Publication Date: 2018
4OR (16142411)16(4)pp. 343-377
In this paper, we consider a multi-source Weber problem of m new facilities with respect to n demand regions in order to minimize the sum of the transportation costs between these facilities and the demand regions. We find a point on the border of each demand region from which the facilities serve the demand regions at these points. We present an algorithm including a location phase and an allocation phase in each iteration for solving this problem. An algorithm is also proposed for carrying out the location phase. Moreover, global convergence of the new algorithm is proved under mild assumptions, and some numerical results are presented. © 2017, Springer-Verlag GmbH Germany, part of Springer Nature.
Publication Date: 2014
Mathematica Bohemica (24647136)139(1)pp. 1-23
For an ordered set W = {w1, w2,..., wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v{pipe}W):= (d(v, w1), d(v, w2),..., d(v, wk)) is called the metric representation of v with respect to W, where d(x, y) is the distance between vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we characterize all graphs of order n with metric dimension n - 3.
Publication Date: 2016
Ars Combinatoria (03817032)127pp. 357-372
For an ordered set W = {ω1, ω1,···, ωk} of vertices and a vertex v in a connected graph G, the ordered fc-vector r(ν|W):= (d(ν, ω1), d(ν, ω1),···, d{ν, ωk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. a minimum resolving set for G is a basis of G and its cardinality is the metric dimension of G. The resolving number of a connected graph G is the minimum k, such that every fc-set of vertices of G is a resolving set. a connected graph G is called randomly k-dimensional if each k-set of vertices of G is a basis. In this paper, along with some properties of randomly k-dimensional graphs, we prove that a connected graph G with at least two vertices is randomly fc-dimensional if and only if G is complete graph Kk+1 or an odd cycle.
Publication Date: 2011
Journal of Algebra (00218693)345(1)pp. 257-265
A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, "a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring". Therefore, an interesting natural question of this sort is "whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?" The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. © 2011.
Publication Date: 2023
Communications in Algebra (00927872)51(11)pp. 4821-4829
Recall that an R-module M is pure-semisimple if every module in the category (Formula presented.) is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that “a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring”. Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is “whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?” The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals (Formula presented.) which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple. © 2023 Taylor & Francis Group, LLC.
Publication Date: 2017
Journal of Algebra (00218693)486pp. 422-424
The statement of Theorem 2.1 in [4] is not correct. We give a counterexample to this statement and following to this error we give further corrections. We also offer two better substitutions for Proposition 2.9 and Theorem 2.11 in [4]. © 2017 Elsevier Inc.
Publication Date: 2013
Communications in Algebra (00927872)41(12)pp. 4559-4575
This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective. Also, over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N ⊕ P, where N is a direct sum of cyclic modules and P is a projective left R-module. The relationship Cpurity with purity and RD-purity are also studied. It is shown that if R is a local duoring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring. If R is a left perfect duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is left Köthe (i.e., every left R-module is a direct sum of cyclic modules). Also, it is shown that for a ring R, if every C-pure projective left R-module is RD-projective, then R is left Noetherian, every p-injective left R-module is injective and every p-flat right R-module is flat. Finally, it is shown that if R is a left p.p-ring and every C-pure projective left R-module is RD-projective, then R is left Noetherian hereditary. The converse is also true when R is commutative, but it does not hold in the noncommutative case. © Taylor & Francis Group, LLC.
Publication Date: 2022
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas (15791505)116(4)
A theorem due to Warfield states that “a ring R is left serial if and only if every (finitely generated) projective left R-module is serial” and a theorem due to Tuganbaev states that “a ring R is a finite direct product of uniserial Noetherian rings if and only if R is left duo, and all injective left R-modules are serial”. Most recently, in our previous paper [Virtually uniserial modules and rings, J Algebra 549:365–385, 2020], we introduced and studied the concept of virtually uniserial modules as a nontrivial generalization of uniserial modules. We say that an R-module M is virtually uniserial if, for every finitely generated submodule 0 ≠ K⊆ M, K/ Rad (K) is virtually simple (an R-module M is virtually simple if, M≠ 0 and M≅ N for every nonzero submodule N of M). Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules. The above results of Warfield and Tuganbaev motivated us to study the following questions: “Which rings have the property that every projective module is virtually serial?” and “Which rings have the property that every injective module is virtually serial?”. The goal of this paper is to answer these questions. © 2022, The Author(s) under exclusive licence to The Royal Academy of Sciences, Madrid.
Publication Date: 2025
Electronic Journal Of Graph Theory And Applications (23382287)13(1)pp. 217-230
Two vertices u, v in a connected graph G are doubly resolved by vertices x, y of G if (Formula presented). A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by ψ(G), is the minimum cardinality of a doubly resolving set for G. In this paper, using adjacency resolving sets and dominating sets of graphs, we study doubly resolving sets in the corona product of graphs G and H, G ☉ H. First, we obtain the upper and lower bounds for the doubly resolving number of the corona product G☉H in terms of the order of G and the adjacency dimension of H, then we present several conditions that make each of these bounds feasible for the doubly resolving number of G ☉ H. Also, for some important families of graphs, we obtain the exact value of the doubly resolving number of the corona product. © (2025), (Indonesian Combinatorics Society). All rights reserved.
Publication Date: 2011
Match (03406253)65(1)pp. 71-78
The Wiener index of a simple graph is defined as the sum of distances between all vertices of the graph. It is well known that the Wiener index of a tree can be obtained as an edge additive quantity where edge contributions are given as the product of the number of vertices closer to each of the two end points of each edge. Thus the distances between vertices are not used for computing the Wiener index of trees. In a similar manner we introduce new topological indices which yields the Wiener, hyper-Wiener, Schultz and modified Schultz indices as special cases for trees. One advantage of this method is that in computing Schultz and modified Schultz of trees we need not take in to account the distances between vertices.
Publication Date: 2022
Discrete Mathematics, Algorithms and Applications (17938317)14(4)
For a set W of vertices and a vertex v in a graph G, the k-vector r2(v|W) = (aG(v,w1),⋯,aG(v,wk)) is the adjacency representation of v with respect to W, where W = {w1,⋯,wk} and aG(x,y) is the minimum of 2 and the distance between the vertices x and y. The set W is an adjacency resolving set for G if distinct vertices of G have distinct adjacency representations with respect to W. The minimum cardinality of an adjacency resolving set for G is its adjacency dimension. It is clear that the adjacency dimension of an n-vertex graph G is between 1 and n - 1. The graphs with adjacency dimension 1 and n - 1 are known. All graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n - 2 are studied in this paper. In terms of the diameter and order of G, a sharp upper bound is found for adjacency dimension of G. Also, a sharp lower bound for adjacency dimension of G is obtained in terms of order of G. Using these two bounds, all graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n - 2 are characterized. © 2022 World Scientific Publishing Company.
Publication Date: 2023
Discrete Applied Mathematics (0166218X)339pp. 178-183
Two vertices u,v in a connected graph G are doubly resolved by x,y∈G if d(v,x)−d(u,x)≠d(v,y)−d(u,y).A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by ψ(G), is the minimum cardinality of a doubly resolving set for the graph G. In this paper all graphs G with ψ(G)=2 are characterized by using 2-connected subgraphs of G. © 2023 Elsevier B.V.
Publication Date: 2024
Hacettepe Journal of Mathematics and Statistics (2651477X)53(2)pp. 342-355
In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are consid-ered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring R is two-sided Köthe if all right R-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with J(R)2 = 0, it is proven that all R-modules are min-pure projective if and only if R is either a field or a quasi-Frobenius ring of composition length 2. © 2024, Hacettepe University. All rights reserved.
Publication Date: 2009
Applied Mathematics Letters (18735452)22(10)pp. 1571-1576
In this work we show that among all n-vertex graphs with edge or vertex connectivity k, the graph G = Kk ∨ (K1 + Kn - k - 1), the join of Kk, the complete graph on k vertices, with the disjoint union of K1 and Kn - k - 1, is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique n-vertex graph with edge or vertex connectivity k whose hyper-Wiener index is minimum. © 2009 Elsevier Ltd. All rights reserved.
Publication Date: 2012
Archivum Mathematicum (00448753)48(4)pp. 291-299
In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring (R, Ai), the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R-modules; (2) M =⊕λχλ Rωλ where λ is an index set and R/Ann(ωλ) is a principal ideal ring for each λ χ λ; (3) Every prime ideal of H is a direct sum of at most |A| cyclic fi-modules where A is an index set and M =⊕λχλ Rωλ; and (4) Every prime ideal of R is a summand of a direct sum of cyclic R-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring (R, M) is a direct sum of (at most n) principal ideals, it suffices to test only the maximal ideal M.
Publication Date: 2022
Bulletin Of The Malaysian Mathematical Sciences Society (01266705)45(5)pp. 2041-2052
Two vertices u, v in a connected graph G are doubly resolved by vertices x, y of G if d(v,x)-d(u,x)≠d(v,y)-d(u,y).A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by ψ(G) , is the minimum cardinality of a doubly resolving set for the graph G. The aim of this paper is to investigate doubly resolving sets in graphs. An upper bound for ψ(G) is obtained in terms of order and diameter of G. ψ(G) is computed for some graphs, and all graphs G of order n with the property ψ(G) = n- 1 are determined. Also, doubly resolving sets for unicyclic graphs are studied and it is proved that the difference between the number of leaves and doubly resolving number of a unicyclic graph is at most 2. © 2022, This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply.
Publication Date: 2014
Communications in Algebra (00927872)42(5)pp. 2061-2081
In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module. © 2014 Copyright Taylor and Francis Group, LLC.
Publication Date: 2014
Proceedings of the American Mathematical Society (10886826)142(8)pp. 2625-2631
In this paper, we obtain a partial solution to the following question of Köthe: For which rings R is it true that every left (or both left and right) R-module is a direct sum of cyclic modules? Let R be a ring in which all idempotents are central. We prove that if R is a left Köthe ring (i.e., every left R-module is a direct sum of cyclic modules), then R is an Artinian principal right ideal ring. Consequently, R is a Köthe ring (i.e., each left and each right R-module is a direct sum of cyclic modules) if and only if R is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem. © 2014, American Mathematical Society.
Publication Date: 2011
Applied Mathematics Letters (18735452)24(10)pp. 1625-1629
For an ordered set W=w1,w2,⋯,wk of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W):=(d(v,w1),d(v,w2),⋯,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set for G with minimum cardinality is called a basis of G and its cardinality is the metric dimension of G. A connected graph G is called a randomly k-dimensional graph if each k-set of vertices of G is a basis of G. In this work, we study randomly k-dimensional graphs and provide some properties of these graphs. © 2011 Elsevier Ltd. All rights reserved.
Publication Date: 2020
Optimization Letters (18624472)14(8)pp. 2539-2547
In this paper, we consider the Cooper’s alternate location and allocation algorithm for solving the constrained uncapacitated multi-source Weber problem. The convergence of the sequence generated by Cooper’s algorithm to a local optimal solution proved under mild assumption. Furthermore, an approach for finding the global optimal solution is proposed. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
Publication Date: 2023
Kyungpook Mathematical Journal (12256951)63(1)pp. 123-129
For an ordered set W = {w1,w2,…,wk} of vertices and a vertex v in a connected graph G, the k-vector (Formula Presented) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, (Formula Presented) of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(Formula Presented), characterize all graphs G with dim(Formula Presented), and prove that dim(Formula Presented) if and only if G is the complete graph Kn or the star graph K1,n−1. © Kyungpook Mathematical Journal
Publication Date: 2017
Journal of Pure and Applied Algebra (00224049)221(4)pp. 935-947
We study direct-sum decompositions of pure-projective modules over some classes of rings. In particular, we determine several classes of rings over which every pure-projective left module is a direct sum of cyclic modules. Finally, the relationship between our results and FGC rings is also studied. © 2016 Elsevier B.V.
Publication Date: 2022
Journal of Pure and Applied Algebra (00224049)226(9)
We say that a ring R is a right RDV-ring if each simple right R-module is RD-injective. In this paper, we study the notion of RDV-rings which is a non-trivial generalization of V-rings and Köthe rings. For instance, commutative RD-rings, serial rings and right duo right uniserial rings are RDV-rings. Several characterizations of right RDV-rings are given. Also, it is shown that over a semilocal ring R with Jacobson radical J, each simple right R-module is RD-flat if and only if R is a left RDV-ring, if and only if R(R/J) is RD-injective, if and only if (R/J)R is RD-flat. As a consequence, we show that a local ring R is a principal ideal ring if and only if R satisfies the ascending chain condition on principal left ideals and R(R/J) is RD-injective. In the case of R being either a local left perfect ring or a normal left perfect ring, we have obtained results which state that to check whether every left R-module is RD-injective (or, R is Köthe), it suffices to test only the RD-injectivity of the simple left R-modules. Finally, we give some characterizations of quasi-Frobenius rings by using these concepts. © 2022 Elsevier B.V.
Publication Date: 2016
Colloquium Mathematicum (00101354)145(2)pp. 167-177
A classical question due to Yoneda is, “When is the tensor product of any two injective modules injective?” Enochs and Jenda gave a complete and explicit answer to this question in 1991. Since RD-injective modules are a generalization of injective modules, it is natural to ask whether the tensor product of any two RD-injective modules is RD-injective. In this paper we deal with this question. © Instytut Matematyczny PAN, 2016.
Publication Date: 2024
Rendiconti del Circolo Matematico di Palermo (0009725X)73(8)pp. 3175-3193
Given modules M and A, M is said to be A-RD-subinjective if for every RD-extension B of A, every f∈Hom(A,M) extends to Hom(B,M). For a module M, the RD-subinjectivity domain of M is defined to be the collection of all modules A such that M is A-RD-subinjective. We investigate basic properties of RD-subinjectivity domains and provide characterizations for various types of rings and modules including p-injective modules, RD-coflat modules, von Neumann regular rings, RD-rings, Köthe rings, right Noetherian rings, and quasi-Frobenius rings in terms of RD-subinjectivity domains. Finally, we study the properties of RD-indigent modules and consider the structure of rings over which every (resp. simple) right module is RD-injective or RD-indigent. © The Author(s), under exclusive licence to Springer-Verlag Italia S.r.l., part of Springer Nature 2024.
Publication Date: 2017
Journal of Algebra (00218693)478pp. 419-436
A ring R is said to be left pure-hereditary (resp. RD-hereditary) if every left ideal of R is pure-projective (resp. RD-projective). In this paper, some properties and examples of these rings, which are nontrivial generalizations of hereditary rings, are given. For instance, we show that if R is a left RD-hereditary left nonsingular ring, then R is left Noetherian if and only if u.dim(RR)<∞. Also, we show that a ring R is quasi-Frobenius if and only if R is a left FGF, left coherent right pure-injective ring. A ring R is said to be left FP-hereditary if every left ideal of R is FP-projective. It is shown that if R is a left CF ring, then R is left Noetherian if and only if R is left pure-hereditary, if and only if R is left FP-hereditary, if and only if R is left coherent. It is shown that every left self-injective left FP-hereditary ring is semiperfect. Finally, it is shown that a ring R is left FP-hereditary (resp. left coherent) if and only if every submodule (resp. finitely generated submodule) of a projective left R-module is FP-projective, if and only if every pure factor module of an injective left R-module is injective (resp. FP-injective), if and only if for each FP-injective left R-module U, E(U)/U is injective (resp. FP-injective). © 2017 Elsevier Inc.
