Showing papers in "Communications in Algebra in 2005"
TL;DR: The structure theory of finite-dimensional Leibniz algebras with additional properties is studied in this paper, where the notion of Leibiz superalgebra is introduced.
Abstract: Leibniz algebras that are noncommutative generalizations of Lie algebras are considered. Nilpotent and simple Leibniz algebras are investigated. The structure theory of finite-dimensional Leibniz algebras with additional properties is studied. We also introduce the notion of Leibniz superalgebra.
121 citations
[...]
TL;DR: In this article, the authors studied the problem of generating a graded K-algebra with projective resolution, such that each P i can be generated in a single degree, where P i is a finite product of copies of the field K, A is generated in degrees 0 and 1, and dim K A 1 ǫ < ∞.
Abstract: Let A = A 0 ⊕ A 1 ⊕ A 2 ⊕ ··· be a graded K -algebra such that A 0 is a finite product of copies of the field K, A is generated in degrees 0 and 1,and dim K A 1 < ∞. We study those graded algebras A with the property that A 0 , viewed as a graded A -module, has a graded projective resolution, , such that each P i can be generated in a single degree. The paper describes necessary and sufficient conditions for the Ext-algebra of A , , to be finitely generated. We also investigate classes of modules over such algebras and Veronese subrings of the Ext-algebra.
116 citations
TL;DR: In this paper, the preservation of the diameter and girth of a zero-divisor graph under extension to polynomial and power series rings was examined. But the preservation was not studied.
Abstract: We recall several results about zero-divisor graphs of commutative rings. Then we examine the preservation of diameter and girth of the zero-divisor graph under extension to polynomial and power series rings.
110 citations
TL;DR: A semisimple element s of a connected reductive group G is called quasi-isolated if C G ( s ) (respectively ( s )) is not contained in a Levi subgroup of a proper parabolic subgroup as mentioned in this paper.
Abstract: A semisimple element s of a connected reductive group G is called quasi-isolated (respectively isolated) if C G ( s ) (respectively ( s )) is not contained in a Levi subgroup of a proper parabolic subgroup of G . We study properties of quasi-isolated semisimple elements and give a classification in terms of the affine Dynkin diagram of G . Tables are provided for adjoint simple groups.
94 citations
TL;DR: A linear finite dynamical system (LFDS) as discussed by the authors is a pair (E, f) where E is a finite dimensional space over a finite field k. A complete description of the dynamics of f, the behavior of f's iterates, is g...
Abstract: A Linear Finite Dynamical System (LFDS) is a pair (E, f) where E is a finite dimensional space over a finite field k. A complete description of the dynamics of f, the behavior of f's iterates, is g...
86 citations
TL;DR: In this paper, the irreducible components of the nilpotent complex Leibniz algebras varieties of dimension less than 5 were described and degenerated.
Abstract: The aim of this work is to describe the irreducible components of the nilpotent complex Leibniz algebras varieties of dimension less than 5. We construct degenerations between one-parametric families of nilpotent Leibniz algebras and study the rigidity of these families. #Communicated by I. Shestakov.
79 citations
TL;DR: The applicability of the Zappa-Szep product to multiplicative structures more general than groups with emphasis on categories and monoids was explored in this article. But the preservation of various properties of the multiplication under Zappa Szep product was not explored.
Abstract: The Zappa-Szep product of a pair of groups generalizes the semidirect product in that neither factor is assumed to be normal in the result. We extend the applicability of the Zappa-Szep product to multiplicative structures more general than groups with emphasis on categories and monoids. We also explore the preservation of various properties of the multiplication under the Zappa-Szep product.
78 citations
TL;DR: In this article, a nilpotent Lie algebra of dimension n and a central extension by an ideal M of maximal dimension such that M is contained in the intersection of the center and the derived algebra of C is considered.
Abstract: Let L be a nilpotent Lie algebra of dimension n and C be a central extension by an ideal M of maximal dimension such that M is contained in the intersection of the center and the derived algebra of C. Then M is called the multiplier of L. We denote by . The classification of L is already known for t(L) ≤ 6. In this article, we classify L for t(L) ≤ 8.
67 citations
TL;DR: In this article, Facchini et al. investigated integral domains in which each ideal is a w-ideal (i.e. the d-and w-operations are the same), called the DW-domains.
Abstract: In this paper, we investigate integral domains in which each ideal is a w-ideal (i.e. the d- and w-operations are the same), called the DW-domains. In some sense this study is similar to that one given in Houston and Zafrullah (1988) [Houston, E., Zafrullah, M. (1988). Integral domains in which each t-ideal is divisorial. Michigan Math. J. 35:291–300.] for the TV-domains. We prove that a domain R is a DW-domain if and only if each maximal ideal of R is a w-ideal, and if R is a domain such that R M is a DW-ideal for each maximal ideal M of R, then so is R, and the equivalence holds when R is v-coherent. We describe the w-operation on pull–backs in order to provide original examples. #Communicated by A. Facchini.
62 citations
TL;DR: In this paper, the authors define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideal ideals are primary and in regular domains almost primes are precisely primes.
Abstract: In studying unique factorization of domains we encountered a property of ideals. Using that we define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideals are primary. We also prove that in a regular domain almost primes are precisely primes. Further, we define strictly nonprime ideals and study some inter relations between almost prime ideals, strictly nonprime ideals and factorization of ideals.
61 citations
TL;DR: In this article, the authors studied the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra algebras, and classified all degenerations of (almost all) five-step and six-step nil-potent seven-dimensional complex Lie alges.
Abstract: We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimension.
TL;DR: In this article, a generalization of the theory of flatness to ordered monoids acting on partially ordered sets (S -posets) is presented. And a unique decomposition theorem for S-posets is given.
Abstract: For a monoid S , a (left) S -act is a nonempty set B together with a mapping S ×B→B sending (s, b) to sb such that S (tb) = lpar;st)b and 1b = b for all S , t ∈ S and B ∈ B. Right S -acts A can also be defined, and a tensor product A ⊗ s B (a set)can be defined that has the customary universal property with respect to balanced maps from A × B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (New York: Walter de Gruyter, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets ( S -posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S ...
TL;DR: In this article, the authors formulate basic results on outer generalized inverses of elements in rings and characterize elements which have the same idempotents related to their particular outer generalized inverse.
Abstract: In this article we formulate basic results on outer generalized inverses of elements in rings. We characterize elements which have the same idempotents related to their particular outer generalized inverses and investigate positive generalized inverses in C*-algebras.
TL;DR: Villarreal et al. as discussed by the authors studied invariants of the edge-ring of a simple graph that can be interpreted as invariants for vertex-critical graphs and showed that if G has a cover by maximum stable sets, then α ≤ √(n−−|A|)/2, where A is the intersection of all the minimum vertex covers of G.
Abstract: Let G be a simple graph with |V(G)| = n and no isolated vertices. Let α be its stability number. We study invariants of the edge-ring of G that can be interpreted as invariants of G. If G has a cover by maximum stable sets we are able to prove the inequality . As a byproduct we prove that if G is vertex-critical, then α ≤ (n − |A|)/2, where A is the intersection of all the minimum vertex covers of G. We estimate the smallest number of vertices in any maximal stable set of G to obtain a bound for the depth of the edge-ring of G. #Communicated by R. Villarreal.
TL;DR: In this paper, the regularity of elements and Green's relations for the transformation semigroup of continuous selfmaps of a set X are discussed. But they do not discuss the relation between elements and transformation semigroups.
Abstract: Let 𝒯 X denote the full transformation semigroup on a set X. For an equivalence E on X, let Then T E (X) is exactly the semigroup of continuous selfmaps of the topological space X for which the collection of all E-classes is a basis. In this paper, we discuss regularity of elements and Green's relations for T E (X).
TL;DR: In this paper, the authors introduced M-Armendariz rings, which are generalizations of Armendariz ring, and investigated their properties for a monoid M and showed that every reduced ring is M-armendariz for any unique product monoid, where N is a unique monoid.
Abstract: For a monoid M, we introduce M-Armendariz rings, which are generalizations of Armendariz rings; and we investigate their properties. Every reduced ring is M-Armendariz for any unique product monoid M. We show that if R is a reduced and M-Armendariz ring, then R is M × N-Armendariz, where N is a unique product monoid. It is also shown that a finitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-Armendariz. Moreover, we study the relationship between the Baerness and the PP-property of a ring R and those of the monoid ring R[M] in case R is M-Armendariz.
TL;DR: In this paper, the authors describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A ∞ as a subquiver, that is, the so-called rooted quivers.
Abstract: In the first part of this article, we describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A ∞ as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category of representations by modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver A ∞, which will allow us to characterize the projective representations of A ∞. This will improve some previous results and make more accurate the statement made in Benson (1991). We think this technique can be applied in many other general situations to provide information about the decomposition of a projective module.
TL;DR: In this paper, the notions of generalized regular sequence and generalized depth are introduced as extensions of the known notions of regular sequence, respectively, and some properties of GCS and depth are given.
Abstract: Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R -module. The two notions of generalized regular sequence and generalized depth are introduced as extensions of the known notions of regular sequence and depth, respectively. Some properties of generalized regular sequence and generalized depth, which are closely related to those of regular sequence and depth, are given. If x1 ,…, xr is a generalized regular sequence of M , then is a finite set. Some finiteness properties for associated primes of local cohomology modules are presented.
TL;DR: In this article, a (unital) extension R⊆T of (commutative) rings is said to have FIP if there are only finitely many (respectively no) rings S such that R ⊆S⊂T of S ⊂ T of ⌈⌉⌈ ⌉ ⌂ T such that
Abstract: A (unital) extension R ⊆ T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ⊂ S ⊂ T. Transfer r...
TL;DR: In this article, the authors give alternative proofs to Smith's result on base-point-freeness of adjoint bundles in characteristic p > 0 and results on uniform behavior of symbolic powers in a regular local ring due to Ein, Lazarsfeld and Smith.
Abstract: We present two applications of a characteristic p analog of multiplier ideals, which is a generalization of the test ideal in the theory of tight closure. Namely, we give alternative proofs to Smith's result on base-point-freeness of adjoint bundles in characteristic p > 0 and results on uniform behavior of symbolic powers in a regular local ring due to Ein, Lazarsfeld and Smith, and Hochster and Huneke.
TL;DR: In this article, the obstruction theory for lifting complexes up to quasi-isomorphism was developed, up to derived categories of flat nilpotent deformations of abelian categories.
Abstract: In this article, we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case w...
TL;DR: In this paper, the Stenstrom-Govorov-Lazard theorem was shown in the context of S-posets and the flatness properties of the S-act over a partially ordered monoid were studied.
Abstract: For a monoid S, a (left) S-act is a nonempty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Over the past three decades, an extensive theory of flatness properties has been developed (involving free acts, projective acts, strongly flat acts, Condition (P), flat acts, weakly flat acts, principally weakly flat acts, and torsion free acts). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by Kilp et al. (2000). Partially ordered acts over a partially ordered monoid S, or S-posets appear naturally in the study of mappings between posets. Preliminary work on flatness properties of S-poset, was done by Fakhruddin in the 1980s (see Fakhruddin, 1986 1988), and continued in recent (Bulman-Fleming and Laan, 2005; Shi et al., 2005). In Bulman-Fleming and Laan (2005), the Stenstrom-Govorov-Lazard theorem was shown in the context of S-posets. Tensor products of S-posets, free, p...
TL;DR: In this article, an upper bound for the Castelnuovo-Mumford regularity of associated graded modules in terms of dimension and relative extended degree was given. But this bound was not extended to the associated graded ring of Rossi, Trung, and Valla.
Abstract: We will give an upper bound for the Castelnuovo-Mumford regularity of associated graded modules in terms of dimension and relative extended degree. This result extends the bound for regularity of the associated graded ring of Rossi,Trung,and Valla.
TL;DR: In this article, the authors give several properties of pseudo-injective modules, and discuss the question of when a pseudo-implicitive module is injective or quasi-impliant.
Abstract: Pseudo-injectivity is a generalization of injectivity. In this paper, we give several properties of pseudo-injective modules, and discuss the question of when a pseudo-injective module is injective or quasi-injective.
TL;DR: In this article, the authors studied the relation between semistar operations on an integral domain D and star operations on the overring of D and showed that there is a bijection between the set of all semistar operation on a domain D with respect to a star operation on the overrings of D.
Abstract: The purpose of this article is to deepen the study of the relation between semistar operations on an integral domain D and the (semi) star operations (that is, the semistar operations, that restricted to the set of the fractional ideals, are star operations (on the overrings of D . First, we define the composition of two semistar operations and study when this composition is a semistar operation. Then we show that there is a bijection between the set of all semistar operations on a domain D and the set of all (semi) star operations on the overrings of D . To do this, we prove that semistar operations on D have a canonical decomposition as the composition of a semistar operation given by the extension to an overring and a (semi) star operation on this overring. Moreover, we study which properties of semistar operations are preserved by this bijection. Finally, we give some applications to the study of semistar operations on valuation and Prufer domains and we give, by using the techniques introduc...
TL;DR: In this paper, it was shown that a Yang-Baxter system can be constructed from any entwining structure and that certain types of Yang-baxter systems of certain types lead to entwined structures.
Abstract: It is shown that a Yang–Baxter system can be constructed from any entwining structure. It is also shown that,conversely,Yang–Baxter systems of certain types lead to entwining structures. Examples of Yang–Baxter systems associated to entwining structures are given,and a Yang–Baxter operator of Hecke type is defined for any bijective entwining map.
TL;DR: In this paper, the authors used an explicit resolution of the diagonal for the variety V 5 and provided cohomological characterizations of universal and quotient bundles, and a splitting criterion for bundles over V 5 is also proved.
Abstract: Using an explicit resolution of the diagonal for the variety V 5, we provide cohomological characterizations of the universal and quotient bundles. A splitting criterion for bundles over V 5 is also proved. The presentation of semistable aCM bundles is shown, together with a resolution–theoretic classification of low rank aCM bundles.
TL;DR: In this paper, the relative projective stratifying system is introduced and a result from which the Theorem 2 in Dlab and Ringel (1992) and Proposition 2.1 in Ringel and Platzeck and Reiten (2001) follows.
Abstract: In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (1992), and also the “Comparison method” given in Platzeck and Reiten (2001) can be generalized. To do so, we introduce the concept of relative projective stratifying system and prove a result from which the Theorem 2 in Dlab and Ringel (1992) and Proposition 2.1 in Ringel (1991) follows. #Communicated by A. Happel. ‡Dedicated to Raymundo Bautista on his 60th birthday.
TL;DR: In this article, the associated primes of Matlis duals of local cohomology modules (MDLCM) were studied, and partial answers to questions which were left open in [1] were obtained.
Abstract: In continuation of [1] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Zoschinger on coassociated primes of arbitrary modules with results from [1 4-6], and obtain partial answers to questions which were left open in [1]. These partial answers give further support for conjecture (*) from [1] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Zoschinger, we prove some non-finiteness results of local cohomology.
TL;DR: In this paper, Enochs et al. studied V-Gorenstein modules and showed that under the finiteness of projective dimension for flat modules (a property that holds for many rings), covers and envelopes for these classes of modules exist.
Abstract: In this article, we study V-Gorenstein modules relatives to a dualizing module (Enochs et al., in press). These modules constitute a generalization of the well-known Gorenstein modules and at the same time an extension to the noncommutative case of Ω-Gorenstein modules (cf. Enochs and Jenda, 2000a, in press). We show that, under hypothesis of finiteness of projective dimension for flat modules (a property that holds for many rings), covers and envelopes for these classes of modules exist.