Showing papers in "Communications in Algebra in 2013"
TL;DR: In this article, it was shown that each biderivation of the Schr dinger-Virasoro Lie algebra over the complex field ℂ is inner and every linear commuting map ψ on 𝔏 has the form ψ(x) ǫ = Ω(x), where Ω is a basis of the one-dimensional center of the center.
Abstract: Let 𝔏 be the Schr dinger–Virasoro Lie algebra over the complex field ℂ. In this article, we prove that each biderivation of 𝔏 is inner. As an application of biderivations, we show that every linear commuting map ψ on 𝔏 has the form ψ(x) = λx + f(x)M 0, where λ ∈ ℂ, M 0 is a basis of the one-dimensional center of 𝔏, and f is a linear function from 𝔏 to ℂ.
69 citations
TL;DR: In this article, the authors characterized all isomorphism classes of finite commutative rings whose total graph has genus at most two, and showed that genus-two is not a sufficient condition for any isomorphic ring.
Abstract: Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by TΓ(R), is the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Maimani et al. [13] determined all isomorphism classes of finite commutative rings whose total graph has genus at most one. In this article, after enumerating certain lower and upper bounds for genus of the total graph of a commutative ring, we characterize all isomorphism classes of finite commutative rings whose total graph has genus two.
60 citations
TL;DR: In this paper, the authors characterized all commutative rings whose total graph (or its complement) is in some known class of graphs and obtained necessary conditions for to be connected whenever is connected.
Abstract: Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T Γ(R), is the undirected graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). denotes the complement of T Γ(R). The study on total graphs has been initiated by D. F. Anderson and A. Badawi [2]. In this article, we characterize all commutative rings whose total graph (or its complement) is in some known class of graphs. Also we determine the structure whenever |Reg(R)| = 2. Further, we obtain certain necessary conditions for to be connected whenever is connected and prove that . It is also proved that if diam(T Γ(R)) = 2, then T Γ(R) is Hamiltonian, which is a generalization of a characterization proved by S. Akbari et al. [1].
43 citations
TL;DR: In this article, a universal approach to the moduli of Li's expanded pairs and expanded degenerations is provided, which enables us to prove algebraicity results, compare with Li's approach and with the approach of Graber and Vakil [16], and generalize to the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneration formulas.
Abstract: We provide a universal approach to the moduli of Jun Li's expanded pairs and expanded degenerations [20] This enables us to prove algebraicity results, compare with Li's approach and with the approach of Graber and Vakil [16], and generalize to the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneration formulas
41 citations
TL;DR: In this paper, the relation between the G C -syzygy with the C-syyzygy of a module was investigated, as well as the relationship between G C-projective resolution and the projective resolution of the module.
Abstract: Let S and R be rings and S C R a semidualizing bimodule. We investigate the relation between the G C -syzygy with the C-syzygy of a module as well as the relation between the G C -projective resolution and the projective resolution of a module. As a consequence, we get that if is an exact sequence of S-modules with all G i , G i G C -projective, such that Hom S (𝔾, T) is still exact for any module T which is isomorphic to a direct summand of direct sums of copies of S C, then Im(G 0 → G 0) is also G C -projective. We obtain a criterion for computing the G C -projective dimension of modules. When S C R is a faithfully semidualizing bimodule, we study the Foxby equivalence between the subclasses of the Auslander class and that of the Bass class with respect to C.
40 citations
TL;DR: Corach, Gustavo, et al. this paper present a paper by Corach et al., this paper, which describes the work of the Departamento de Matematica of Argentina.
Abstract: Fil: Corach, Gustavo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica; Argentina
39 citations
TL;DR: In this article, the authors studied the total acyclicity of projective and injective representations of quivers and gave a classification for such complexes in terms of associated vertex-complexes.
Abstract: We study total acyclicity for complexes of projective and injective representations of quivers. A classification is given for such complexes in terms of associated vertex-complexes. When the base ring or the quiver is nice enough, this classification is used to prove the existence of Gorenstein projective precovers in the category of representations of quivers. Furthermore, we exploit this local description to obtain some criteria for the category of representations of a quiver to be Gorenstein or virtually Gorenstein. If Λ is an artin algebra it is proved that, for an arbitrary quiver 𝒬, the representation category Rep(𝒬, Λ) is virtually Gorenstein whenever Λ is virtually Gorenstein. A description of Gorenstein projective and Gorenstein injective representations of quivers over general rings is also provided.
34 citations
TL;DR: In this article, it was shown that a direct summand of an endoregular module inherits the property of being von Neumann regular, while the direct sum of the modules does not.
Abstract: Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of rings R are characterized in terms of endoregular R-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending an endoregular abelian group is also provided.
33 citations
TL;DR: In this paper, it was shown that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ǫ R (M) is a prime ideal of R and R/Q is a divisible right (R/Q)-module.
Abstract: Let R be an arbitrary ring. A nonzero unital right R-module M is called a second module if M and all its nonzero homomorphic images have the same annihilator in R. It is proved that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ann R (M) is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies the ascending chain condition on two-sided ideals, then every nonzero R-module has a nonzero homomorphic image which is a second module. Every nonzero Artinian module contains second submodules and there are only a finite number of maximal members in the collection of second submodules. If R is a ring and M is a nonzero right R-module such that M contains a proper submodule N with M/N a second module and M has finite hollow dimension n, for some positive integer n, then there exist a positive integer k ≤ n and prime ideals P i (1 ≤ i ≤ k) such that if L is a proper submodule of M...
31 citations
TL;DR: In this paper, the authors introduced the notation of the nth finiteness dimension for all n ∈ ℕ0, and proved the following results: (i) ; (ii) The R-modules are I-cofinite for all and for all minimax submodules N of, the R -modules are finitely generated, whenever is finite.
Abstract: Let R be a commutative Noetherian ring, I an ideal of R, and M a non-zero R-module. The purpose of this article is to introduce the notation of the nth finiteness dimension for all n ∈ ℕ0, and to prove the following results: (i) ; (ii) The R-modules are I-cofinite for all and for all minimax submodules N of , the R-modules are finitely generated, whenever is finite. This implies that if I has dimension one, then is I-cofinite for every i ≥ 0, which is a generalization of the main results of Delfino–Marley, Yoshida, and Bahmanpour–Naghipour. (iii) , whenever R is semilocal. (iv) The R-modules are weakly Laskerian for all j ≥ 0 and all , whenever (R, 𝔪) is a complete Noetherian local ring. Moreover, in this situation for all weakly Laskerian submodules N of , the R-modules are weakly Laskerian, whenever is finite. In addition, some examples about , for n = 0, 1, 2, 3, are included.
27 citations
TL;DR: In this paper, it was shown that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra S over a field, and for an arbitrary intersection of (P i ) i∈[s] of S such that each P i is not contained in the sum of the other (P j ) j≠i.
Abstract: We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra S over a field and for an arbitrary intersection of monomial prime ideals (P i ) i∈[s] of S such that each P i is not contained in the sum of the other (P j ) j≠i .
TL;DR: In this article, it was shown that a Schubert variety X(w) is a toric variety if and only if the Weyl group element w is a product of distinct simple reflections.
Abstract: In this article we prove that a Schubert variety X(w) is a toric variety if and only if the Weyl group element w is a product of distinct simple reflections.
TL;DR: In this article, the authors give the categorisation of Leibniz algebras, which is equivalent to 2-term sh Leibbniz algesbras and show that the Dirac structures of twisted Courant algebroids give rise to two-term L ∞-algebras and geometric structures behind them are exactly H-twisted Lie algesbroids.
Abstract: In this article, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in [26] as well as twisted Courant algebroids by closed 4-forms introduced in [13]. We also prove that Dirac structures of twisted Courant algebroids give rise to 2-term L ∞-algebras and geometric structures behind them are exactly H-twisted Lie algebroids introduced in [10].
TL;DR: In this article, the theory of Schunck classes and projectors for soluble Leibniz algebras, parallel to that for Lie algesbras is presented.
Abstract: I set out the theory of Schunck classes and projectors for soluble Leibniz algebras, parallel to that for Lie algebras. Primitive Leibniz algebras come in pairs, one (Lie) symmetric, the other antisymmetric. A Schunck formation containing one member of a pair also contains the other. If ℌ is a Schunck formation and H is an ℌ-projector of the Leibniz algebra L, then H is intravariant in L. An example is given to show that the assumption that the Schunck class ℌ is a formation cannot be omitted.
TL;DR: In this paper, it was shown that the quantum group inclusion is maximally optimal, where On is the usual orthogonal group and is the half-liberated quantum group, and there is no intermediate compact quantum group.
Abstract: We prove that the quantum group inclusion is “maximal,” where On is the usual orthogonal group and is the half-liberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group . In order to prove this result, we use the following: (1) the isomorphism of projective versions , (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.
TL;DR: In this paper, the authors investigated a category of Virasoro-algebra modules that includes Whittaker modules, and provided a classification of simple modules in the category and a description of certain induced modules that are a natural generalization of simple Whitterer modules.
Abstract: This article builds on work from [16], where the authors described Whittaker modules for the Virasoro algebra. Using the framework outlined in [3], the current article investigates a category of Virasoro-algebra modules that includes Whittaker modules. Results in this article include a classification of the simple modules in the category and a description of certain induced modules that are a natural generalization of simple Whittaker modules.
TL;DR: A Frattini theory for non-associative algebras was developed in [13] and results for particular classes of algesbras have appeared in various articles as mentioned in this paper.
Abstract: A Frattini theory for non-associative algebras was developed in [13] and results for particular classes of algebras have appeared in various articles. Especially plentiful are results on Lie algebras. It is the purpose of this paper to extend some of the Lie algebra results to Leibniz algebras.
TL;DR: In this article, the authors give an upper bound for the degree of the reduced Jacobian scheme when Y is a free rank 3 central essential arrangement, and investigate the connections between the first syzygies on J F and the generators of.
Abstract: Let I ⊂ ℂ[x, y, z] be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If I is a locally complete intersection we give a criterion for ℂ[x, y, z]/I to be arithmetically Cohen–Macaulay. Since the setup above is most commonly used when I = J F is the Jacobian ideal of the defining polynomial of a “quasihomogeneous” reduced curve Y = V(F) in ℙ2, our main result becomes a criterion for freeness of such divisors. As an application we give an upper bound for the degree of the reduced Jacobian scheme when Y is a free rank 3 central essential arrangement, as well as we investigate the connections between the first syzygies on J F , and the generators of .
TL;DR: In this article, the authors obtained the structure of groups having the maximum sum of element orders on noncyclic nilpotent groups of the same order and proved that the maximum order on such groups occurs in cyclic groups.
Abstract: In [1], the authors proved that the maximum sum of element orders on finite groups of the same order occurs in cyclic group. In this paper we obtain the structure of groups having maximum sum of element orders on noncyclic nilpotent group of the same order.
TL;DR: In this paper, it was shown that a reversible ring with an (α, δ)-condition satisfies a McCoy-type property, in the context of the Ore extension R[x; α, ϴ], and provided a rich class of reversible (semicommutative) ϴ-compatible rings.
Abstract: Nielsen [29] proved that all reversible rings are McCoy and gave an example of a semicommutative ring that is not right McCoy. When R is a reversible ring with an (α, δ)-condition, namely (α, δ)-compatibility, we observe that R satisfies a McCoy-type property, in the context of Ore extension R[x; α, δ], and provide rich classes of reversible (semicommutative) (α, δ)-compatible rings. It is also shown that semicommutative α-compatible rings are linearly α-skew McCoy and that linearly α-skew McCoy rings are Dedekind finite. Moreover, several extensions of skew McCoy rings and the zip property of these rings are studied.
TL;DR: In this paper, the zero divisor graph of a commutative ring with identity 1 ǫ ≥ 0 was described, and bounds for the number of edges were given.
Abstract: Let R be a commutative ring with identity 1 ≠ 0 and T be the ring of all n × n upper triangular matrices over R. In this paper, we describe the zero divisor graph of T. Some basic graph theory properties of are given, including determination of the girth and diameter. The structure of is discussed, and bounds for the number of edges are given. In the case that R is a finite integral domain and n = 2, the structure of is fully described and an explicit formula for the number of edges is given.
TL;DR: T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules as mentioned in this paper, and many equivalent conditions for a module M to be t-semiple are found.
Abstract: We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z 2(M) ⊕ S(M) (where Z 2(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.
TL;DR: In this paper, an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph was given, and as an application, the same authors gave a lower bound for a monomial ideal in a polynomial ring S.
Abstract: We give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We also give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph.
TL;DR: In this paper, a notion of mutation of subcategories in a right triangulated category is defined, and a triangulation of a quotient category with a right-triangulated subcategory is defined.
Abstract: A notion of mutation of subcategories in a right triangulated category is defined in this article. When (𝒵, 𝒵) is a 𝒟-mutation pair in a right triangulated category 𝒞, the quotient category 𝒵/𝒟 carries naturally a right triangulated structure. Moreover, if the right triangulated category satisfies some reasonable conditions, then the right triangulated quotient category 𝒵/𝒟 becomes a triangulated category. When 𝒞 is triangulated, our result unifies the constructions of the quotient triangulated categories by Iyama-Yoshino and by Jorgensen, respectively.
TL;DR: In this article, the Ding projective, Ding injective complexes are introduced, and the homotopy theory on the category of modules can be extended to a homotropic theory on complexes.
Abstract: The so-called Ding–Chen ring is an n-FC ring which is both left and right coherent, and has both left and right self FP-injecitve dimensions at most n for some non-negative integer n. In this article, we introduce the so-called Ding projective, Ding injective complexes, and show that over Ding–Chen rings the homotopy theory on the category of modules can be extended to a homotopy theory on the category of complexes.
TL;DR: In this article, the authors classify the reflection subgroups of a finite Coxeter group up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacies class of its Coxeter elements.
Abstract: Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy.
TL;DR: In this article, it was shown that every locally compact Hausdorff topological right gyrogroup appears as a topological gyrotransversal to a closed subgroup in some universal sense.
Abstract: In this article we study topological right gyrogroups and gyrotransversals and discuss the situations in which it is unique. It is shown that the Einstein gyrogroup is the unique gyrotransversal to its gyroautomorphism group in its group of gyrogroup motions. Further, we show that every locally compact Hausdorff topological right gyrogroup appears as a topological gyrotransversal to a closed subgroup of a topological group in some universal sense. We also show that there are right gyrogroups (also gyrogroups) which can not be embedded as a gyrotransversal to a closed subgroup in a connected topological group to which it generates.
TL;DR: In this paper, the authors study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algesbras.
Abstract: B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions.
TL;DR: In this article, the authors provide a detailed structure description of derived subpolygroups of polygroups, and introduce the concept of perfect and solvable polygroups and give some results in this respect.
Abstract: The purpose of this paper is to provide a detailed structure description of derived subpolygroups of polygroups. We introduce the concept of perfect and solvable polygroups and we give some results in this respect. Finally, we discuss on τ-multi-semi-direct hyperproduct of polygroups.
TL;DR: In this paper, the existence of covers and envelopes by some special functors on the category of finitely presented modules is investigated. And the relationship between phantoms and ext-phantoms is obtained.
Abstract: We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every left R-module M has an Ext-phantom preenvelope f: M → N with coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by the FP-injective objects.