Showing papers in "Communications in Algebra in 2003"
TL;DR: In this article, the authors studied the structure of the stacks of twisted stable maps to the classifying stack of a finite group G, which they called the stack of twisted G-covers, or twisted g-bundles.
Abstract: We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.
365 citations
TL;DR: For a commutative ring R with identity, the zero-divisor graph of R, denoted Γ(R), is the graph whose vertices are the non-zero zero divisors of R with two distinct vertices joined by an edge when the product of the vertices is zero as discussed by the authors.
Abstract: For a commutative ring R with identity, the zero-divisor graph of R, denoted Γ(R), is the graph whose vertices are the non-zero zero-divisors of R with two distinct vertices joined by an edge when the product of the vertices is zero. We will generalize this notion by replacing elements whose product is zero with elements whose product lies in some ideal I of R. Also, we determine (up to isomorphism) all rings R such that Γ(R) is the graph on five vertices.
171 citations
TL;DR: For a ring endomorphism α, the authors introduced α-skew Armendariz rings which are a generalization of α-rigid rings and Armenderiz rings, and investigated their properties.
Abstract: For a ring endomorphism α, we introduce α-skew Armendariz rings which are a generalization of α-rigid rings and Armendariz rings, and investigate their properties. Moreover, we study on the relationship between the Baerness and p.p.-property of a ring R and these of the skew polynomial ring R[x; α] in case R is α-skew Armendariz.
142 citations
TL;DR: In this paper, Anderson, D. D., Papick, I. J., and Fontana, M., Loper, K. A. (2001a) extended the notion of star operation and generalized Kang's notion of a star Nagata r...
Abstract: In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book (Gilmer, R. (1972). Multiplicative Ideal Theory. New York: Marcel Dekker) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prufer and P. Lorenzen from 1930's. Fontana and Loper investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain D (Fontana M., Loper K. A. (2001a). Kronecker function rings: a general approach. In Anderson, D. D., Papick, I. J., eds. Ideal Theoretic Methods in Commutative Algebra. Lecture Notes Pure Appl. Math. 220, Marcel Dekker, pp. 189–205 and Fontana, M., Loper, K. A. (2001b). A Krull-type theorem for the semistar integral closure of an integral domain. ASJE Theme Issue “Commutative Algebra” 26:89–95). In this paper we extend that study and also generalize Kang's notion of a star Nagata r...
126 citations
TL;DR: In this paper, the Quot functor Quot(ℱ/𝒳/S) is represented by a separated and locally finitely-presented algebraic space over S.
Abstract: Given a separated and locally finitely-presented Deligne-Mumford stack 𝒳 over an algebraic space S, and a locally finitely-presented 𝒪𝒳-module ℱ, we prove that the Quot functor Quot(ℱ/𝒳/S) is represented by a separated and locally finitely-presented algebraic space over S Under additional hypotheses, we prove that the connected components of Quot(ℱ/𝒳/S) are quasi-projective over S Dedicated to Steven L Kleiman on the occasion of his 60th birthday
122 citations
TL;DR: In this paper, a subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and HN∩N≤H G ǫ = core(H).
Abstract: A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are: Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ℱ. Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ℱ.
91 citations
TL;DR: In this paper, it was shown that the representation category of the quantum group of a non-degenerate bilinear form is monoidally equivalent to that of the classical quantum group SL q (2) for a well chosen non-zero parameter q. The key ingredient for the proof is the direct and explicit construction of an appropriate Hopf bigalois extension.
Abstract: We show that the representation category of the quantum group of a non-degenerate bilinear form is monoidally equivalent to the representation category of the quantum group SL q (2) for a well chosen non-zero parameter q. The key ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, when the base field is of characteristic zero, a full description of cosemisimple Hopf algebras whose representation semi-ring is isomorphic to the one of SL(2).
70 citations
TL;DR: In this article, the main purpose of the paper is to determine finite groups satisfying Γ(G) = ǫ(S) and to give applications which generalize Abe (Abe, S. Preprint) and Chen (Chen, G.3:49-58).
Abstract: Let Gbe a finite group and Sa sporadic simple group. We denote by π(G) the set of all primes dividing the order of G. The prime graph Γ(G) of Gis defined in the usual way connecting pand qin π(G) when there is an element of order pqin G. The main purpose of this paper is to determine finite group Gsatisfying Γ(G) = Γ(S) (See Theorem 3) and to give applications which generalize Abe (Abe, S. Two ways to characterize 26 sporadic finite simple groups. Preprint) and Chen (Chen, G. (1996). A new characterization of sporadic simple groups. Algebra Colloq.3:49–58). The results are elementary but quite useful.
64 citations
TL;DR: In this paper, it was shown that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup.
Abstract: We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the c ...
60 citations
TL;DR: Anderson and Camillo as mentioned in this paper gave an example of a commutative ring which is a pm-ring yet not clean, e.g., C(ℝ).
Abstract: An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently,Anderson and Camillo (Anderson,D. D.,Camillo,V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover,every clean ring is a pm-ring,that is every prime ideal is contained in a unique maximal ideal. In the same article,the authors give an example of a commutative ring which is a pm-ring yet not clean,e.g.,C(ℝ). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1.
59 citations
TL;DR: In this article, the authors studied the birational action of S n ×S m on the space of matrices defined by Braverman and Kazhdan using geometric crystals, from the viewpoint of the theory of set-theoretical solutions of the quantum Yang-Baxter equations.
Abstract: We study the birational action of S n ×S m on the space of matrices defined by Braverman and Kazhdan using geometric crystals, from the viewpoint of the theory of set-theoretical solutions of the quantum Yang-Baxter equations. In particular, we obtain a direct proof of the well-definedness of this action, which does not use the theory of geometric crystals.
TL;DR: In this article, the authors define a stratifying system for special biserial self-injective algebras and show that this produces a module Y, whose endomorphism ring A is standardly stratified.
Abstract: Let A be a finite dimensional algebra over an algebraically closed field k. For any fixed partial ordering of an index set,Λ say,labelling the simple A-modules L(i),there are standard modules,denoted by Δ(i),i ∈ Λ. By definition,Δ(i) is the largest quotient of the projective cover of L(i) having composition factors L(j) with j ≤ i. Denote by ℱ(Δ) the category of A-modules which have a filtration whose quotients are isomorphic to standard modules. The algebra A is said to be standardly stratified if all projective A-modules belong to ℱ(Δ). In this paper we define a “stratifying system” and we show that this produces a module Y,whose endomorphism ring A is standardly stratified. In particular,we construct stratifying systems for special biserial self-injective algebras.
TL;DR: In this article, the authors introduce a new class of rings called Nonnil-Noetherian rings, which are closely related to the class of Noetherian ring and show that many of the properties of nonnil-noetherians are also true for nonnil noetherians.
Abstract: Let R be a commutative ring with 1 such that Nil(R) is a divided prime ideal of R. The purpose of this paper is to introduce a new class of rings that is closely related to the class of Noetherian rings. A ring R is called a Nonnil-Noetherian ring if every nonnil ideal of R is finitely generated. We show that many of the properties of Noetherian rings are also true for Nonnil-Noetherian rings; we use the idealization construction to give examples of Nonnil-Noetherian rings that are not Noetherian rings; we show that for each n ≥ 1, there is a Nonnil-Noetherian ring with Krull dimension n which is not a Noetherian ring.
TL;DR: In this paper, the existence of minimal level artinian graded algebras having socle degree r and type t and their h-vector in terms of the r-binomial expansion of t were proved.
Abstract: In this paper we prove the existence of minimal level artinian graded algebras having socle degree r and type t and describe their h-vector in terms of the r-binomial expansion of t. We also investigate the graded Betti numbers of such algebras and completely describe their extremal resolutions. We also show that any set of points in ℙ n whose Hilbert function has first difference as described above, must satisfy the Cayley-Bacharach property.
TL;DR: In this article, it was shown that a monomial ideal in a polynomial ring in nindeterminates over a field is normal if and only if the first n−− 1 positive powers of the ideal are integrally closed.
Abstract: In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an ℕ-graded ring Aof the form A ≥m ≔ ⊕l≥m A land monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in nindeterminates over a field is normal if and only if the first n − 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I( λ ) ≔ , where J( λ ) = ( ,…, ) ⊆ K[x 1,…, x n ]. To state our main result in this setting, we let l = lcm(λ1,…, ,…λ n ), for 1 ≤ i ≤ n, and set λ ′ = (λ1,…, λ i−1, λ i + l, λ i+1,…, λ n ). We prove that if I( λ ′) is normal then I( λ ) is normal and that the converse holds with a small additional assumption.
TL;DR: In this paper, an axiomatic concept of Kronecker function rings was introduced and applied to algebraic field extensions and its connection with the defining valuation domains, and the authors investigated its behavior in algebraic fields.
Abstract: We provide an axiomatic concept of Kronecker function rings and apply it to associate a Kronecker function ring to any integral domain D and any ideal system (star operation) on D. We investigate its behavior in algebraic field extensions and its connection with the defining valuation domains.
TL;DR: For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies as discussed by the authors.
Abstract: For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.
TL;DR: In this paper, the local quiver is used as a tool to investigate the etale local structure of moduli spaces of θ-semistable representations of quivers.
Abstract: In this paper we introduce and study the local quiver as a tool to investigate the etale local structure of moduli spaces of θ-semistable representations of quivers. As an application we determine the dimension vectors associated to irreducible representations of the torus knot groups G p,q = ⟨a, b ∣ a p = b q ⟩.
TL;DR: In this article, the existence of n-ary systems playing the role of enveloping algebras for Filippov (n-Lie) algeses is investigated.
Abstract: The question of existence of n-ary systems playing the role of enveloping algebras for Filippov (n-Lie) algebras is investigated. We consider the ternary algebras whose reduced binary algebras are associative, the associative algebras, and two other classes of ternary algebras as possible enveloping algebras.
TL;DR: In this article, the authors lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings and apply these results mainly to categories of entwined modules.
Abstract: The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gomez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A → C*. Such pairings, satisfying the so called α-condition, were called in the author's dissertation measuring α-pairings and can be considered as the corner stone in his study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g., Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Doi, Koppinen, Menini et al.
TL;DR: In this article, it was shown that X is a P-space if and only if C F (X) is a ℵ0-selfinjective ring or equivalently, if X is selfinherent.
Abstract: Let C F (X) denote the socle of C(X). It is shown that X is a P-space if and only if C(X) is a ℵ0-selfinjective ring or equivalently, if and only if is ℵ0-selfinjective. We also prove that X is an extremally disconnected P-space with only a finite number of isolated points if and only if is selfinjective. Consequently, if X is a P-space, then X is either an extremally disconnected space with at most a countable number of isolated points or both C(X) and have uncountable Goldie-dimensions. Prime ideals of are also studied.
TL;DR: In this article, an associative algebra A k is constructed and shown to be the full centralizer of the Lie superalgebra on V ⊗k, where V is the natural 2n-dimensional representation.
Abstract: We construct an associative algebra A k and show that there is a representation of A k on V ⊗k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A k is the full centralizer of 𝔭(n) on V ⊗k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A k we decompose the tensor space V ⊗k , for k = 2 or 3, and show that V ⊗k is not completely reducible for any k ≥ 2.
TL;DR: In this article, it was shown that a coalgebra is co-Frobenius if and only if C generates every left and every right C-comodule in the coalgebra.
Abstract: We prove new characterizations of Quasi-co-Frobenius (QcF) coalgebras and co-Frobenius coalgebras. Among them, we prove that a coalgebra is QcF if and only if C generates every left and every right C-comodule. We also prove that every QcF coalgebra is Morita-Takeuchi equivalent to a co-Frobenius coalgebra.
TL;DR: In this article, a basis for the ℤ n ǫℤ2-graded identities of the algebras M n (E) and M 2(E) was given, and it was shown that M n is PI-equivalent to M 1,1(E)-⊗E. This fact is a particular case of a general result obtained by Kemer.
Abstract: Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ⊗ E, then deduce its G × ℤ2-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ℤ n × ℤ2-graded identities of the algebras M n (E). Moreover we give the graded cocharacter sequence of M 2(E), and show that M 2(E) is PI-equivalent to M 1,1(E) ⊗ E. This fact is a particular case of a more general result obtained by Kemer.
TL;DR: In this paper, it was shown that a submodule of a module M over a commutative ring R such that M/N is finitely generated is a prime submodule minimal over N if and only if it is the saturation of N+pM for certain prime ideal p of R.
Abstract: Let N be a submodule of a module M over a commutative ring R such that M/N is finitely generated. It is shown that a submodule of M is a prime submodule minimal over N if and only if it is the saturation of N + pM for certain prime ideal p of R. The bearing of this result upon the M-radical of N is discussed.
TL;DR: Altschuler and Coste as mentioned in this paper showed that for a quasitriangular quasi-Hopf algebra with an R-matrix R, this condition is unnecessary and also the condition of invertibility of R.
Abstract: Following (Drinfeld, V. G. (1990a). Quasi-Hopf algebras.Leningrad Math. J. 1:1419–1457) a quasi-Hopf algebra has, by definition, its antipode bijective. In this note, we will prove that for a quasitriangular quasi-Hopf algebra with an R-matrix R, this condition is unnecessary and also the condition of invertibility of R. Finally, we will give a characterization for a ribbon quasi-Hopf algebra. This characterization has already been given in Altschuler and Coste (Altschuler, D., Coste, A. (1992). Quasi-quantum groups, knots, three-manifolds and topological field theory. Comm. Math. Phys. 150:83–107.), but with an additional condition. We will prove that this condition is unnecessary.
TL;DR: In this article, the authors investigated the additivity of maps φ from 𝒜 onto and#x 1d 49c;′ that are bijective and satisfy for all a, b ∈ and&#x 49c′, and concluded that φ is additive.
Abstract: Let 𝒜 and 𝒜′ are associative algebras over the field ℚ of rational numbers and k ∈ ℚ. In this paper, we investigate the additivity of maps φ from 𝒜 onto 𝒜′ that are bijective and satisfy for all a, b ∈ 𝒜. If 𝒜 is a unital prime algebra containing a non-trivial idempotent, or 𝒜 is a unital algebra which has a system of matrix units, or 𝒜 is a standard operator algebra on a Banach space, then we can conclude that φ is additive.
TL;DR: In this article, the authors give a description of the indecomposable objects in the derived category of a finite-dimensional skewedgentle algebra, which is a generalization of the notion of skewed-gentle algebras.
Abstract: We give a description of the indecomposable objects in the derived category of a finite-dimensional skewed-gentle algebra.
TL;DR: In this article, the authors give some classes of modules where the two concepts π and prime are equivalent and provide conditions under which a given ring R is a Dedekind domain if and only if the R-module M is prime.
Abstract: In the first section of this paper the authors give some classes of modules where the two concepts π and prime are equivalent. Also, they provide conditions under which a given ring R is a Dedekind domain if and only if the R-module M is prime. In the final section they answer a question concerning modules which satisfy the radical formula.
TL;DR: In this article, the Tits-Kantor-Koecher superalgebras associated to Jordan superpairs covered by grids were described and extended to the supercase.
Abstract: In this paper we describe the Tits-Kantor-Koecher superalgebras associated to Jordan superpairs covered by grids,extending results from Neher (Neher,E. (1996). Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid. Amer. J. Math. 118; 439–491) to the supercase. These Lie superalgebras together with their central coverings are precisely the Lie superalgebras graded by a 3-graded root system.