Showing papers in "Communications in Algebra in 2015"
TL;DR: In this paper, it was shown that any Jordan τ-derivation of R is X-inner if either R is not a GPI-ring or R is a PI-ring except when charR = 2 and dimC RC = 4, where C is the extended centroid of R.
Abstract: Let R be a prime ring which is not commutative, with maximal symmetric ring of quotients Q ms (R), and let τ be an anti-automorphism of R. An additive map δ: R → Q ms (R) is called a Jordan τ-derivation if δ(x 2) = δ(x)x τ + xδ(x) for all x ∈ R. A Jordan τ-derivation of R is called X-inner if it is of the form x → ax τ − xa for x ∈ R, where a ∈ Q ms (R). It is proved that any Jordan τ-derivation of R is X-inner if either R is not a GPI-ring or R is a PI-ring except when charR = 2 and dim C RC = 4, where C is the extended centroid of R.
65 citations
TL;DR: In this paper, it was shown that for a commutative ring A and n ≥ 3, the graph ZD(R) is connected with diameter two (at most three) and girth three.
Abstract: Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R∖{(0, 0,…, 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)* = Z(R)∖{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for ...
53 citations
TL;DR: In this paper, it was shown that the endomorphism ring of an R-module M R is not a left-Rickart ring in general, but it is a Baer module.
Abstract: It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M R is a finitely generated 𝔏-Rickart module, we prove that End R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End R (M) is a left extending ring iff M is a Baer module and End R (M) is left cononsingular.
50 citations
TL;DR: In this article, a family of quotient rings of the Rees algebra associated to a commutative ring is studied, which generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring.
Abstract: A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.
36 citations
TL;DR: In this paper, all finite groups whose commuting (noncommuting) graphs can be embed on the plane, torus, or projective plane are classified, and all of them can be found on the Euclidean plane.
Abstract: In this paper, all finite groups whose commuting (noncommuting) graphs can be embed on the plane, torus, or projective plane are classified.
35 citations
TL;DR: In this paper, the super finitely presented dimension of R is defined in terms of only super-finitely presented left R-modules, which is a generalization of the notion of FP-injective modules.
Abstract: Let R be a ring. A left R-module M (resp., right R-module N) is called weak injective (resp., weak flat) if (resp., ) for every super finitely presented left R-module F. By replacing finitely presented modules by super finitely presented modules, we may generalize many results of a homological nature from coherent rings to arbitrary rings. Some examples are given to show that weak injective (resp., weak flat) modules need not be FP-injective (resp., not flat) in general. In addition, we introduce and study the super finitely presented dimension (denote by l.sp.gldim(R)) of R that are defined in terms of only super finitely presented left R-modules. Some known results are extended.
35 citations
TL;DR: In this article, a structural characterization of bounded above cohomologically complete complexes and the Cohomologically Complete Nakayama Theorem are presented. But the results of the analysis are limited to the case of a commutative noetherian ring.
Abstract: Let A be a commutative noetherian ring, and 𝔞 an ideal in it. In this paper we continue the study, begun in [11], of the derived 𝔞-adic completion and the derived 𝔞-torsion functors. Here are our results: (1) a structural characterization of bounded above cohomologically complete complexes; (2) the Cohomologically Complete Nakayama Theorem; and (3) a characterization of cohomologically cofinite complexes.
35 citations
TL;DR: In this article, the depth of the binomial edge ideal of a generalized block graph is computed and generalized block graphs whose binomial edges are Cohen-Macaulay and unmixed are characterized.
Abstract: We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen–Macaulay and unmixed. So that we generalize the results of Ene, Herzog, and Hibi on block graphs. Moreover, we study unmixedness and Cohen–Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs.
34 citations
TL;DR: The main purpose of as discussed by the authors is to define representations and a cohomology of Hom-Lie color algebras and to study some key constructions and properties of hom-lie color algesbras.
Abstract: The main purpose of this paper is to define representations and a cohomology of Hom–Lie color algebras and to study some key constructions and properties. We describe Hartwig–Larsson–Silvestrov Theorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α k -generalized derivations and provide examples.
31 citations
TL;DR: In this article, it was shown that the inclusion ideal graph of a ring R is not connected if and only if R ≤ 3, and if R ≥ M 2(D) or D 1Õ×D 2, for some division rings, D, D 1 and D 2.
Abstract: Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ⊂ J or J ⊂ I. In this paper, we show that In(R) is not connected if and only if R ≅ M 2(D) or D 1 × D 2, for some division rings, D, D 1 and D 2. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings.
30 citations
TL;DR: In this paper, it was shown that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a stable poset is always triangulated and has a cluster structure in many cases.
Abstract: Cyclic posets are generalizations of cyclically ordered sets. In this article, we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14], the infinity-gon of [12], and the m-cluster category of type A ∞ (m ≥ 3) [13] are examples of this construction as well as some new examples such as the cluster category of ℤ2. An extension of this construction and further examples are given in [16].
TL;DR: In this paper, ideal-theoretic and homological extensions of the Prufer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal were investigated.
Abstract: This paper investigates ideal-theoretic as well as homological extensions of the Prufer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prufer conditions.
TL;DR: In this paper, a classification of real or complex Leibniz algebras whose nilradical is the 3-dimensional Heisenberg algebra is presented, which is a generalization of the classification of solvable Lie algesbras with heisenberg nilradical.
Abstract: All solvable Lie algebras with Heisenberg nilradical have already been classified. We extend this result to a classification of solvable Leibniz algebras with Heisenberg nilradical. As an example, we show the complete classification of all real or complex Leibniz algebras whose nilradical is the 3-dimensional Heisenberg algebra.
TL;DR: In this article, the transfer of FCP and FIP properties between E and F is studied for an extension E: R ⊂ S of (commutative) rings and the induced extension F: R(X)⊂ ǫ s of Nagata rings.
Abstract: For an extension E: R ⊂ S of (commutative) rings and the induced extension F: R(X) ⊂ S(X) of Nagata rings, the transfer of the FCP and FIP properties between E and F is studied. Then F has FCP ⇔ E has FCP. The extensions E for which F has FIP are characterized. While E has FIP whenever F has FIP, the converse fails for certain subintegral extensions; it does hold if E is integrally closed, seminormal, or subintegral with R quasi-local having infinite residue field. If F has FIP, conditions are given for the sets of intermediate rings of E and F to be order-isomorphic.
TL;DR: In this paper, the authors studied derivations and generalized derivations satisfying certain identities on semigroup ideals of near-rings and proved the necessity of the 3-primeness hypothesis.
Abstract: The purpose of this paper is to study derivations and generalized derivations satisfying certain identities on semigroup ideals of near-rings. Some well-known results characterizing commutativity of 3-prime near-rings by derivations have been generalized by using semigroup ideals. Moreover, examples proving the necessity of the 3-primeness hypothesis are given.
TL;DR: In this article, the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case was studied and it was shown that it admits a graded, finte-dimensional, connected gradation.
Abstract: We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finte-dimensional, connected gradation. Dually, the vector space IR is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibility which makes IR a Com-pre-Lie algebra. We give a presentation of this object as a Com-pre-Lie algebra and as a pre-Lie algebra.
TL;DR: In this paper, a new family of identities satisfied by the semigroups U n (𝕋) of n × n upper triangular tropical matrices is constructed and an elementary proof is given.
Abstract: A new family of identities satisfied by the semigroups U n (𝕋) of n × n upper triangular tropical matrices is constructed and an elementary proof is given.
TL;DR: In this article, the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a nontrivial partition, and symmetric and alternating groups was studied.
Abstract: We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a nontrivial partition, and symmetric and alternating groups. Also, for such a group, the corresponding proper power graph has diameter at most 26 whenever it is connected.
TL;DR: In this paper, the authors studied the vanishing ideal of an algebraic toric set in a projective space over a finite field, and gave an explicit combinatorial description of a set of generators of I(X) when X is associated to an even cycle or to a connected bipartite graph.
Abstract: Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit combinatorial description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise vertex disjoint even cycles. In this case, a formula for the regularity of I(X) is given. We show an upper bound for this invariant, when X is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph is connected. We are able to show a formula for the length of the parameterized linear code associated with any graph, in terms of the number of bipartite and non-bipartite components.
TL;DR: The authors generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen-Macaulay modules.
Abstract: We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen–Macaulay modules.
TL;DR: In this paper, the authors explore the multiple holomorphs of the dihedral groups D n and quaternionic (dicyclic) groups Q n for n ≥ 3.
Abstract: The holomorph of a group G is Norm B (λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D n and quaternionic (dicyclic) groups Q n for n ≥ 3.
TL;DR: In this article, it was shown that a class of groups is root in the sense of K. W. Gruenberg if, and only if, it is closed under subgroups and Cartesian wreath products.
Abstract: We prove that a class of groups is root in a sense of K. W. Gruenberg if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we obtain a condition which is sufficient for the generalized free product of two nilpotent groups to be residual solvable.
TL;DR: In this paper, the dimension of the bounded derived category of finitely generated modules over a commutative Noetherian ring has been studied, and it is shown that it is finite over a complete local ring containing a field with perfect residue field.
Abstract: Several years ago, Bondal, Rouquier, and Van den Bergh introduced the notion of the dimension of a triangulated category, and Rouquier proved that the bounded derived category of coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension. In this article, we study the dimension of the bounded derived category of finitely generated modules over a commutative Noetherian ring. The main result of this article asserts that it is finite over a complete local ring containing a field with perfect residue field. Our methods also give a ring-theoretic proof of the affine case of Rouquier's theorem.
TL;DR: In this article, the authors investigated groups with the second largest value of ψ on the set of groups of the same order, where ψ(G) denotes the sum of element orders of G.
Abstract: For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group ℤ n . In this paper, we investigate groups with the second largest value of ψ on the set of groups of the same order.
TL;DR: In this paper, the authors give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle and find a criterion to determine when Betti number of this ideal are nonzero.
Abstract: We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are nonzero and give a formula to compute its projective dimension and regularity.
TL;DR: In this article, the Chermak-Delgado lattice of a finite group with subgroups is defined as a moduar sublattice within the subgroup lattice.
Abstract: If G is a finite group with subgroup H, then the Chermak–Delgado measure of H (in G) is defined as |H||C G (H)|. The Chermak–Delgado lattice of G, denoted 𝒞𝒟(G), is the set of all subgroups with maximal Chermak–Delgado measure; this set is a moduar sublattice within the subgroup lattice of G. In this paper we provide an example of a p-group P, for any prime p, where 𝒞𝒟(P) is lattice isomorphic to 2 copies of ℳ2 (a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in a p-group with Chermak–Delgado lattice that, for any positive integers n and l, is a 2l-string of n-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for a p-group with Chermak–Delgado lattice that is a 2l-string of ℳ p+1 (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.
TL;DR: In this article, the authors study transfers of S-Noetherian property to the composite semigroup ring and the composite generalized power series ring, and show that the latter is an S-noetherian ring.
Abstract: Let R be a commutative ring with identity and S a multiplicative subset of R. We say that R is an S-Noetherian ring if for each ideal I of R, there exist an s ∈ S and a finitely generated ideal J of R such that sI ⊆ J ⊆ I. In this article, we study transfers of S-Noetherian property to the composite semigroup ring and the composite generalized power series ring.
TL;DR: In this paper, various classes of monoids of transformations of a finite chain, including those of transformations that preserve or reverse either the order or the orientation, are studied, and the study of the ranks of all their ideals is complete.
Abstract: In this paper we consider various classes of monoids of transformations of a finite chain, including those of transformations that preserve or reverse either the order or the orientation. In line with Howie and McFadden [24], we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals.
TL;DR: In this paper, the authors introduce and study V- and CI-semirings, which are semirings all of whose simple and cyclic semimodules are injective.
Abstract: In this article, we introduce and study V- and CI-semirings—semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and antibounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple antibounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.
TL;DR: In this paper, the authors established characteristic-free criteria for the componentwise linearity of graded ideals and classified them among the Gorenstein ideals, the standard determinantal ideals, and the ideals generated by the submaximal minors of a symmetric matrix.
Abstract: We establish characteristic-free criteria for the componentwise linearity of graded ideals. As applications, we classify the componentwise linear ideals among the Gorenstein ideals, the standard determinantal ideals, and the ideals generated by the submaximal minors of a symmetric matrix.