Showing papers in "Communications in Algebra in 2017"
TL;DR: In this paper, the authors investigated the small support and co-support of X, introduced by Foxby and Benson, over a chain complex over a commutative noetherian ring.
Abstract: Let X be a chain complex over a commutative noetherian ring R, that is, an object in the derived category 𝒟(R). We investigate the small support and co-support of X, introduced by Foxby and Benson,...
39 citations
TL;DR: In this article, it was shown that for every s ≥ 1, the equality reg(Is)=2s+⌈n−12⌉−1 holds.
Abstract: Let G = W(Cn) be a whiskered cycle graph with edge ideal I = I(G). We prove that for every s≥1, the equality reg(Is)=2s+⌈n−12⌉−1 holds.
38 citations
TL;DR: In this paper, the authors investigate commutativity of ring R with involution, which admits a derivation satisfying certain algebraic identities, and provide examples to show that various restrictions imposed in the hypotheses of their theorems are not superfluous.
Abstract: In this paper, we investigate commutativity of ring R with involution ∗ which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.
37 citations
TL;DR: In this paper, the authors studied the metric dimension of zero-divisor graphs associated with commutative rings and showed that the dimension of these graphs scales with the number of vertices.
Abstract: Let R be a commutative ring with unity 1 and let G(V,E) be a simple graph. In this research article, we study the metric dimension in zero-divisor graphs associated with commutative rings. We show ...
34 citations
TL;DR: In this paper, the biderivations without the skew-symmetric condition of W-algebras including the Witt algebra, the algebra W(2,2) and their central extensions are characterized.
Abstract: In this paper, the biderivations without the skew-symmetric condition of W-algebras including the Witt algebra, the algebra W(2,2) and their central extensions are characterized. Some classes of non-inner biderivations are presented. As applications, the forms of linear commuting maps and the commutative post-Lie algebra structures on aforementioned W-algebras are given.
34 citations
TL;DR: In this article, a natural generalization of locally noetherian and locally coherent categories leads to define locally type FP∞ categories, which include not just all categories of modules over a ring, but also...
Abstract: A natural generalization of locally noetherian and locally coherent categories leads us to define locally type FP∞ categories. They include not just all categories of modules over a ring, but also ...
29 citations
TL;DR: In this paper, it was shown that the set of all σ-subnormal (σ-nilpotent) subgroups of G forms a sublattice of the lattice of all subgroups.
Abstract: Let G be a finite group and σ = {σi|i∈I} be some partition of the set of all primes. We say that G is: σ-primary if G is a σi-group, for some i; σ-nilpotent if G=G1×⋯×Gn for some σ-primary groups G1,…,Gn. A subgroup A of G is called σ-subnormal in G if there is a subgroup chain A=A0≤A1≤⋯≤Am=G such that either Ai−1⊴Ai or Ai∕(Ai−1)Ai is σ-primary for all i = 1,…,m. In this paper we prove that the set of all σ-subnormal (σ-nilpotent) subgroups of G forms a sublattice of the lattice of all subgroups of G and, being based on this result, we prove also that if every Schmidt subgroup of G is σ-subnormal in G, then the commutant subgroup G′ is σ-nilpotent.
28 citations
TL;DR: In this paper, it was shown that if R is a finite presented Koszul 𝕂-algebra then every graded skew PBW extension of R is KoszUL.
Abstract: Pre-Koszul and Koszul algebras were defined by Priddy [15]. There exist some relations between these algebras and the skew PBW extensions defined in [8]. In [24] we gave conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul 𝕂-algebra then every graded skew PBW extension of R is Koszul.
25 citations
TL;DR: In this paper, the classification of four-dimensional non-Lie nilpotent Leibniz algebras is given, using the canonical forms for the congruence classes of matrices of bilinear forms.
Abstract: Leibniz algebras are certain generalization of Lie algebras. In this paper, we give the classification of four-dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms and some other techniques to obtain our result.
25 citations
TL;DR: In this paper, a survey of the structure of relatively free dimonoids is presented, including some new ideas in the last sections of the paper, and it is shown that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.
Abstract: Loday introduced the notion of a dimonoid and constructed the free dimonoid. The paper concerns the variety theory of dimonoids. We recall and summarize the results obtained by Loday, the author as well as others on the structure of some relatively free dimonoids. This part of the paper should be viewed as a survey, however we include some new ideas in the last sections. Namely, we construct a free (lz;rs)-dimonoid, a free (rs;rz)-dimonoid, a free rs-dimonoid, a free (rb;rs)-dimonoid, a free (rs;rb)-dimonoid and characterize the least (lz;rs)-congruence, the least (rs;rz)-congruence, the least rs-congruence, the least (rb;rs)-congruence and the least (rs;rb)-congruence on the free dimonoid. We also establish that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.
24 citations
TL;DR: In this paper, the authors consider the case of a polynomial f a Lie polynomials in M2(K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0.
Abstract: Kaplansky asked about the possible images of a polynomial f in several noncommuting variables. In this paper, we consider the case of f a Lie polynomial. We describe all the possible images of f in M2(K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial sk is not a Lie polynomial, for k>2.
Abstract: Let ℛ be a commutative ring with identity and let 𝔄 = Tri(𝒜,ℳ,ℬ) be a triangular algebra consisting of unital algebras 𝒜,ℬ over ℛ and an (𝒜,ℬ)-bimodule ℳ which is faithful as a left 𝒜-module as well as a right ℬ-module. In this paper, we prove that under certain assumptions every nonlinear generalized Lie triple derivation GL:𝔄→𝔄 is of the form GL = δ+τ, where δ:𝔄→𝔄 is an additive generalized derivation on 𝔄 and τ is a mapping from 𝔄 into its center which annihilates all Lie triple products [[x,y],z].
TL;DR: In this article, the authors define right n-angulated categories, which are analogous to right triangulated classes, and show that under certain conditions, they can be defined under certain c...
Abstract: We define right n-angulated categories, which are analogous to right triangulated categories. Let 𝒞 be an additive category and 𝒳 a covariantly finite subcategory of 𝒞. We show that under certain c...
TL;DR: In this article, a structure theory of parakahler hom-Lie algebras in terms of hom-left-symmetric (HLS) algebases is presented.
Abstract: In this paper, the parakahler Hom-Lie algebras or phase space of Hom-Lie algebras in terms of Hom-left-symmetric algebras are studied. A structure theory of parakahler Hom-Lie algebras in terms of ...
TL;DR: In this paper, a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two is given, where the characteristic of the ground field is not two.
Abstract: A nonassociative algebra is defined to be zeropotent if the square of any element is zero. Zeropotent algebras are exactly the same as anticommutative algebras when the characteristic of the ground field is not two. The class of zeropotent algebras properly contains that of Lie algebras. In this paper, we give a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional complex Lie algebras, which is in accordance with the conventional one.
TL;DR: In this article, the structure of groups and algebras that can be represented as automorphisms, respectively derivations, of bilinear maps is considered. And exact sequences that capture structure and prove theorems of Morita and Skolem-Noether type are introduced.
Abstract: We consider the structure of groups and algebras that can be represented as automorphisms, respectively derivations, of bilinear maps. Representations of that sort arise when we attempt to describe the automorphisms of groups, rings, and algebras that are nilpotent. We introduce exact sequences that capture structure and prove theorems of Morita and Skolem–Noether type. We apply these results to compute automorphisms of groups and rings.
TL;DR: In this article, it was shown that if cd(I,R) = t > 0 and the R-module HomR(R∕I,HIt(R)) is finitely generated, then the H-module is faithful, i.e., AnnHit(R)=0.
Abstract: Let R be a commutative Noetherian domain and let I be an ideal of R. In this paper it is shown that if cd(I,R) = t>0 and the R-module HomR(R∕I,HIt(R)) is finitely generated, then the R-module HIt(R) is faithful, i.e., AnnHIt(R)=0. This result provides a partially affirmative answer to Lynch’s conjecture in [10]. Moreover, in this paper we construct a counterexample to this conjecture.
TL;DR: In this article, it was shown that an ideal generated by m'th powers of generic forms of degree d ≥ 2 gives the same Hilbert series as an ideal created by generic form of degree md.
Abstract: There is a longstanding conjecture by Froberg about the Hilbert series of the ring R∕I, where R is a polynomial ring, and I an ideal generated by generic forms. We prove this conjecture true in the case when I is generated by a large number of forms, all of the same degree. We also conjecture that an ideal generated by m’th powers of generic forms of degree d≥2 gives the same Hilbert series as an ideal generated by generic forms of degree md. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.
TL;DR: The catenary degree of elements contained in numerical monoids generated by arithmetic sequences is computed by describing each element in terms of the cardinality of its length set and of its set of factorizations, which allows us to define and compute the dissonance number.
Abstract: We compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.
TL;DR: In this article, the authors classify finite groups with exactly two supercharacter theories and show that the solvable groups with two super-character theories are Ω3 and S3.
Abstract: In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are ℤ3 and S3. We also show that the only nonsolvable group with two supercharacter theories is Sp(6,2).
TL;DR: In this paper, it was shown that the number of cyclic subgroups is minimal for the cyclic group and the product of the orders of the elements is maximal for the group.
Abstract: We prove several results detecting cyclicity or nilpotency of a finite group G in terms of inequalities involving the orders of the elements of G and the orders of the elements of the cyclic group of order |G|. We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group, and the product of the orders of the elements is maximal for the cyclic group.
TL;DR: In this article, the Euclidean distance data singular locus and the EDF data isotropic locus are connected to the dual cone of an affine cone.
Abstract: The generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree (or ED degree). The two special loci of the data points where the number of critical points is smaller than the ED degree are called the Euclidean distance data singular locus and the Euclidean distance data isotropic locus. In this article, we present connections between these two special loci of an affine cone and its dual cone.
TL;DR: In this article, it was shown that the power graph of the alternating group An is 2-connected if and only if the type graph of an alternating group is 2connected, and if either n = 3 or none of n, n−1,n−1/n−2,n2 and n−12 is a prime.
Abstract: Let 𝒫0(An),𝒫0(An),𝒯0(An) and 𝒪0(An) be, respectively, the proper power graph, the proper quotient power graph, the proper type graph and the proper order graph of the alternating group An, for n≥3. We determine number and typology of the components of those graphs. In particular, we prove that the power graph 𝒫(An) is 2-connected if and only if the type graph 𝒯 (An) is 2-connected, if and only if either n = 3 or none of n,n−1,n−2,n2 and n−12 is a prime.
TL;DR: The k-zero-divisor hypergraph of R, denoted by ℋk(R), is a hypergraph with vertex vertices as discussed by the authors, where k is a constant.
Abstract: Let R be a commutative ring with identity and let Z(R,k) be the set of all k-zero-divisors in R and k>2 an integer. The k-zero-divisor hypergraph of R, denoted by ℋk(R), is a hypergraph with vertex...
TL;DR: In this article, the relative Tate cohomology of complexes is studied for detecting the finiteness of the relative homological dimensions of complexes with respect to the Tate resolution dimension.
Abstract: Let (𝒳,𝒴) be a complete and hereditary cotorsion pair in a bicomplete abelian category 𝒜. We introduce a Gorenstein category 𝒢(𝒳) and 𝒢(𝒳)-resolution dimension of complexes with respect to (𝒳,𝒴). For complexes with finite 𝒢(𝒳)-resolution dimension, Tate 𝒳-resolutions are constructed. Furthermore, we study relative Tate cohomology of complexes, which is useful for detecting the finiteness of the relative homological dimensions of complexes.
TL;DR: In this article, Grobner-Shirshirshov bases technique for pre-associative (dendriform) algebras and composition-diamond lemma were developed.
Abstract: We develop Grobner–Shirshov bases technique for pre-associative (dendriform) algebras and prove a version of composition-diamond lemma.
TL;DR: In this article, the first steps in classifying finite solvable groups having Property A, which is a property involving abelian normal subgroups, were provided, and this classification is reduced to a lower bound.
Abstract: This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced t...
TL;DR: In this article, the simplicity of quadratic Lie conformal algebras is investigated from the view of the corresponding Gel'fand-Dorffman bialges.
Abstract: In this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some sufficient conditions and necessary condition...
TL;DR: In this article, it was shown that τ is a commuting map of a semiprime ring with an anti-automorphism τ, which is of finite order, for all x ∈ R, where n 1,n 2, n 3,nk are k fixed positive integers.
Abstract: Let R be a semiprime ring with an anti-automorphism τ, which is of finite order. It is proved that if [[…[τ(x),xn1],…],xnk]=0 for all x∈R, where n1,n2,…,nk are k fixed positive integers, then τ is a commuting map. Moreover, commuting anti-automorphisms of semiprime rings are also characterized.
Abstract: For sequences of naturally graded quasi-filiform Leibniz algebras of second type ℒ1 and ℒ3 introduced by Camacho et al., all possible right and left solvable indecomposable extensions over the field ℝ are constructed so that these algebras serve as the nilradicals of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.