Showing papers in "Communications in Algebra in 2008"
TL;DR: In this paper, the authors studied the problem of nonzero zero-divisor graphs of commutative rings with identity and proved that they satisfy certain divisibility conditions between elements of R or comparability conditions between ideals of R.
Abstract: Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ⊆ zR for all z ∈ Z(R)\Nil(R).
127 citations
TL;DR: In this article, it was shown that φ-prime ideals enjoy analogs of many of the properties of prime ideals, such as weakly prime, almost prime, and almost prime.
Abstract: Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I − {0} (resp., ab ∈ I − I 2) implies a ∈ I or b ∈ I. Let φ:ℐ(R) → ℐ(R) ∪ {∅} be a function where ℐ(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I − φ(I) implies a ∈ I or b ∈ I. So taking φ∅(J) = ∅ (resp., φ0(J) = 0, φ2(J) = J 2), a φ∅-prime ideal (resp., φ0-prime ideal, φ2-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.
101 citations
TL;DR: In this paper, it was shown that the only probability measure on [0, 1] n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure.
Abstract: MV-algebras can be viewed either as the Lindenbaum algebras of Łukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] n . The maximal spectrum of Free n is canonically homeomorphic to [0, 1] n , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] n which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic l...
95 citations
TL;DR: In this article, the notion of lax coring was introduced, generalizing Wisbauer's notion of weak coring, and several duality results were given for partial H-comodule algebras.
Abstract: We introduce partial (co)actions of a Hopf algebra H on an algebra. To this end, we introduce first the notion of lax coring, generalizing Wisbauer's notion of weak coring. We also have the dual notion of lax ring. Several duality results are given, and we develop Galois theory for partial H-comodule algebras.
87 citations
TL;DR: In this paper, the structure of all covers of Lie algebras that their Schur multipliers are finite-dimensional is given, which generalizes the work of Batten and Stitzinger (1996).
Abstract: In this article, we give the structure of all covers of Lie algebras that their Schur multipliers are finite dimensional, which generalizes the work of Batten and Stitzinger (1996). Also, similar to a result of Yamazaki (1964) in the group case, it is shown that each stem extension of a finite dimensional Lie algebra is a homomorphic image of a stem cover for it. Moreover, we introduce an ideal in every Lie algebra, which is the smallest ideal contained in the center whose factor algebra is capable, and give some different forms of this ideal. Finally, we study the connection between this ideal and the concept of the Schur multiplier.
85 citations
TL;DR: In this paper, the authors define a G-regular local ring as a commutative, noetherian, local ring over which all totally reflexive modules are free, and show that it behaves similarly to regular local rings.
Abstract: In this article, we define a G-regular local ring as a commutative, noetherian, local ring, over which all totally reflexive modules are free. We study G-regular local rings and observe that they behave similarly to regular local rings. We extend Eisenbud's matrix factorization theorem and Knorrer's periodicity theorem to G-regular local rings.
69 citations
TL;DR: In this paper, it was shown that a finite-dimensional algebra over an algebraically closed field is of derived dimension 0 if and only if it is an iterated tilted algebra of Dynkin type.
Abstract: We prove that a finite-dimensional algebra over an algebraically closed field is of derived dimension 0 if and only if it is an iterated tilted algebra of Dynkin type.
68 citations
TL;DR: In this paper, the zero-divisor graph of a commutative ring and its genus were investigated, and all isomorphism classes of finite rings with identity with genus one were determined.
Abstract: This paper investigates properties of the zero-divisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zero-divisor graph has genus one.
65 citations
TL;DR: In this paper, the authors describe the finite-dimensional irreducible ⊠-modules from multiple points of view and show an isomorphism from the Lie algebra to the three-point loop algebra by generators and relations.
Abstract: Recently Brian Hartwig and the second author found a presentation for the three-point 2 loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra ⊠ by generators and relations, and displayed an isomorphism from ⊠ to the three-point 2 loop algebra. In this article, we describe the finite-dimensional irreducible ⊠-modules from multiple points of view.
64 citations
TL;DR: In this paper, all finite-dimensional indecomposable solvable Lie algebras with the quasifiliform Lie algebra as the nilradical were constructed and classified, and it was shown that the dimension of 𝔤 is at most.
Abstract: All finite-dimensional indecomposable solvable Lie algebras 𝔤, having the quasifiliform Lie algebra as the nilradical, are constructed and classified. It turns out that the dimension of 𝔤 is at most .
56 citations
TL;DR: In this paper, the authors define many new examples of modules of equations for secant varieties of Segre varieties that generalize Strassen's commutation equations, obtained by constructing subspaces of matrices from tensors that satisfy various commutation properties.
Abstract: We define many new examples of modules of equations for secant varieties of Segre varieties that generalize Strassen's commutation equations (Strassen, 1988). Our modules of equations are obtained by constructing subspaces of matrices from tensors that satisfy various commutation properties.
TL;DR: In this paper, the authors define conditions on the ring R which guarantee that the class of absolutely pure modules will be left coherent, which they show implies a number of other necessary properties.
Abstract: Absolutely pure modules act in ways similar to injective modules. There are conditions on a ring which guarantee that the class of injective modules will be covering. In this article, we define conditions on the ring R which guarantee that the class of absolutely pure modules will be covering. These include R being left coherent, which we show implies a number of other necessary properties.
TL;DR: In this article, the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S was constructed, and it was shown that the von Neumann regularity of R is equivalent to its semisimplicity.
Abstract: In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S∖{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].
TL;DR: In this article, the modular properties of nodal curves on a low genus K3 surface were investigated, and it was shown that a general genus g curve C is the normalization of a � -nodal curve X sitting on a primitively polarized K 3 surface S of degree 2p − 2, for 2 ≤ g = p − �
Abstract: We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a � -nodal curve X sitting on a primitively polarized K3 surface S of degree 2p − 2, for 2 ≤ g = p − �
TL;DR: In this paper, it was shown that Γ(M n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field.
Abstract: The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL n (F) and SL n (F). We show that Γ(M n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL n (F)) and Γ(SL n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M n (F))≃Γ(M m (E)), then n = m and |F|=|E|.
TL;DR: For the edge ideals of a certain class of forests, this article showed that the arithmetical rank equals the projective dimension of the edge ideal of a given forest, and that the rank of an edge ideal is the same as that of a tree.
Abstract: We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.
TL;DR: The quasi-Jordan algebras as mentioned in this paper are a generalization of the Jordan type where the commutative law is changed by a quasi-commutative identity and the Jordan identity is retained.
Abstract: In this article we introduce a new algebraic structure of Jordan type and we show several examples. This new structure, called “quasi-Jordan algebras,” appears in the study of the product where x, y are elements in a dialgebra (D, ⊣, ⊢). The quasi-Jordan algebras are a generalization of Jordan algebras where the commutative law is changed by a quasi-commutative identity and a special form of the Jordan identity is retained. We show a few results about the relationship between Jordan algebras and quasi-Jordan algebras. Also, we compare quasi-Jordan algebras with some structures. In particular, we find a special relation with Leibniz algebras. We attach a quasi-Jordan algebra to any ad-nilpotent element of index of nilpotence at most 3 in a Leibniz algebra.
TL;DR: In this article, it was shown that the alternating groups, the sporadic groups and the exceptional covering groups of groups of Lie type satisfy the inductive condition required to verify the McKay conjecture for finite groups.
Abstract: We show that the alternating groups, the sporadic groups and the exceptional covering groups of groups of Lie type satisfy the inductive condition required to verify the McKay conjecture for finite...
TL;DR: In this article, the notion of skew-armendariz rings was introduced, which are a generalization of α-skew Armendariz ring and α-rigid rings.
Abstract: Let α be an endomorphism and δ an α-derivation of a ring R. We introduce the notion of skew-Armendariz rings which are a generalization of α-skew Armendariz rings and α-rigid rings and extend the classes of non reduced skew-Armendariz rings. Some properties of this generalization are established, and connections of properties of a skew-Armendariz ring R with those of the Ore extension R[x; α, δ] are investigated. As a consequence we extend and unify several known results related to Armendariz rings.
TL;DR: In this article, the influence of SS-quasinormal on maximal or minimal subgroups of Sylow subgroups is investigated for the generalized Fitting subgroup of a finite group.
Abstract: A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.
TL;DR: In this paper, the authors classify surfaces of general type with pg = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian, and they belong to four families: surfaces of type I, II, III, IV.
Abstract: A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F)/G. In this article, we classify the surfaces of general type with pg = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV. The moduli spaces 𝔐I, 𝔐II, 𝔐IV are irreducible, whereas 𝔐III is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples.
TL;DR: In this article, the authors introduced the concept of nilpotent submodules and proved that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nil-potent elements and its Krull dimension is zero.
Abstract: All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.
TL;DR: In this article, it was shown that every left R-module over a left P-coherent ring R has a divisible cover, and a left Rmodule M is D-injective if and only if M is the kernel of a precover A→B with A injective.
Abstract: A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext1(G, M) = 0 (resp. Tor1(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.
TL;DR: In this article, a multiplicative Lie isomorphism is defined for a prime ring with 1 containing a nontrivial idempotent E and another prime ring ℛ with E.
Abstract: Let ℛ be a prime ring with 1 containing a nontrivial idempotent E, and let ℛ′ be another prime ring. If Φ:ℛ → ℛ′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ T,S for all T, S ∈ ℛ, where Z′ T,S is an element in the center 𝒵′ of ℛ′ depending on T and S.
TL;DR: The automorphism group of the special linear group is analyzed and it is shown that this MOR cryptosystem has better security than the ElGamal cryptos system over finite fields.
Abstract: In this article we study the MOR cryptosystem. We use the group of unitriangular matrices over a finite field as the non-abelian group in the MOR cryptosystem. We show that a cryptosystem similar to the ElGamal cryptosystem over finite fields can be built using the proposed groups and a set of automorphisms of these groups. We also show that the security of this proposed MOR cryptosystem is equivalent to the ElGamal cryptosystem over finite fields.
TL;DR: In this paper, the authors investigated the Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple groups M 23 using the Luthar-Passi method.
Abstract: We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M 23 using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M 11 and M 12. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.
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TL;DR: In this article, the authors considered the class of perfect GMV-algebras, which includes all noncommutative analogs of perfect MV-algeses.
Abstract: The focus of this article is the class of perfect GMV-algebras, which includes all noncommutative analogs of perfect MV-algebras. As in the commutative case, we show that each perfect GMV-algebra possesses a single negation, it is generated by its infinitesimal elements, and can be uniquely realized as an interval in a lexicographical product of the lattice-ordered group of integers and an arbitrary lattice-ordered group. Further, we establish that the category of perfect GMV-algebras is equivalent to the category of all lattice-ordered groups. The variety of GMV-algebras generated by the class of perfect GMV-algebras plays a key role in our considerations. Among other results, we describe a finite equational basis for this variety and prove that it fails to satisfy the amalgamation property. In fact, we show that uncountably many of its subvarieties fail this property.
TL;DR: In this article, a generalized derivation of R such that [g(r k ), r k ] n ǫ n  = 0 for all r ∈ I, where k, n are fixed positive integers.
Abstract: Let R be a noncommutative prime ring and I a nonzero left ideal of R Let g be a generalized derivation of R such that [g(r k ), r k ] n = 0 for all r ∈ I, where k, n are fixed positive integers Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c − α) = 0 for a suitable α ∈ C In particular we have that g(x) = α x, for all x ∈ I
TL;DR: In this paper, Guralnick, Penttila, Praeger, Saxl, and Saxl used a theorem of Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup.
Abstract: We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines
TL;DR: In this article, the fundamental relation on a hypermodule over a hyperring is defined as the smallest equivalence relation so that the quotient would be the module over a ring.
Abstract: The main tools in the theory of hyperstructues are the fundamental relations. The fundamental relation on a hypermodule over a hyperring was already introduced by Vougiouklis. The fundamental relation on a hypermodule over a hyperring is defined as the smallest equivalence relation so that the quotient would be the module over a ring. Note that generally the commutativity with respect to both sum in the (fundemental) module and product in the (fundamental) ring are not assumed. In this article we introduce a new strongly regular equivalence relation on hypermodules so that the quotient is module (with abelin group) over a commutative ring. Also we state the conditions that is equivalent with the transitivity of this relation and finally we characterize the complete hypermodules over hyperrings.