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.
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.
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.
Match (03406253)65(1)pp. 27-32
Gutman and Zhou (Relations between Wiener, hyper-Wiener and Zagreb indices, Chemical Physics Letters 394 (2004) 93-95) obtained some bounds on Wiener and hyper-Wiener indices, in term of the first Zagreb index in molecular graphs with girth greater than four. We obtain new inequalities for Wiener and hyperWiener indices, in terms of the first and the second Zagreb indices and the number of hexagons in these graphs. These inequalities improve the bounds obtained by Gutman and Zhou and are the best possible bounds. Using these relations we obtain a bound on the second Zagreb index in terms of the first Zagreb index, for hexagon-free graphs.
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.
Discrete Mathematics (0012365X)312(22)pp. 3349-3356
For a set W of vertices and a vertex v in a connected graph G, the k-vector rW(v)=(d(v,w1),⋯,d(v,wk)) is the metric representation of v with respect to W, where W=w1,⋯,wk and d(x,y) is the distance between the vertices x and y. The set W is 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. In this paper, we study the metric dimension of the lexicographic product of graphs G and H, denoted by G[H]. First, we introduce a new parameter, the adjacency dimension, of a graph. Then we obtain the metric dimension of G[H] in terms of the order of G and the adjacency dimension of H. © 2012 Elsevier B.V. All rights reserved.
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.
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.
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.
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.
Journal of Algebra (00218693)401pp. 179-200
We study direct-sum decompositions of RD-projective modules. In particular, we investigate the rings over which every RD-projective right module is a direct sum of cyclically presented right modules, or a direct sum of finitely presented cyclic right modules, or a direct sum of right modules with local endomorphism rings (SSP rings). SSP rings are necessarily semiperfect. For instance, the superlocal rings introduced by Puninski, Prest and Rothmaler in [28] and the semilocal strongly π-regular rings introduced by Kaplansky in [24] are SSP rings. In the case of a Noetherian ring R (with further additional hypotheses), an RD-projective R-module M turns out to be either a direct sum of finitely presented cyclic modules or of the form M = T(M) ⊕ P, where T(M) is the torsion part of M (elements of M annihilated by a regular element of R) and P is a projective module. © 2013 Elsevier Inc.
Bulletin Of The Iranian Mathematical Society (10186301)41(3)pp. 633-638
A set W ⊆ V (G) is called a resolving set for G, if for each two distinct vertices u, v ∈ V (G) there exists w 2 W such that d(u,w) ≠ d(v,w), where d(x, y) is the distance between the vertices x and y. The minimum cardinality of a resolving set for G is called the metric dimension of G, and denoted by dim(G). In this paper, it is proved that in a connected graph G of order n which has a cycle, dim(G) ≤ n-g(G)+2, where g(G) is the length of the shortest cycle in G, and the equality holds if and only if G is a cycle, a complete graph or a complete bipartite graph Ks,t, s, t ≥ 2. © 2015 Iranian Mathematical Society.
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.
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.
Journal of Algebra (00218693)460pp. 128-142
A famous theorem of algebra due to Osofsky states that "if every cyclic left R-module is injective, then R is semisimple". Therefore, a natural question of this sort is: "What is the class of rings R for which every cyclic left R-module is pure-injective or pure-projective?" The goal of this paper is to answer this question. For instance, we show that if every cyclic left R-module is pure-injective, then R is a left perfect ring. As a consequence, a commutative coherent ring R is Artinian if and only if every cyclic R-module is pure-injective. Also, a commutative ring R is pure-semisimple (i.e., every R-module is pure-injective) if and only if all cyclic R-modules and all indecomposable R-modules are pure-injective. We obtain some generalizations of Osofsky's theorem in the cases R is semiprimitive or commutative coherent or a commutative semiprime Goldie ring. Finally, we show that a ring R is left Noetherian if and only if every cyclic left R-module is pure-projective. As a corollary of this result we obtain: if every cyclic left R-module is pure-injective and pure-projective, then R is a left Artinian ring. The converse is also true when R is commutative. © 2016 Elsevier Inc.
Ars Combinatoria (03817032)129pp. 249-259
A set W C V(G) is called a resolving set, if for each two distinct vertices u v € V(G) there exists w € W such that d(uw) d(v,w), where d(x,y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a unique basis graph. In this paper, we study some properties of unique basis graphs.
Discrete Mathematics, Algorithms and Applications (17938317)9(2)
A set W ∪ V (G) is called a resolving set, if for each pair of distinct vertices u,v ϵ V (G) there exists t ϵ W such that d(u,t)=d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dimM(G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k ≤ j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) =deg(v) = j. In this paper, we prove that if G is a k-path, then dimM(G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two. © 2017 World Scientific Publishing Company.
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.
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.
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.
Mathematical Methods of Operations Research (14325217)88(1)pp. 81-98
In this paper, we consider the rectilinear distance location problem with box constraints (RDLPBC) and we show that RDLPBC can be reduced to the rectilinear distance location problem (RDLP). A necessary and sufficient condition of optimality is provided for RDLP. A fast algorithm is presented for finding the optimal solution set of RDLP. Global convergence of the method is proved by a necessary and sufficient condition. The new proposed method is provably more efficient in finding the optimal solution set. Computational experiments highlight the magnitude of the theoretical efficiency. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
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.
Journal of Algebra and its Applications (17936829)18(1)
A left R-module M is said to be left singly injective if ExtR1(F/K,M) = 0 for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we study the notion of singly injective modules which is generalization of injective modules and absolutely pure modules. In this direction, we give conditions which guarantee that each singly injective left R-module is either injective or absolutely pure. Finally, we study rings whose simple modules are singly injective (SSI-rings). © 2019 World Scientific Publishing Company.
Bulletin Of The Iranian Mathematical Society (10186301)45(2)pp. 495-514
This paper considers the classical multi-source Weber problem (MWP), which is to find M new facilities with respect to N customers to minimize the sum of transportation costs between these facilities and the customers. We propose a modified algorithm in the spirit of Cooper’s work for solving the MWP including a location phase and an allocation phase. The task of location phase is to find the optimal solution sets of many single-source Weber problems (SWPs), which are reduced by the heuristic of the nearest center reclassification for the customers in the previous allocation phase. Some examples are stated to clarify the proposed algorithms. Moreover, we present an algorithm with O(dlog d) time for finding the optimal solution set of SWP in the collinear case; where d is the number of customers. © 2018, The Author(s).
Journal of Pure and Applied Algebra (00224049)223(2)pp. 713-720
A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules. © 2018 Elsevier B.V.
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.
Bulletin of the Korean Mathematical Society (10158634)57(2)pp. 371-381
A ring R is called right pure-injective if it is injective with respect to pure exact sequences. According to a well known result of L. Melkersson, every commutative Artinian ring is pure-injective, but the converse is not true, even if R is a commutative Noetherian local ring. In this paper, a series of conditions under which right pure-injective rings are either right Artinian rings or quasi-Frobenius rings are given. Also, some of our results extend previously known results for quasi-Frobenius rings. ©2020 Korean Mathematial Soiety.
