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Showing papers in "Communications in Algebra in 2004"


Journal ArticleDOI
TL;DR: In this paper, the resolution of a facet ideal associated with simplicial complexes is studied, and it is shown that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles.
Abstract: In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determine the projective dimension and the regularity of I if the tree is 1-dimensional, show that the graded Betti numbers of I satisfy an alternating sum property if the tree is connected in codimension 1, and classify all trees whose facet ideal has a linear resolution.

172 citations


Journal ArticleDOI
TL;DR: In this article, the notions of Baer and quasi-Baer properties were introduced in a general module theoretic setting, and it was shown that a module M is Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M.
Abstract: We introduce the notions of Baer and quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property and every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the (FI-) extending property are investigated and it is shown that a module M is (quasi-) Baer and (FI-) 𝒦-cononsingular if and only if it is (FI-) extending and (FI-) 𝒦-nonsingular. We prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. Applications of results are provided.

137 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigate coherent-like conditions and related properties that a trivial extension R ∈ A ∈ E might inherit from the ring A for some classes of modules E. The results capture previous results dealing primarily with coherence and also establish satisfactory analogues of well-known coherence-like results on pullback constructions.
Abstract: This paper investigates coherent-like conditions and related properties that a trivial extension R ≔ A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition.

116 citations


Journal ArticleDOI
TL;DR: A ring R is called Armendariz if, whenever in R[x], a i b j ǫ = 0 for all i and j, and a necessary and sufficient condition for a trivial extension of a ring R to be Armenderiz is obtained as mentioned in this paper.
Abstract: A ring R is called Armendariz if, whenever in R[x], a i b j = 0 for all i and j In this paper, some “relatively maximal” Armendariz subrings of matrix rings are identified, and a necessary and sufficient condition for a trivial extension to be Armendariz is obtained Consequently, new families of Armendariz rings are presented

89 citations


Journal ArticleDOI
TL;DR: In this article, a method to obtain the primitive central idempotent of the rational group algebra ℚG over a finite group associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group was given.
Abstract: We give a method to obtain the primitive central idempotent of the rational group algebra ℚG over a finite group G associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group. We also show that for abelian-by-supersolvable groups this method takes a particularly easy form that can be used to compute the Wedderburn decomposition of ℚG.

82 citations


Journal ArticleDOI
TL;DR: In this paper, a general theory of tilting modules for graded Lie superalgebras is developed, extending the work of Soergel for grading Lie algebra, and the main result of the article gives a twisted version of BGG reciprocity relating multiplicities in Δ-flags of indecomposable tilting module to composition multiplicity of costandard modules.
Abstract: We develop a general theory of tilting modules for graded Lie superalgebras, extending work of Soergel for graded Lie algebras. The main result of the article gives a twisted version of BGG reciprocity relating multiplicities in Δ-flags of indecomposable tilting modules to composition multiplicities of costandard modules. We then discuss the examples 𝔤l(m|n) and 𝔮(n) in detail.

65 citations


Journal ArticleDOI
TL;DR: In this paper, a necessary and sufficient condition such that the geometric space (G, P σ(G)) of a 0-simple semigroup is strongly transitive is given.
Abstract: In this paper we determine a family P σ(H) of subsets of a hypergroup H such that the geometric space (H, P σ(H)) is strongly transitive and we use this fact to characterize the hypergroups such that the derived hypergroup D(H) of H coincides with an element of P σ(H). In this case a n-tuple (x 1, x 2,…, x n ) ∈ H n exists such that . Moreover, in the last section, we prove that in every semigroup the transitive closure γ* of the relation γ is the smallest congruence such that is a commutative semigroup. We determine a necessary and sufficient condition such that the geometric space (G, P σ(G)) of a 0-simple semigroup is strongly transitive. Finally, we prove that if G is a simple semigroup, then the space (G, P σ(G)) is strongly transitive and the relation γ of G is transitive.

64 citations


Journal ArticleDOI
TL;DR: For standard graded Artinian K-algebras defined by componentwise linear ideals and Gotzmann ideals, the authors gave conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.
Abstract: For standard graded Artinian K-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.

61 citations


Journal ArticleDOI
TL;DR: In this article, a semigroup T(X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ and R is defined.
Abstract: For a set X, an equivalence relation ρ on X, and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T(X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T(X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and image R. We determine the structure of T(X, ρ, R) in terms of Green's relations, describe the regular elements of T(X, ρ, R), and determine the following classes of the semigroups T(X, ρ, R): regular, abundant, inverse, and completely regular.

53 citations


Journal ArticleDOI
TL;DR: In this paper, a conjugacy closed loop is considered, and the associators of elements of the loop have order 2 if in addition the loop is diassociative (i.e., an extra loop).
Abstract: Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is power-associative and |Q| is finite and relatively prime to 6, then Q is a group. If Q is a finite non-associative extra loop, then 16 ∣ |Q|.

52 citations


Journal ArticleDOI
TL;DR: The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics as discussed by the authors. But it is not a theory suited to number-theoretical applications.
Abstract: The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrere symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed field, namely sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law due to Witt. Needless to say, we have been much influenced by the work of Tate on sum-of-residues-equals-zero and the work of Arbarello-De Concini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

Journal ArticleDOI
TL;DR: In this article, the i-th sectional geometric genus of a quasi-polarized variety of a given light variable (X, L) was given, which is a generalization of the degree and the sectional genus of the light variable.
Abstract: Let (X, L) be a quasi-polarized variety of dim X = n. In this paper we give a new invariant (the i-th sectional geometric genus) of (X, L), which is a generalization of the degree and the sectional genus of (X, L). Furthermore we study some properties of the sectional geometric genus.

Journal ArticleDOI
TL;DR: Aldrich et al. as mentioned in this paper proved the existence of flat covers and cotorsion envelopes for any quasi-coherent sheaf over the projective line, where R is any commutative ring.
Abstract: In this paper we prove the existence of a flat cover and a cotorsion envelope for any quasi-coherent sheaf over the projective line , where R is any commutative ring. We first prove a general result that guarantees the existence of ℱ-covers and ℱ⊥-envelopes in the general setting of a Grothendieck category (not necessarily with enough projectives) provided that the class ℱ satisfies some “standard” conditions. This will generalize some results of the earlier work. [Aldrich, S. T., Enochs, E., Garcia Rozas, J. R., Oyonarte, L. (2001). Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications. J. Algebra 243:615–630].

Journal ArticleDOI
TL;DR: In this paper, the authors studied flat module representations for quivers not containing the path and proved that for every such quiver any representation has a flat cover and a cotorsion envelope.
Abstract: This paper is devoted to the study of flat module representations. We characterize such representations for quivers not containing the path ·• → • (the so called rooted quivers). Then, we prove that for every such quiver any representation has a flat cover and a cotorsion envelope. We finally observe that if Q is one of those quivers and if ℱ denotes the class of all flat representations of Q, then the pair (ℱ, ℱ⊥) is a cotorsion theory.

Journal ArticleDOI
TL;DR: The notions of DoiHopf group module and group twisted smash product are defined as a respective generalization of an ordinary Doi-Hopf module and a usual twisted product as mentioned in this paper.
Abstract: The notions of Doi–Hopf group module and group twisted smash product are defined as a respective generalization of an ordinary Doi–Hopf module and a usual twisted smash product. It is shown that the Zunino's Yetter-Drinfel'd modules are special cases as these new Doi–Hopf group modules and that the Zunino's Drinfel'd double appears as a type of such a group twisted smash product, respectively. Furthermore, the concepts of group skew pair and generalized group smash product are introduced. As for a group skew pair σ for a T- coalgebra H and a T-algebra B, it is proved that a group twisted smash product is a Hopf group coalgebra if H has a bijective antipode, and that is a special case of such a generalized group smash product.

Journal ArticleDOI
TL;DR: In this article, the notion of ℋ-subgroups was introduced and supersolvability of finite groups was shown to exist for cyclic subgroups of G of prime order or of order 4.
Abstract: A finite group G is called G a 𝒯-group if each subnormal subgroup of G is normal in G and a subgroup K of G is called an ℋ-subgroup of G if N G (K) ∩ K g ⊆ K for all g ∈ G. Using the notion of ℋ-subgroups, we present some new conditions for supersolvability and we characterize supersolvable groups, which are either 𝒯-groups or A-groups (i.e., all their Sylow subgroups are abelian). For example, we prove that if all cyclic subgroups of G of prime order or of order 4 are ℋ- subgroups of G, then G is supersolvable with a well defined structure. We also show, that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic ℋ-subgroups of G.

Journal ArticleDOI
TL;DR: In this paper, a formal deformation functor of A from the category of artinian local dg algebras to simplicial sets is proposed, which is a generalization of the classical deformationfunctor for an algebra over a linear operad.
Abstract: Let k be a field of characteristic zero, 𝒪 be a dg operad over k and let A be an 𝒪-algebra. In this note we suggest a definition of a formal deformation functor of A from the category of artinian local dg algebras to the category of simplicial sets. This functor generalizes the classical deformation functor for an algebra over a linear operad. In the case 𝒪 and A are non-positively graded, we prove that Def A is governed by the tangent Lie algebra T A which can be calculated as the Lie algebra of derivations of a cofibrant resolution of A. An example shows that the result does not necessarily hold without the non-positivity condition.

Journal ArticleDOI
TL;DR: Bounds for the diameter of commuting involution graphs of special linear groups over fields of characteristic 2 are given in this article, where the disc sizes of these graphs are shown to be unbounded.
Abstract: Bounds are given for the diameter of commuting involution graphs of special linear groups over fields of characteristic 2. For 2- and 3-dimensional special linear groups over any finite field the disc sizes are determined. Examples are given of commuting involution graphs which have unbounded diameter.

Journal ArticleDOI
TL;DR: In this article, the notions of semistar linkedness and semistar flatness are introduced and studied for semistar multiplication domains. But the authors focus on the semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prufer domains.
Abstract: In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the “classical” concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural generalizations, to the semistar setting, of their corresponding “classical” concepts. As an application, among other results, we obtain a semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prufer domains for Prufer semistar multiplication domains.

Journal ArticleDOI
TL;DR: In this paper, the irregularity of the image of the Iitaka fibration in terms of the dimension of certain cohomological support loci was characterized in the image.
Abstract: We characterize the irregularity of the image of the Iitaka fibration in terms of the dimension of certain cohomological support loci.

Journal ArticleDOI
Dexu Zhou1
TL;DR: In this paper, the concepts of (n, d)-injective and (n-d)-flat modules and homological dimensions were introduced and used to characterize right n-coherent rings and right (weak) n-d-rings in various way.
Abstract: We introduce the concepts of (n, d)-injective and (n, d)-flat as generalizations of injective, flat modules and homological dimensions, and use them to characterize right n-coherent rings and right (weak) (n, d)-rings in various way. Some known results can be obtained as corollaries.

Journal ArticleDOI
TL;DR: For the vector product n-Lie algebra with commutator, this article showed that any finite-dimensional irreducible n-lie V n -module is isomorphic to an n--Lie extension of s o n+1-module with highest weight tπ1 for some nonnegative integer t.
Abstract: Let V n = ⟨e 1,…, e n+1⟩ be the vector product n-Lie algebra with n-Lie commutator [e 1,…, ˆe i ,…, e n+1] = (− 1) i e i over the field of complex numbers. Any finite-dimensional n-Lie V n -module is completely reducible. Any finite- dimensional irreducible n-Lie V n -module is isomorphic to an n-Lie extension of s o n+1-module with highest weight tπ1 for some nonnegative integer t.

Journal ArticleDOI
TL;DR: In this paper, the notions of group coalgebra Galois extension and group entwining structure are defined, and it is proved that any coalgebra coalgebra with a unique group-entwining map ψ = ψα, β, α, β∈π compatible with the right group coaction is a coalgebra of Galois type.
Abstract: The notions of group coalgebra Galois extension and group entwining structure are defined. It is proved that any group coalgebra Galois extension induces a unique group-entwining map ψ = {ψα, β}α, β∈π compatible with the right group coaction, generalizing the recent work of Brzezinski and Hajac [Brzezinski, T., Hajac, P. M. (1999). Coalgebra extensions and algebra coextensions of Galois type. Comm. Algebra 27:1347–1368].

Journal ArticleDOI
TL;DR: In this article, it was shown that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity.
Abstract: We show that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated. Information depending on the Classification of Finite Simple Groups is used throughout.

Journal ArticleDOI
TL;DR: In this paper, the Lie superalgebra of superderivations of H type Lie super algebras is obtained, and the relation between skew superderivation space SkDer(L) and central extension H 2(L, F) on some Lie super algebra L is discussed.
Abstract: In this paper, we obtain the Lie superalgebra of superderivations of H type Lie superalgebra and discuss the relation between skew superderivations space SkDer(L) and central extension H 2(L, F) on some Lie superalgebra L. As an application, we obtain the dimension of central extension of Lie superalgebras with a nondegenerate associative forms.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the fixed subgroup of an arbitrary family of endomorphisms ψ i, i ∈ ǫ of a finitely generated free group F is F-supercompressed.
Abstract: In this paper, we prove that the fixed subgroup of an arbitrary family of endomorphisms ψ i , i ∈ I, of a finitely generated free group F, is F-super-compressed. In particular, r(∩ i∈I Fix ψ i ) ≤ r(M) for every subgroup M ≤ F containing ∩ i∈I Fix ψ i . This provides new evidence towards the inertia conjecture for fixed subgroups of free groups. As a corollary, we show that, in the free group of rank n, every strictly ascending chain of fixed subgroups has length at most 2n. This answers a question of Levitt.

Journal ArticleDOI
TL;DR: In this article, the authors show that faithful multiplication and projective modules have common properties and use a method based on Anderson's characterizations of multiplication modules which enables new proofs of theorems on multiplication modules of El Bast, Low, Smith and Smith.
Abstract: The associated ideal θ(M) and the trace ideal T(M) of a module M play analogous but distinct roles in the study of multiplication and projective modules respectively. We further investigate both and show in particular that faithful multiplication and projective modules have common properties. We use a method based on Anderson's characterizations of multiplication modules which enables new proofs of theorems on multiplication modules of El Bast, Low, Smith and Smith, and then, also enables similar results for projective modules.

Journal ArticleDOI
TL;DR: The extended centroid of associative superalgebras is studied in this article, where the superalgebra versions of some known theorems on derivations and functional identities in associative prime rings are obtained.
Abstract: The extended centroid of prime associative superalgebras is studied As applications we obtain the superalgebra versions of some known theorems on derivations and functional identities in associative prime rings

Journal ArticleDOI
TL;DR: Kostrikin and Shafarevich as mentioned in this paper showed that the relation between the Hom-functor and the Ext-functors of an artin algebra over a local commutative artinian ring k is equivalent to a relation between almost split sequences in p(Λ) and its relation with the almost split sequence for left Λ-modules.
Abstract: Let Λ be an artin algebra over a local commutative artinian ring k. We consider the category P(Λ) whose objects are morphims f : P → Q with P and Q projective left Λ-modules. In P(Λ), we introduce an exact structure in the sense of Gabriel and Roiter [Gabriel, P., Roiter, A. V. (1992). Representations of finite-dimensional algebras. In: Kostrikin, A. I., Shafarevich, I. V. eds. Encyclopaedia of the Mathematical Sciences. Vol. 73. Springer. Algebra VIII.] or equivalently Quillen [Quillen, D. (1973). Higher Algebraic K-Theory, 1. SLNM 341. Berlin: Springer, pp. 85–147], then we describe the corresponding projectives and injectives. For p(Λ), the full subcategory whose objects are morphisms f : P → Q with P and Q finitely generated, we prove a relation between the Hom-functor and the Ext-functor. From here, we can prove the existence of almost split sequences in p(Λ) and its relation with the almost split sequences for left Λ-modules.

Journal ArticleDOI
TL;DR: In this paper, it was shown that G is a Dedekind group if d = 1, where d is the number of defects in the subgroups of G. This extends a famous theorem of Roseblade and holds for groups G that belong to the class 𝔛 introduced by Cernikov.
Abstract: A special case of the main result is as follows: Let G be a locally (soluble-by-finite) group of infinite rank in which every subgroup of infinite rank is subnormal of defect at most d. Then G is nilpotent of class bounded in terms of d only. This extends a famous theorem of Roseblade and holds, more generally, for groups G that belong to the class 𝔛 introduced by Cernikov. It is also shown that G is a Dedekind group if d = 1.