Showing papers in "Communications in Algebra in 1999"
TL;DR: In this paper, it was shown that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0.
Abstract: Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called a generalized derivation if there exists a derivation δ of ρ into U such that for all x,y∈ρ. In this note, we prove that there exists an element a∈ U such that for all x ∈ ρ. From this characterization, it is proved that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0. Analogous results are also obtained for the case of generalized derivations with nilpotent values on Lie ideals or one-sided ideals.
286 citations
TL;DR: In this paper, a natural generalization of strongly π-regular rings, called strongly clean regular rings, is presented. But their relationship to Fitting's lemma is not discussed.
Abstract: A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. These rings are shown to be a natural generalization of the strongly π-regular rings, and several properties of strongly π-regular rings are extended, including their relationship to Fitting's lemma.
190 citations
TL;DR: In this article, the authors considered the problem of finding the map π from the partial informations provided by the map and the factorization of the map into a product (i.e., composition) of tame automorphisms O i's.
Abstract: Let K be a finite field of 2m elements. Let O4O3O2O1be tame automorphisms of the n + r-dimensional affine space Kn+r . Let the composition O4O3O2O1 be π. The automorphism π and some of the O i ’S will be hidden. Let the component expression of π be . Let the restriction of π to a subspace be as . The field K and the polynomial map will be announced as the public key. Given a plaintext , then the ciphertext will be . Given O i and , it is easy to find . Therefore the plaintext can be recovered by . The private key will be the set of maps O4O3O2O1. The security of the system rests in part on the difficulty of finding the map π from the partial informations provided by the map and the factorization of the map π into a product (i.e., composition) of tame automorphisms O i 's.
156 citations
TL;DR: In this paper, a semigroup with zero 0 and n ≥ 2 is shown to satisfy ZCn for a fixed n ≥ 3, and a ring R with identity which satisfies ZC2 but does not satisfy Zc3.
Abstract: Let S be a semigroup with zero 0 and let n ≥ 2. We say that S satisfies for each permutation σ ∈ S n A ring R satisfies ZCn if (.R, .) satisfies ZCn. We show that if S satisfies ZCn for a fixed n ≥ 3, then S also satisfies ZCn+1, but we give an example of a ring R with identity which satisfies ZC2 but does not satisfy ZC3 We show that a semigroup with no nonzero nilpotents satisfiesZCn for all n ≥ 2 and investigate rings that satisfy ZCn.
106 citations
TL;DR: In this paper, Fintte two-arc transitive graphs admitting a suzuki simple group admit a simple group in the form of a group-simple group (SGP).
Abstract: (1999). Fintte two-arc transitive graphs admitting a suzuki simple group. Communications in Algebra: Vol. 27, No. 8, pp. 3727-3754.
101 citations
TL;DR: The notion of coalgebra-Galois coextensions was introduced in this article as a natural generalisation of a Hopf Galois extension, and it is shown that any coalgebra G-extension induces a unique entwining map ψ compatible with the right coaction.
Abstract: The notion of a coalgebra-Galois extension is defined as a natural generalisation of a Hopf-Galois extension. It is shown that any coalgebra-Galois extension induces a unique entwining map ψ compatible with the right coaction. For the dual notion of an algebra-Galois coextension it is also proven that there always exists a unique entwining structure compatible with the right action.
101 citations
TL;DR: In this article, the authors extended the theory of actions of Hopf algebras to actions of multiplier Hopf algebra and showed a duality theorem for the smash product.
Abstract: For an action a of a group G on an algebra R (over C), the crossed product Rxα G is the vector space of R-valued functions with finite support in G, together with the twisted convolution product given by where p∈G This construction has been extended to the theory of Hopf algebras Given an action of a Hopf algebra A on an algebra R, it is possible to make the tensor productR⊗A into an algebra by using a twisted product, involving the action In this case, the algebra is called the smash product and denoted by R#A In the group case, the action a of G on R yields an action of the group algebra CG as a Hopf algebra on R and the crossed Rxα G coincides with the smash product R#CG In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras The main result in the paper is a duality theorem for such actions We consider dual pairs of multiplier Hopf
86 citations
TL;DR: In this paper, Poisson enveloping algebras have been studied in the context of algebraic geometry and they have been shown to be polynomially enveloping.
Abstract: (1999). Poisson enveloping algebras. Communications in Algebra: Vol. 27, No. 5, pp. 2181-2186.
83 citations
TL;DR: In this article, the authors carried out a systematic study of various ring theoretic properties of triangular matrix rings and proved that being strong left Kasch or strong right mininjective are Morita invariant properties.
Abstract: In this paper we carry out a systematic study of various ring theoretic properties of formal triangular matrix rings. Some definitive results are obtained on these rings concerning properties such as being respectively left Kasch, right mininjective, clean, potent, right PF or a ring of stable rank ≤ n. The concepts of a strong left Kasch ring, a strong right mininjective ring are introduced and it is determined when the triangular matrix rings are respectively strong left Kasch or strong right mininjective. It is also proved that being strong left Kasch or strong right mininjective are Morita invariant properties.
74 citations
TL;DR: In this article, the associated primes of local cohomology modules are studied and the authors show that the primes can be computed in polynomial time with respect to local cohoms.
Abstract: (1999). On the associated primes of local cohomology modules. Communications in Algebra: Vol. 27, No. 12, pp. 6191-6198.
71 citations
TL;DR: The notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings as mentioned in this paper.
Abstract: It is well-known that a ring Ris semiperfect if and only if RR (orRR ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (orRR ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).
TL;DR: In this paper, it was shown that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1, and if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided then it is maximal and R is quasilocal.
Abstract: Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a nonprincipal divided prime, then P-1 = { x ∊ T : xP ⊃ P} is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1. Also, we show that if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided, then it is maximal and R is quasilocal.
TL;DR: The notion of double extension to quadratic Lie superalgebras was introduced by Medina and Revoy [8] as mentioned in this paper, who gave a sufficient condition for a non-degenerate super-symmetric, consistent g-invariant bilinear form B to be a double extension.
Abstract: In this paper we study Lie superalgebras g with a non-degenerate super-symmetric, consistentg-invariant bilinear form B. Such a (g,B) is called quadratic Lie superalgebra. Our first result generalizes the notion of double extension to quadratic Lie superalgebras. This notion was introduced by Medina and Revoy [8] to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Lie superalgebra to be a double extension. Any nonsimple quadratic Lie superalgebra such that dim is a double extension; also we give an inductive classification of this class of quadratic Lie superalgebras.
TL;DR: In this article, the Symplectic ideals of poisson algebras and the poisson structure associated to quantum matrices are discussed, and a poisson algebraic model is proposed.
Abstract: (1999). Symplectic ideals of poisson algebras and the poisson structure associated to quantum matrices. Communications in Algebra: Vol. 27, No. 5, pp. 2163-2180.
TL;DR: In this paper, the schur multiplier of p -groups is studied. But the authors focus on the Schur multiplier in the context of p-groups and do not consider the Schulman multiplier in general.
Abstract: (1999). On the schur multiplier of p -groups. Communications in Algebra: Vol. 27, No. 9, pp. 4173-4177.
TL;DR: In this paper, it is proved that each integral domain can be embedded as a subring of some antimatter domain which is not a field, and a detailed study is made of the passage of the an-timatter property between the partners within an overring extension.
Abstract: Antimatter domains are defined to be the integral domains which do not have any atoms. It is proved that each integral domain can be em-bedded as a subring of some antimatter domain which is not a field. Any fragmented domain is an antimatter domain, but the converse fails in each positive Krull dimension. A detailed study is made of the passage of the“an-timatter”property between the partners within an overring extension. Special attention is given to characterizing antimatter domains in classes of valuation domains, pseudo-valuation domains, and various types of pullbacks.
TL;DR: In this paper, a systematic method to calculate cleft extensions for pointed Hopf algebras is developed and applied to Uq(sl 2) and the Frobenius-Lusztig kernel Uq (sl 2).
Abstract: A systematic method to calculate cleft extensions for pointed Hopf algebras is developed and applied to Uq(sl 2) and the Frobenius-Lusztig kernel Uq(sl 2)'.
TL;DR: In this article, it was shown that a ring; R is strongly regular iff R is a weakly right duo ring whose simple singular right R-modules are GP-injective.
Abstract: We investigate von Neumann regularity of rings whose simple singular right R-modules are GP-injective. It is proved that a ring; R is strongly regular iff R is a weakly right duo ring whose simple ...
TL;DR: In this paper, the Jacobson radical, the socle and the singular submodule of a module M were defined, respectively, and the notation N ⊆ess M was used to indicate that N is an essential submodule.
Abstract: An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ⊆ess M means that N is an essential submodule of M.
TL;DR: In this article, the Wedderburn-Malcev theorem holds for a large class of Hopf algebras, such as coordinate rings of completely reducible affine algebraic groups.
Abstract: Let H be a Hopf algebra over a field k. Under some assumptions on H we state and prove a generalization of the Wedderburn-Malcev theorem for i7-comodule algebras. We show that our version of this theorem holds for a large enough class of Hopf algebras, such as coordinate rings of completely reducible affine algebraic groups, finite dimensional Hopf algebras over fields of characteristic 0 and group algebras. Some dual results are also included.
TL;DR: In this paper, the authors classify diagonal locally simple Lie algebras of countable dimension over an algebraically closed field of zero characteristic, and give some remarks on classification of locally simple associative algesbras.
Abstract: The aim of this article is to classify diagonal locally simple Lie algebras of countable dimension over an algebraically closed field of zero characteristic. We also give some remarks on classification of locally simple associative algebras. Recall that an algebra A is called locally finite if any finite subset of A is contained in a finite-dimensional subalgebra. If these subalgebras can be chosen simple, A is called locally simple. Observe that A is simple in this case. Let F be an algebraically closed field of zero characteristic, A be a locally simple associative algebra of countable dimension over F . It follows from the definition that there is an increasing sequence of simple subalgebras M1 ⊂ M2 ⊂M3 ⊂ . . . of A such that A = ∪i=1Mi. It is more convenient to write M1 →M2 →M3 → . . . (1)
TL;DR: In this article, it was shown that any simple simple Lie algebra with root system ▵ is graded by ▵, and that any classical double D (g) is graded with ▵.
Abstract: Let g be a complex simple Lie algebra with root system ▵. We prove that any classical double D (g) is graded by ▵.As a consequence of this fact we obtain that D (g)≅g⊗A, where A is a unital com-muative associative algebra of dimension 2. Therefore we have two possibilities for A nilpotent and semisimple. The first case leads to solutions of CYBE and the second case leads to solutions of mCYBE. We obtain an explicit description of Lie bialgebra structures on g in both cases.
TL;DR: In this article, the authors consider the case where the quadric has rank 3 and give sufficient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be the graph of an automorphism.
Abstract: We continue the classification, begun in [11], [14] and [12], of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurfcice in P3. In this paper, we consider those cases where the quadric has rank 3. We also give sufficient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be the graph of an automorphism.
TL;DR: In this article, the authors construct parametrized Yang-Baxter operators from algebra struc-tures, transfer the theory to coalgebras, and find the cases where such operators arising from (co) algesbras are isomorphic.
Abstract: We construct parametrized Yang-Baxter operators from algebra struc-tures, transfer the theory to coalgebras, and find the cases where such operators arising from (co) algebras are isomorphic. We give examples in dimension 2.
TL;DR: In this paper, it was shown that for every finite group G, if R(G) is not trivial, then the normalizer property holds for G. In this paper, we extend the result of the authors.
Abstract: Let R(G) denote the intersection of all nonnormal subgroups of a group G. In this note, we prove that for every finite group G, if R(G) is not trivial, then the normalizer property holds forG.
TL;DR: In this paper, rings described by various purities are described in terms of different purities and purities of rings in the context of algebraic ring descriptions in algebraic geometry.
Abstract: (1999). Rings described by various purities. Communications in Algebra: Vol. 27, No. 5, pp. 2127-2162.
TL;DR: In this article, the authors investigated general principally injective rings satisfying additional conditions, and various results were developed, many extending known results, and the aim of this paper is to investigate the general injective ring satisfying these additional conditions.
Abstract: The aim of this paper is to investigate general principally injective rings satisfying additional conditions. Various results are developed, many extending known results.
TL;DR: In this article, the authors used Lusztig's description of the degrees of irreducible characters of reductive groups and the determination of Brauer trees by Fong and Srinivasan to handle the case of Lie types.
Abstract: Let G be a covering group of a finite almost simple group. We determine those faithful irreducible complex characters x of G for which x ⊗ x ∗ - 1 is again irreducible. This gives a classification of the quasi-simple absolutely irreducible subgroups of GLn (q) of order prime to q which act irreducibly on the Lie algebra of type An -1 via the adjoint representation. The proof uses Lusztig’s description of the degrees of irreducible characters of reductive groups and the determination of Brauer trees by Fong and Srinivasan to handle the case of groups of Lie type. It turns out that the only infinite series of examples are characters of Weyl representations for SU n (Fn) and Sp2n (F3).
TL;DR: In this article, the Frobenius functors are characterized for categories of modules and categories of comodules, and they have been applied to coalgebras and Hopf modules.
Abstract: We investigate functors between abelian categories having usomor-phic left and right adjoint functors (these functors are called Frobenius Functors). They are characterized for categories of modules and categories of comodules. We give some applications in coalgebras and Hopf modules. In particular, we introduce the notion of Frobenius homomorphism of coalgebras. The set of isomorphism classes of Frobenius functors between quite general Grothendieck categories is endowed with an abelian group structure. This gives a functorial notion of Grothendieck group which behaves satisfactorily.