial rings, and the ring of Gaussian integers ℤ m i modulo m . Then, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of ζ S . In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.

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36

Ghashghaei, E., M. Tamer Koşan, M. Namdari, and T. Yildirim. "Rings in which every left zero-divisor is also a right zero-divisor and conversely." Journal of Algebra and Its Applications 18, no. 05 (May 2019): 1950096. http://dx.doi.org/10.1142/s0219498819500968.

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Анотація:

A ring [Formula: see text] is called eversible if every left zero-divisor in [Formula: see text] is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that [Formula: see text] is eversible if and only if its upper triangular matrix ring [Formula: see text] is eversible, and if [Formula: see text] is eversible then [Formula: see text] is eversible.

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37

Kumar, Ravindra, та Om Prakash. "Divisor graph of the complement of Γ(R)". Asian-European Journal of Mathematics 12, № 04 (2 липня 2019): 1950057. http://dx.doi.org/10.1142/s1793557119500578.

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Анотація:

Let [Formula: see text] be the complement of the zero-divisor graph of a finite commutative ring [Formula: see text]. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that [Formula: see text] is a divisor graph if [Formula: see text] is a local ring. It is shown that when [Formula: see text] is a product of two local rings, then [Formula: see text] is a divisor graph if one of them is an integral domain. Further, if [Formula: see text], then [Formula: see text] is a divisor graph.

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38

Tian, Yanzhao, and Lixiang Li. "Comments on the Clique Number of Zero-Divisor Graphs of Z n." Journal of Mathematics 2022 (March 29, 2022): 1–11. http://dx.doi.org/10.1155/2022/6591317.

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Анотація:

In 2008, J. Skowronek-kazi o ´ w extended the study of the clique number ω G Z n to the zero-divisor graph of the ring Z n , but their result was imperfect. In this paper, we reconsider ω G Z n of the ring Z n and give some counterexamples. We propose a constructive method for calculating ω G Z n and give an algorithm for calculating the clique number of zero-divisor graph. Furthermore, we consider the case of the ternary zero-divisor and give the generation algorithm of the ternary zero-divisor graphs.

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39

Caldeira, Cristina, and João Filipe Queiró. "Invariant factors of products over elementary divisor domains." Linear Algebra and its Applications 485 (November 2015): 345–58. http://dx.doi.org/10.1016/j.laa.2015.07.038.

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40

Das, Biswajit, and Shreemayee Bora. "Nearest Matrix Polynomials With a Specified Elementary Divisor." SIAM Journal on Matrix Analysis and Applications 41, no. 4 (January 2020): 1505–27. http://dx.doi.org/10.1137/19m1286505.

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41

Akgunes, Nihat, and Yasar Nacaroglu. "Some properties of zero divisor graph obtained by the ring Zp × Zq × Zr." Asian-European Journal of Mathematics 12, no. 06 (October 14, 2019): 2040001. http://dx.doi.org/10.1142/s179355712040001x.

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Анотація:

The concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra 116 (1988) 208–226]. In this paper, we present some properties of zero divisor graphs obtained from ring [Formula: see text], where [Formula: see text] and [Formula: see text] are primes. Also, we give some degree-based topological indices of this special graph.

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42

KUZMINA, A. S. "FINITE RINGS WITH EULERIAN ZERO-DIVISOR GRAPHS." Journal of Algebra and Its Applications 11, no. 03 (May 24, 2012): 1250055. http://dx.doi.org/10.1142/s0219498812500557.

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Анотація:

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.] In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

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43

Jafari Rad, Nader, Sayyed Heidar Jafari, and Doost Ali Mojdeh. "On Domination in Zero-Divisor Graphs." Canadian Mathematical Bulletin 56, no. 2 (June 1, 2013): 407–11. http://dx.doi.org/10.4153/cmb-2011-156-1.

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Анотація:

AbstractWe first determine the domination number for the zero-divisor graph of the product of two commutative rings with 1. We then calculate the domination number for the zero-divisor graph of any commutative artinian ring. Finally, we extend some of the results to non-commutative rings in which an element is a left zero-divisor if and only if it is a right zero-divisor.

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44

Ansari-Toroghy, Habibollah, Faranak Farshadifar, and Farideh Mahboobi-Abkenar. "An ideal-based cozero-divisor graph of a commutative ring." Boletim da Sociedade Paranaense de Matemática 40 (January 18, 2022): 1–8. http://dx.doi.org/10.5269/bspm.43261.

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Анотація:

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this article, we introduce the cozero-divisor graph $\acute{\Gamma}_I(R)$ of $R$ and explore some of its basic properties. This graph can be regarded as a dual notion of an ideal-based zero-divisor graph.

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45

Liu, H. Q. "The Dirichlet Divisor Problem of Arithmetic Progressions." Canadian Mathematical Bulletin 59, no. 3 (September 1, 2016): 592–98. http://dx.doi.org/10.4153/cmb-2016-029-5.

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46

Kuzmina, A. S., and Yu N. Maltsev. "ON VARIETIES OF RINGS WHOSE FINITE RINGS ARE DETERMINED BY THEIR ZERO-DIVISOR GRAPHS." Asian-European Journal of Mathematics 05, no. 02 (June 2012): 1250019. http://dx.doi.org/10.1142/s1793557112500192.

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Анотація:

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

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47

Alkam, Osama, and Emad Abu Osba. "Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n." Algebra 2014 (December 15, 2014): 1–6. http://dx.doi.org/10.1155/2014/146873.

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Анотація:

Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

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48

AFKHAMI, M., K. KHASHYARMANESH, and M. R. KHORSANDI. "ZERO-DIVISOR GRAPHS OF ÖRE EXTENSION RINGS." Journal of Algebra and Its Applications 10, no. 06 (December 2011): 1309–17. http://dx.doi.org/10.1142/s0219498811005191.

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Анотація:

Let R be an associative ring with two-sided multiplicative identity. In this paper, in the case that R is a commutative α-compatible ring, we compare the diameter (and girth) of the zero-divisor graphs Γ(R) and Γ(R[x;α, δ]). Moreover, we study the zero-divisors of the Öre extension ring R[x;α, δ], whenever R is reversible and (α, δ)-compatible.

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49

Hashemi, E., R. Amirjan, and A. Alhevaz. "On zero-divisor graphs of skew polynomial rings over non-commutative rings." Journal of Algebra and Its Applications 16, no. 03 (March 2017): 1750056. http://dx.doi.org/10.1142/s0219498817500566.

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Анотація:

In this paper, we continue to study zero-divisor properties of skew polynomial rings [Formula: see text], where [Formula: see text] is an associative ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. For an associative ring [Formula: see text], the undirected zero-divisor graph of [Formula: see text] is the graph [Formula: see text] such that the vertices of [Formula: see text] are all the nonzero zero-divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if [Formula: see text] or [Formula: see text]. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text]. Our goal in this paper is to give a characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text], when the base ring [Formula: see text] is reversible and also have the [Formula: see text]-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

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50

FACCHINI, ALBERTO, and FRANZ HALTER-KOCH. "PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS." Journal of Algebra and Its Applications 02, no. 04 (December 2003): 435–49. http://dx.doi.org/10.1142/s0219498803000593.

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Анотація:

We study some applications of the theory of commutative monoids to the monoid [Formula: see text] of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

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