Showing papers in "Communications in Algebra in 2016"
TL;DR: In this paper, the authors studied the prime spectrum of the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the A+XB[X], the A + XB[[X]] and the D+M constructions).
Abstract: Let f: A → B be a ring homomorphism, and let J be an ideal of B. In this article, we study the amalgamation of A with B along J with respect to f (denoted by A ⋈fJ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the A + XB[X], the A + XB[[X]] and the D + M constructions). In particular, we completely describe the prime spectrum of the amalgamation A ⋈fJ and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen–Macaulay ring or a Gorenstein ring.
54 citations
TL;DR: In this paper, it was shown that any super-skewsymmetric super-biderivation of SVir is inner and every linear super-commuting map ψ on SVir can be computed in the form ψ(x) = ǫ(ǫ)c, where c is the central charge of the super-Virasoro algebras.
Abstract: Let SVir be the well-known super-Virasoro algebras. In this paper, we first prove that any super-skewsymmetric super-biderivation of SVir is inner. Based on this, we show that every linear super-commuting map ψ on SVir is of the form ψ(x) = f(x)c, where f is a linear function from SVir to ℂ mapping the odd part of SVir to zero, and c is the central charge of SVir.
48 citations
TL;DR: In this article, the authors introduced a graph structure called subspace inclusion graph ℐn(𝕍) on a finite dimensional vector space, where the vertex set is the collection of nontrivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other.
Abstract: In this paper, the authors introduce a graph structure, called subspace inclusion graph ℐn(𝕍) on a finite dimensional vector space 𝕍 where the vertex set is the collection of nontrivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other. The diameter, girth, clique number, and chromatic number of ℐn(𝕍) are studied. It is shown that two subspace inclusion graphs are isomorphic if and only if the base vector spaces are isomorphic. Finally, some properties of subspace inclusion graph are studied when the base field is finite.
48 citations
TL;DR: In this article, a graph structure called a nonzero component graph on finite dimensional vector spaces is introduced, and the graph is connected and the domination number and independence number are found.
Abstract: In this article, we introduce a graph structure, called a nonzero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms, and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite.
47 citations
TL;DR: In this article, the authors investigated distant properties of commuting graphs on D2n and studied the metric dimension of the commuting graph on D 2n and computed its resolving polynomial.
Abstract: Let Γ be a non-abelian group and Ω ⊆ Γ. We define the commuting graph G = 𝒞(Γ, Ω) with vertex set Ω and two distinct elements of Ω are joined by an edge when they commute in Γ. In this article, among some properties of commuting graphs, we investigate distant properties as well as detour distant properties of commuting graph on D2n. We also study the metric dimension of commuting graph on D2n and compute its resolving polynomial.
40 citations
TL;DR: In this article, the additivity of n-multiplicative isomorphisms from ℜ onto ℕ and derivations of ℘ into ℞ was studied.
Abstract: Let ℜ and ℜ′ be alternative rings. We study the additivity of n-multiplicative isomorphisms from ℜ onto ℜ′ and of n-multiplicative derivations of ℜ. We prove that, if ℜ contains a family of nontrivial idempotents satisfying Martindale's conditions, then these two maps are additives.
33 citations
TL;DR: In this article, the authors define, characterize and deduce properties of chain complexes of R-modules and show that the category Ch(R) has a cofibrantly generated model structure where every object is co-fibrant and the fibrant objects are exactly the Gorenstein AC-injective chain complexes.
Abstract: Absolutely clean and level R-modules were introduced in [2] and used to show how Gorenstein homological algebra can be extended to an arbitrary ring R. This led to the notion of Gorenstein AC-injective and Gorenstein AC-projective R-modules. Here we study these concepts in the category of chain complexes of R-modules. We define, characterize and deduce properties of absolutely clean, level, Gorenstein AC-injective, and Gorenstein AC-projective chain complexes. We show that the category Ch(R) of chain complexes has a cofibrantly generated model structure where every object is cofibrant and the fibrant objects are exactly the Gorenstein AC-injective chain complexes.
30 citations
TL;DR: In this article, it was shown that any abelian Schur group belongs to one of several explicitly given families only and is isomorphic to any noncyclic ABG of odd order.
Abstract: A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ℤ3 × ℤ3 k or ℤ3 × ℤ3 × ℤ p where k ≥ 1 and p is a prime. In addition, we prove that ℤ2 × ℤ2 × ℤ p is a Schur group for every prime p.
26 citations
TL;DR: In this paper, the authors give an S-version of Eakin-Nagata-Formanek Theroem [7], in the case where S is finite, and prove that any increasing chain of extended submodules of an R-module is S-stationary.
Abstract: Let R be a commutative ring with unity, S a multiplicative subset of R, and M an R-module. In this article, we investigate S-Noetherian modules. We give an S-version of Eakin–Nagata–Formanek Theroem [7], in the case where S is finite. We prove that if M is an S-finite R-module and any increasing chain of extended submodules of M is S-stationary then M is S-Noetherian.In the second part of this article, we define S-accr modules. An R-module M is said to satisfy S-accr if any ascending chain of residuals of the form (N: B) ⊆ (N: B2) ⊆ (N: B3) ⊆ … is S-stationary where N is a submodule of M and B is a finitely generated ideal of R. We investigate the class of such modules M, and we generalize some known results of P. C. Lu ([5], [6]).
25 citations
TL;DR: In this article, a monoid of all transformations of the finite set X preserving a uniform partition of X into m subsets of size n, where m, n ≤ 2, is defined.
Abstract: Denote by 𝒯n and 𝒮n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ℰn = {1} ∪ (𝒯n∖𝒮n). Let 𝒯(X, 𝒫) denote the monoid of all transformations of the finite set X preserving a uniform partition 𝒫 of X into m subsets of size n, where m, n ≥ 2. We enumerate the idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ⟨ E ⟩ generated by the idempotents E = E(𝒯(X, 𝒫)). We show that S = S1 ∪ S2, where S1 is a direct product of m copies of ℰn, and S2 is a wreath product of 𝒯n with 𝒯m∖𝒮m. We calculate the rank and idempotent rank of S, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of ℰn.
23 citations
TL;DR: In this article, the authors generalize some results about the graded Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal J of dimension one.
Abstract: We generalize some results about the graded Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal J of dimension one. When the saturation I of J is a complete intersection, we get formulas for some invariants. Examples of hypersurfaces V: f = 0 in ℙn whose Jacobian ideals J satisfy this property and with nontrivial Alexander polynomials are given in any dimension. A Lefschetz property for the graded quotient I/J is proved for n = 2 and a counterexample due to A. Conca is given for such a property when n = 3.
TL;DR: For an R-module M, projective in σ[M] and satisfying ascending chain condition on left annihilators, the concept of Goldie module was introduced in this paper.
Abstract: For an R-module M, projective in σ[M] and satisfying ascending chain condition on left annihilators, the authors introduce the concept of Goldie module. The authors also use the concept of semiprime module defined by Raggi et al. in [15] to give necessary and sufficient conditions for an R-module M, to be a semiprime Goldie module. This theorem is a generalization of Goldie's theorem for semiprime left Goldie rings. Moreover, the authors prove that M is a semiprime (prime) Goldie module if and only if the ring S = EndR(M) is a semiprime (prime) right Goldie ring. Also, we study the case when M is a duo module.
TL;DR: In this article, the cycle structure of a partial isometry was investigated and the Green's relations on the cycle was characterized for a 0-E-unitary inverse semigroup.
Abstract: Let ℐn be the symmetric inverse semigroup on Xn = {1, 2,…, n}, and let 𝒟𝒫n and 𝒪𝒟𝒫n be its subsemigroups of partial isometries and of order-preserving partial isometries of Xn, respectively. In this article, we investigate the cycle structure of a partial isometry and characterize the Green's relations on 𝒟𝒫n and 𝒪𝒟𝒫n. We show that 𝒪𝒟𝒫n is a 0-E-unitary inverse semigroup. We also investigate the ranks of 𝒟𝒫n and 𝒪𝒟𝒫n.
TL;DR: In this article, the authors classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank over an algebraically closed field of characteristic 0.
Abstract: We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank over an algebraically closed field of characteristic 0. In case of rank 0, we describe conditions to obtain non trivial ℤ k -gradings.
TL;DR: In this article, the authors consider generalized skew derivations of R, with the same associated automorphism, and describe all possible forms of F and G, and show that for all r 1, 1, 2, rn ∈ R, f(x1, n, xn) is a noncentral multilinear polynomial over C with n noncommuting variables.
Abstract: Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, and C its extended centroid. Suppose that F, G are generalized skew derivations of R, with the same associated automorphism, and f(x1,…, xn) a noncentral multilinear polynomial over C with n noncommuting variables, such that for all r1,…, rn ∈ R. Then we describe all possible forms of F and G.
TL;DR: The existence of the largest left quotient ring Ql(R) of an arbitrary ring R of a ring R is proved in this article, where S 0 (R) is the largest regular denominator set of R and S ∈ R ∩ Ql (R).
Abstract: The left quotient ring (ie, the left classical ring of fractions) Qcl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens In this article, existence of the largest left quotient ring Ql(R) of an arbitrary ring R is proved, ie, Ql(R) = S0(R)−1R where S0(R) is the largest left regular denominator set of R It is proved that Ql(Ql(R)) = Ql(R); the ring Ql(R) is semisimple iff Qcl(R) exists and is semisimple; moreover, if the ring Ql(R) is left Artinian, then Qcl(R) exists and Ql(R) = Qcl(R) The group of units Ql(R)* of Ql(R) is equal to the set {s−1t | s, t ∈ S0(R)} and S0(R) = R ∩ Ql(R)* If there exists a finitely generated flat left R-module which is not projective, then Ql(R) is not a semisimple ring We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denomina
TL;DR: In this article, the loci corresponding to nonsingular degree 5 projective plane curves, which are nonempty, were determined. But these loci do not cover the full automorphism group of genus g curves.
Abstract: Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite nontrivial group) is isomorphic to a subgroup of Aut(δ), and let be the subset of curves δ such that G ≅ Aut(δ), where Aut(δ) is the full automorphism group of δ. Now, for an integer d ≥ 4, let be the subset of Mg representing smooth, genus g, plane curves of degree d, i.e. smooth curves that admits a plane non-singular model of degree d, (in this case, g = (d − 1)(d − 2)/2), and consider the sets and .Henn in [7] and Komiya and Kuribayashi in [10], listed the groups G for which is nonempty. In this article, we determine the loci , corresponding to nonsingular degree 5 projective plane curves, which are nonempty. Also, we present the analogy of Henn's results for quartic curves concerning nonsingular plane model equations associated to these loci (see Table 2 for more details). Similar arguments can be applied to deal with h...
TL;DR: In this paper, it was shown that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field.
Abstract: In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.
TL;DR: In this paper, the largest regular subsemigroup of 𝒪&#x 1d4ab; n (Y) of all transformations with range contained in Y is described.
Abstract: Let X n be a chain with n elements (n ∈ ℕ), and let 𝒪𝒫 n be the monoid of all orientation-preserving transformations of X n . In this article, for any nonempty subset Y of X n , we consider the subsemigroup 𝒪𝒫 n (Y) of 𝒪𝒫 n of all transformations with range contained in Y: We describe the largest regular subsemigroup of 𝒪𝒫 n (Y), which actually coincides with its subset of all regular elements. Also, we determine when two semigroups of the type 𝒪𝒫 n (Y) are isomorphic and calculate their ranks. Moreover, a parallel study is presented for the correspondent subsemigroups of the monoid 𝒪ℛ n of all either orientation-preserving or orientation-reversing transformations of X n .
TL;DR: In this paper, the authors investigate properties and describe examples of tilt-stable objects on a smooth complex projective, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.
Abstract: We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarization ω gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.
TL;DR: In this paper, it was shown that relations between functors are preserved between their twists, and deduced that various relations hold between derived Hochschild (co)-homology and the f! functor.
Abstract: Let be a regular ring, and let A, B be essentially finite type -algebras. For any functor F: D(ModA) × ⋅ × D(ModA) → D(ModB) between their derived categories, we define its twist F!: D(ModA) × ⋅ × D(ModA) → D(ModB) with respect to dualizing complexes, generalizing Grothendieck's construction of f!. We show that relations between functors are preserved between their twists, and deduce that various relations hold between derived Hochschild (co)-homology and the f! functor. We also deduce that the set of isomorphism classes of dualizing complexes over a ring (or a scheme) form a group with respect to derived Hochschild cohomology, and that the twisted inverse image functor is a group homomorphism.
TL;DR: In this article, the authors give a new characterization of strongly Gorenstein projective modules, and prove that a strongly projective left R-module of countable type is a strongly girded projective module.
Abstract: In this article, we give a new characterization of Gorenstein projective modules. As applications of our result, we prove that a strongly Gorenstein projective module of countable type is Gorenstein flat, and each left R-module has a special Gorenstein projective precover whenever all projective left R-modules have finite injective dimension.
TL;DR: In this paper, the same authors generalized the geometric sequence {ap, ap −1b, ap−2b2, bp} to allow the p copies of a (resp. b) to all be different.
Abstract: We generalize the geometric sequence {ap, ap−1b, ap−2b2,…, bp} to allow the p copies of a (resp. b) to all be different. We call the sequence {a1a2a3… ap, b1a2a3…ap, b1b2a3…ap,…, b1b2b3…bp} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apery sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.
TL;DR: In this paper, a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C is considered, and f(x1,xn) is a multilinear polynomial over C, which is not central valued on R.
Abstract: Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x1,…, xn) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) − f(r)G(f(r))) = 0 for all r = (r1,…, rn) ∈ Rn, then one of the following holds: There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a − q] = 0 and f(x1,…, xn)2 is central valued on R;There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa − bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;R satisfies s4 and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa − bx and G(x) = ax + xb for all x ∈ R;There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb − δ(x), G(x) = bx + δ(x) and d(x) = [c, x...
TL;DR: In this article, the authors investigated various properties of the pure virtual braid group PV3 and showed that it is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands.
Abstract: We investigate various properties of the pure virtual braid group PV3. Out of its presentation, we get a free product decomposition of PV3. As a consequence, we show that PV3 is residually torsion free nilpotent, what implies that the set of the finite type invariants in the sense of Goussarov–Polyak–Viro is complete for virtual pure braids with three strands. Moreover, we prove that the presentation of PV3 is aspherical. We determine also the cohomology ring and the associated graded Lie algebra of PV3.
TL;DR: In this article, the authors modify the method of little groups to construct supercharacter theories of semidirect products with abelian normal subgroups, and apply this construction to reproduce known super character theories of several families of unipotent groups.
Abstract: The “method of little groups” describes the irreducible characters of semidirect products with abelian normal subgroups in terms of the irreducible characters of the factor groups. We modify this method to construct supercharacter theories of semidirect products with abelian normal subgroups. In particular, we apply this construction to reproduce known supercharacter theories of several families of unipotent groups. We also utilize our method to construct a collection of new supercharacter theories of the unipotent upper-triangular matrices.
TL;DR: In this article, the minimal free resolution of the Veronese modules, Sn, d, k = ⊕i≥0Sk+id, where S = 𝕂[x, y, z](d), for d ǫ = 4, 5, and S 2, dǫ is pure.
Abstract: We study the minimal free resolution of the Veronese modules, Sn, d, k = ⊕i≥0Sk+id, where S = 𝕂[x1,…, xn], by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We prove that Sn, d, k is Cohen–Macaulay if and only if k d(n − 1) − n. We prove combinatorially that the resolution of S2, d, k is pure. We show that . As an application, we calculate the complete Betti diagrams of the Veronese rings 𝕂[x, y, z](d), for d = 4, 5, and 𝕂[x, y, z, u](3).
TL;DR: The octopus inequality conjecture was recently proven by Caputo, Liggett, and Richthammer as discussed by the authors using a nonlinear mapping in the group algebra which permits a proof by induction.
Abstract: A conjecture by D. Aldous, which can be formulated as a statement about the first nontrivial eigenvalue of the Laplacian of certain Cayley graphs on the symmetric group generated by transpositions, has been recently proven by Caputo, Liggett, and Richthammer. Their proof is a subtle combination of two ingredients: a nonlinear mapping in the group algebra which permits a proof by induction, and a quite hard estimate named the octopus inequality. In this article we present a simpler and more transparent proof of the octopus inequality, which emerges naturally when looking at the Aldous’ conjecture from an algebraic perspective.
TL;DR: In this article, it was shown that every cycle-finite artin algebra with finitely many isomorphism classes of τ-rigid indecomposable modules is of finite representation type.
Abstract: We prove that every cycle-finite artin algebra with finitely many isomorphism classes of τ-rigid indecomposable modules is of finite representation type.
TL;DR: In this paper, it was shown that there is only a unique non-alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.
Abstract: Simple right alternative superalgebras which have a simple algebra as even part and, as odd part, an irreducible bimodule over the even part are investigated. Under these conditions, superalgebras with one dimensional even part are classified, as well as superalgebras having M 2(F) as even part and a unital irreducible bimodule over M 2(F) of dimension less than or equal to 6 as odd part. It is shown that there is only a unique non alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.