Showing papers in "Communications in Algebra in 1997"
TL;DR: In this article, the authors propose a homological algebra of homotopy algebras, which is a generalization of homology of homophily of homologies.
Abstract: (1997). Homological algebra of homotopy algebras. Communications in Algebra: Vol. 25, No. 10, pp. 3291-3323.
430 citations
167 citations
TL;DR: In this paper, the spectrum of a module over a commutative ring has been studied in the context of algebraic communication in algebraic networks, where the spectrum is defined as
Abstract: (1997). On the spectrum of a module over a commutative ring. Communications in Algebra: Vol. 25, No. 1, pp. 79-103.
131 citations
TL;DR: The ring of Hurwitz series as mentioned in this paper is a ring with identity over a commu- tative ring, and its structure and applications in differential algebra have been examined in detail.
Abstract: This paper introduces the ring of Hurwitz series over a commu- tative ring with identity, and examines its structure and applications, especially to the study of differential algebra. In particular, we see that rings of Hurwitz series bear a resemblance to rings of formal power se- ries, and that for rings of positive characteristic, the structure of the ring of Hurwitz series closely mirrors that of the ground ring.
98 citations
TL;DR: For a homogeneous ideal I of S = A[Xo,...,Xn] as mentioned in this paper, the Castelnuovo-Mumford regularities of the powers of I are compared to the regularity reg I of I itself.
Abstract: For a homogeneous ideal I of S = A[Xo,... ,Xn] we compare the Castelnuovo-Mumford regularities of the powers of I to the regularity reg I of I itself.
82 citations
TL;DR: In this paper, the theory of local cohomology and local duality was generalized to a large class of non-commutative N-graded noetherian algebras.
Abstract: We generalize the theory of local cohomology and local duality to a large class of non-commutative N‐graded noetherian algebras; specifically, to any algebra, B, that can be obtained as graded quotient of some noetherian AS‐Gorenstein algebra, A. As an application, we generalize three “classical” commutative results. For any graded module M over B we have the Bass-numbers ui (M) = dimk Exti b(k, M), and we can then prove that for M finitely generated, we have • id(M) =sup{i|ui(M)≠0}; • the Bass-theorem: if id(M) < ∞, then id(M) = depth(B); • the “No Holes”-theorem: if depth(M) ≤i≤(M), then μi (M) ≠ 0, where id(M) is M's injective dimension as an object in the category of graded modules, while depth(M) is the smallest i such that Exti B(k, M) ≠ 0. As a further application, we also generalize a non‐vanishing result for local cohomology. It states that if M is a finitely generated graded B‐ module, then Here is the i'th local cohomology‐module of M. To prove this result, we need the AS‐Gorenstein algebra, A,...
78 citations
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TL;DR: A short, elementary proof is given in this article that right exchange rings are left exchange, with an application to exchange rings with one in the stable range, with the application to replace a left exchange ring with a right exchange ring.
Abstract: A short, elementary proof is given that right exchange rings are left exchange, with an application to exchange rings with one in the stable range.
75 citations
72 citations
71 citations
TL;DR: In this paper, the authors introduced the notion of coalgebra crossed products, which is a generalisation of the cross product of Hopf algebras by a coalgebra C. The result of such a crossed product is an algebra which is also a right C-comodule.
Abstract: We introduce the notion of a crossed product of an al¬gebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.
60 citations
TL;DR: Theorem 1.1.1 as mentioned in this paper is a generalization of a number of results proved earlier, such as Theorem 2.1, Theorem 3.1 and Theorem 4.2.
Abstract: Let A be a prime ring with nonzero right ideal R and f : R → A an additive map. Next, let k,n1, n2,…,nk be natural numbers. Suppose that […[[(x), xn1], xn2],…, xnk]=0 for all x ∈ R. Then it is proved in Theorem 1.1 that [f(x),x]=0 provided that either char(A)=0 or char (A)> n1+n2+ …+nk Theorem 1.1 is a simultaneous generalization of a number of results proved earlier.
TL;DR: In this article, some categorical remarks on the representation theory of coalgebras are made, and the authors propose a representation theory for the coalgebraic representation of graphs.
Abstract: (1997). Some categorical remarks on the representation theory of coalgebras. Communications in Algebra: Vol. 25, No. 9, pp. 2775-2794.
TL;DR: The duality between cotilting and tilting modules over an arbitrary ring was studied in this article, where it was shown that a module P is partial cotiling iff P is a direct sum-manderer of a COTILING module C such that the left Ext-orthogonal class ⊥P coincides with C.
Abstract: We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P1P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rejp-torsion parts, and the elements of L′ by their Trp-torsion-free parts.
TL;DR: In this paper, the generalized and projective Reed-Muller codes were obtained by using techniques from commutative algebra such as the ideal of a set of points, the a invariant, the Hilbert function, and the Koszul complex.
Abstract: By using techniques from commutative algebra such as the ideal of a set of points, the a‐invariant, the Hilbert function, and the Koszul complex, the main results about the Generalized and Projective Reed‐Muller codes are obtained.
TL;DR: In this article, the authors obtained a representation theorem for bijective additive mappings preserving commutativity in both directions on 2-torsion free unital centrally closed semiprime rings such that the ideal associated with the second Kaplansky polynomial is essential.
Abstract: We obtain a representation theorem for bijective additive mappings preserving commutativity in both directions on 2-torsion free unital centrally closed semiprime rings such that the ideal associated with the second Kaplansky polynomial is essential. The same methods also yield descriptions of Lie automorphisms and Lie derivations.
TL;DR: The canonical module of a one-dimensional reduced local ring has been studied in this paper, where it is shown that the canonical module can be found in the canonical ring of a single-dimensional local ring.
Abstract: (1997). The canonical module of a one-dimensional reduced local ring. Communications in Algebra: Vol. 25, No. 9, pp. 2939-2965.
TL;DR: In this paper, the authors consider an iterated Ore extension k[y][x;σ,δ] of the complex number field k, with δ a k-automorphism of k[x] and δ u-derivation of k [y] vanishing on k.
Abstract: Let R be an iterated Ore extension k[y][x;σ,δ] of the complex number field k, with δ a k-automorphism of k[y] and δ a u-derivation of k[y] vanishing on k. We suppose that the center of R is k. Up to a change of variables, any finite group G of k-automorphisms of R acts linearly on kx⊕D1ky. When the quotient division ring D of R is isomorphic to the Weyl skewfield D1(k)1 , then DG⋍D1 (k). In any other noncommutative case, D is isomorphic to the quantum Weyl skewfield Dq 1(k) for some q∊k∗ not a root of one, and DG⋍Ds 1(k) with s = q‖G‖.
TL;DR: In this paper, it was shown that for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group GMis equal to G.
Abstract: By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group GMis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ⋛ 2. A Galois cover f :X ↪ Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the buildin...
TL;DR: In this article, it was shown that if R is an exchange ring with prime factors Artinian and R/J(R) is homomorphically semipimitive, then it is strongly π-regular and idempotents lift modulo J(R).
Abstract: An associative ring R with identity is called an exchange ring if RR; has the exchange property introduced by Crawley and Jon- sson [5]. We prove, in this paper, that if R is an exchange ring with prime factors Artinian, then R is strongly π-regular. If R is an exchange ring with primitive factors Artinian and R/J(R) is homomorphically semipimitive, then R/J(R) is strongly π-regular and idempotents lift modulo J(R). Also, it is shown that for exchange rings, bounded index of nilpotence implies primitive factors Artinian. These are generalizations of the corresponding results in [16], [11], [8] and [2]. Examples are given showing that the generalizations are nontrivial.
TL;DR: In this paper, the identities of lei algebras with actions of hopf algesbras are discussed. But the authors focus on the action of hopfs.
Abstract: (1997). Identities of lei algebras with actions of hopf algebras. Communications in Algebra: Vol. 25, No. 10, pp. 3179-3187.
TL;DR: In this article, it was shown that in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fix^ for some 4> L Aut(F).
Abstract: We show that, in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fix^ for some 4> L Aut(F). We further show that, in the rank two case, if the intersection of an arbitrary family of proper maximal rank fixed subgroups has rank two then all those subgroups are equal. In particular, every G < Aut(F) with r(FixG) = 2 is either trivial or infinite cyclic.
TL;DR: In this article, the authors describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.
Abstract: The chief aim of this paper is to describe a procedure which, given a d-dimensional absolutely irreducible matrix representation of a finite group over a finite field E, produces an equivalent representation such that all matrix entries lie in a subfield F of E which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time O(|E:F|d3) when |F| is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.
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TL;DR: In this paper, the vine monoid decomposes as a quotient of a general product of these two sub-monoids and is shown to be a submonoid of the free group of rank n.
Abstract: The vine monoids are introduced as equivalence classes of configurations of strings. The strings of a vine can cross over one another, like those of a braid, and can also join together. The vine monoids map homomorphically onto the full transformation monoids. Vines in which strings do not intersect are braids. The vines in which strings only intersect form a monoid isomorphic to the monoid of order-preserving transformations of a finite chain. The vine monoid decomposes as a quotient of a general product of these two submonoids. A presentation for the vine monoids is found. The monoid of n-string vines is exhibited as a submonoid of the endomorphism monoid of the free group of rank n. Finally, some known results about the full transformation monoids are deduced.
TL;DR: In this article, it was shown that holomorphic bundles on O(−k) for k>0 are algebraic, and that bundles on a blow-up of a surface at a point are trivial on a neighborhood of the exceptional divisor.
Abstract: We show that holomorphic bundles on O(−k) for k>0 are algebraic. We also show that holomorphic bundles on O(−1) are trivial outside the zero section. A corollary is that bundles on the blow-up of a surface at a point are trivial on a neighborhood of the exceptional divisor minus the exceptional divisor.
TL;DR: In this article, it was shown that for any q ≥ 3, there are only finitely many r such that △(3,q,r) fails to have property H. This phenomenon has been termed property H by Mushtaq and Servatius.
Abstract: A long standing conjecture (attributed to Graham Higman) asserts that each of the triangle groups △(p,q,r)for 1/p+1/q+1/r>1 contains among its homomorphic images all but finitely many of the alternating or symmetric groups. This phenomenon has been termed property H by Mushtaq and Servatius [9]. The work of several authors over the last decade and a half has shown that for any value of q, there are only finitely many r such that△(2,q,r) fails to have property H. In this paper, the techniques used by these authors are generalised to handle the possinblity that p is odd, and as a result, it is shown that for any q≧3, there are only finitely many r such that △(3,q,r)fails to have property H.
TL;DR: In this paper, the definition of a group of reflexions complexe is discussed. But it is not defined in detail, and the definition is left open to the reader to interpret.
Abstract: (1997). Sur le corps de definition d'un groupe de reflexions complexe. Communications in Algebra: Vol. 25, No. 8, pp. 2703-2716.
TL;DR: In this paper, it was shown that R is a one-sided unit regular ring and that for every x [euro] R, there exists an idempotente and a right or left invertible u such that x [d] eu or x[d] ue is a regular ring.
Abstract: Let R be regular. We show that the following are equivalent:(1) R is a one sided unit regular ring. (2) For every x [euro] R, there exist an idempotente and a right or left invertible u such that x [d] eu or x [d] ue. (3) For every x [euro] R,there exists a right or left invertible u such that xu or ux is an idempotent. Moreover, we give some characterizations of one-sided unit regular rings by group inverses.
TL;DR: In this paper, the necessary and sufficient conditions under which A ⊕ B = A⊕ C implies B and C are comparable relative to ≤ for all finitely generated projective modules over a regular ring.
Abstract: In this paper we will give necessary and sufficient conditions under which A ⊕ B = A ⊕ C implies B and C are comparable relative to ≤ for all finitely generated projective modules A, B and C over a regular ring
TL;DR: DeMeyer and Harada as mentioned in this paper generalized central Galois algebras to the class of Galois Azumaya extensions, where they characterized such extensions in terms of the induced skew group ring being an Azumoya algebra or being H-separable over the ground ring.
Abstract: F. R. DeMeyer ([3]), T. Kanzaki ([7]) and M. Harada ([5]) investigated central Galois Algebras, i.e., Galois algebras Λ over k with group G such that k is the center of Λ. The present authors in [1] generalized the class of central Galois algebras to the class of Galois Azumaya extensions (Galois extensions of an Azumaya algebra), where they characterized such extensions in terms of the induced skew group ring being an Azumaya algebra or being H–separable over the ground ring. In the same paper, the authors gave a description of the ring S as a tensor product of fixed ring and the corresponding commutator subring, which is a central Galois algebra. In the present paper, we shall show two one-to-one correspondence theorems for a Galois Azumaya extension. One of the correspondence is between two sets of separable subextensions, with one set coming from the Azumaya algebra; and the other correspondence is a generalization of the correspondence given by DeMeyer in [3] for separable algebras whose centers are Galois extensions. Moreover, the structure theorem of DeMeyer for central Galois algebras whith an inner Galois group, and the characterization of Kanzaki-Harada of central Galois algebras are generalized to a Galois Azumaya extension.
TL;DR: In this article, a classification of all finite groups in which any two elements of the same order are fused orinverse-fused is given, where the order of the elements is fixed.
Abstract: Elements a, b of a group G are said to be fused or inverse-fused if there exists σeAut(G) such that a = bσ or a = (b-1)σ respectively. This paper gives a classification of all finite groups in which any two elements of the same order are fused orinverse-fused.