Showing papers in "Communications in Algebra in 2007"
TL;DR: Hartwig, Larsson, and Silvestrov as discussed by the authors applied a deformation method to the simple 3-dimensional Lie algebra 𝔰&#x 1d529;2&# x 1d53d;).
Abstract: In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006) and Larsson and Silvestrov (2005a) to the simple 3-dimensional Lie algebra 𝔰𝔩2(𝔽). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 𝔰𝔩2(𝔽) we can, by choosing parameters suitably, deform 𝔰𝔩2(𝔽) into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 𝔰𝔩2(𝔽) is rigid.
128 citations
TL;DR: In this paper, a quantum analog of the three-point 2 loop algebra via generators and relations is introduced, which is called q. The authors show how q is related to the quantum group Uq( √ √ q), the Uq ( √ ǫ) loop algebra, and the positive part of.
Abstract: Recently, Hartwig and the second author found a presentation for the three-point 2 loop algebra via generators and relations. To obtain this presentation they defined an algebra ⊠ by generators and relations, and displayed an isomorphism from ⊠ to the three-point 2 loop algebra. We introduce a quantum analog of ⊠ which we call ⊠q. We define ⊠q via generators and relations. We show how ⊠q is related to the quantum group Uq(2), the Uq(2) loop algebra, and the positive part of . We describe the finite dimensional irreducible ⊠q-modules under the assumption that q is not a root of 1, and the underlying field is algebraically closed.
93 citations
TL;DR: In this article, the authors give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the notion of relative commutative degree of subgroups H of a group G, where H is a subgroup of G and G is a group.
Abstract: The aim of this article is to give a generalization of the concept of commutativity degree of a finite group G (denoted by d(G)), to the concept of relative commutativity degree of a subgroup H of a group G (denoted by d(H, G)). We shall state some results concerning the new concept which are mostly new or improvements of known results given in Gustafson (1973) and Moghaddam et al. (2005). Moreover, we shall define the relative nth nilpotency degree of a subgroup of a group and give some results concerning this at the end of the article.
90 citations
TL;DR: In this article, it was shown that the dual of a pointed semisimple category with respect to a module category is a Grothendieck ring and the associator of the dual is an associator.
Abstract: A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, ω), where G is a finite group and ω ∊ H 3(G, k ×). A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have nonisomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data.
74 citations
TL;DR: The fundamental relation on a hyperring (H v -ring) was introduced by Vougiouklis at the fourth AHA congress as discussed by the authors, which is defined as the smallest equivalence relation so that the quotient would be the (fundamental) ring.
Abstract: The main tools in the theory of hyperstructues are the fundamental relations. The fundamental relation on a hyperring was introduced by Vougiouklis at the fourth AHA congress. The fundamental relation on a hyperring (H v -ring) is defined as the smallest equivalence relation so that the quotient would be the (fundamental) ring. Note that the commutativity with respect to both sum and product in the (fundamental) ring are not assumed. Now, in this article we would like the (fundamental) ring to be commutative with respect to both sum and product, that is, the fundamental ring should be an ordinary commutative ring. Therefore we introduce a new strongly regular equivalence relation on hyperrings (H v -rings). If we consider this relation on a hperring (H v -ring), then the set of quotients is a commutative ring. Some properties of such rings are investigated.
70 citations
TL;DR: In this article, the notion of nonsingularity of a module is introduced and connected to its endomorphism rings, and rings for which all modules are nonsingular are precisely determined.
Abstract: We introduce the notion of 𝒦-nonsingularity of a module and show that the class of 𝒦-nonsingular modules properly contains the classes of nonsingular modules and of polyform modules. A necessary and sufficient condition is provided to ensure that this property is preserved under direct sums. Connections of 𝒦-nonsingular modules to their endomorphism rings are investigated. Rings for which all modules are 𝒦-nonsingular are precisely determined. Applications include a type theory decomposition for 𝒦-nonsingular extending modules and internal characterizations for 𝒦-nonsingular continuous modules which are of type I, type II, and type III, respectively.
62 citations
TL;DR: The strong orthogonal completion of the polycyclic monoid P n on n generators was constructed in this paper, where the group of units is the Thompson group of generators.
Abstract: We construct what we call the strong orthogonal completion C n of the polycyclic monoid P n on n generators. The inverse monoid C n is congruence free and its group of units is the Thompson group V...
44 citations
TL;DR: The notion of orthogonal completion of an inverse monoid with zero was introduced in this paper, where it was shown that the polycyclic monoid on n generators is isomorphic to the right ideal monoid of right ideal isomorphisms between the finitely generated right ideals of the free monoid.
Abstract: We introduce the notion of an orthogonal completion of an inverse monoid with zero. We show that the orthogonal completion of the polycyclic monoid on n generators is isomorphic to the inverse monoid of right ideal isomorphisms between the finitely generated right ideals of the free monoid on n generators, and so we can make a direct connection with the Thompson groups Vn,1.
44 citations
TL;DR: In this paper, it was shown that the clique number of a non-cyclic graph G is finite if and only if G has no infinite clique and if G is a finite nilpotent group.
Abstract: We associate a graph Γ G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ∊ G| 〈x, y〉 is cyclic for all y ∊ G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ G is finite if and only if Γ G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ G ≅ Γ H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ G ≅ Γ H for some group H, then G ≅ H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ G ≅ Γ H for some group H, then |G| = |H|. These suggest the...
43 citations
TL;DR: In this article, it was shown that ( is a perfect cotorsion theory if R is a right coherent ring with FP-id(R R ) ≤ n. This result was proven by Aldrich, Enochs, Jenda, and Oyonarte in Noetherian case.
Abstract: Let R be a ring, n a fixed non-negative integer and ℱ ℐ n (ℱ n ) the class of all right (left) R-modules of FP-injective (flat) dimension at most n. We prove that ( is a perfect cotorsion theory if R is a right coherent ring with FP-id(R R ) ≤ n. This result was proven by Aldrich, Enochs, Jenda, and Oyonarte in Noetherian case. The modules in are also studied. Some applications are given.
42 citations
TL;DR: A generalization of a classical result from the theory of nilpotent Lie algebras to Leibniz algesbras leads to several applications concerning the nil-potent properties both of these two types of leitneral algebraids.
Abstract: A generalization of a classical result from the theory of nilpotent Lie algebras to Leibniz algebras leads to several applications concerning the nilpotent properties both of these two types of algebras.
TL;DR: In this article, a categorical connection between the category of unital l-groups and the variety of generalized generalized MV-algebras was established, which enables us to naturally export equational machinery and terminology like "variety" from the latter category to the former.
Abstract: In spite of the well-know fact that the system of l-groups with strong unit (unital l-groups) does not form a variety, there is a categorical connection between the category of unital l-groups and the variety of generalized MV-algebras which enables us to naturally export equational machinery and terminology like “variety” from the latter category to the former. Using this categorical equivalence, we study varieties, or equationally defined classes, and top varieties, varieties above the normal valued variety, of both structures. We generalize Chang's Completeness Theorem for generalized MV-algebras, and formulate some open questions for both structures.
TL;DR: In this paper, the authors present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings.
Abstract: In this article we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over ring homomorphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensions. Most of these results are new even when the ring homomorphism is local. The main tool for these analyses is a nonlocal version of the amplitude inequality of Iversen, Foxby, and Iyengar. We provide numerous examples demonstrating the need for certain hypotheses and the strictness of many inequalities.
TL;DR: In this paper, given multigraded free resolutions of two monomial ideals, the authors construct a multi-generated free resolution of the sum of the two ideals, which is the same as the one in this paper.
Abstract: Given multigraded free resolutions of two monomial ideals, we construct a multigraded free resolution of the sum of the two ideals
TL;DR: In this article, it was shown that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero divisor graph of some Boolean ring.
Abstract: A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x2 = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of Kn if n + 1 = pq for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.
TL;DR: In this article, the authors considered Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components and showed that all such structures are triangular coboundary.
Abstract: In an article by Michaelis, a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H1(𝒲, 𝒲 ⊗ 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.
TL;DR: In this article, it was shown that there is an Avramov-Martsinkovsky type exact sequence with, Gtor, Tor, and Tor, which can be computed using either a complete resolution of MR or using a full resolution of RN.
Abstract: We show that there is an Avramov–Martsinkovsky type exact sequence with , Gtor, and Tor. We prove that if R is a Gorenstein ring, then the modules , n ≥ 1 can be computed using either a complete resolution of MR or using a complete resolution of RN. We show that over a Gorenstein ring a left R-module N is Gorenstein flat if and only if . We also show that over commutative Gorenstein rings the modules can be computed by the combined use of a flat resolution and a Gorenstein flat resolution of M.
TL;DR: A transitive permutation group with no fixed point free elements of prime order, or equivalently, no nontrivial semiregular subgroups, is an elusive permutation groups as discussed by the authors.
Abstract: An elusive permutation group is a transitive permutation group with no fixed point free elements of prime order, or equivalently, no nontrivial semiregular subgroups. We provide several new constructions of elusive groups, some of which enable us to build elusive groups with new degrees.
TL;DR: In this paper, the smallest equivalence relation on a polygroup P such that P/γ* is an abelian group was introduced and results were obtained on extension polygroups, derived hypergroups, γ-parts, and semi-direct hyperproduct.
Abstract: The γ*-relation was introduced by Freni. In this article, we use the γ*-relation in a given polygroup. In this way, the γ*-relation is the smallest equivalence relation on a polygroup P such that P/γ* is an abelian group. Results are obtained on extension polygroups, derived hypergroups, γ-parts, and semi-direct hyperproduct.
TL;DR: In this article, the results of Artin-Markov on braid groups were presented by using the Grobner-Shirshov basis, and they can reobtain the normal form of ARTIN-MIRIvanovsky as an easy corollary.
Abstract: In this article, we will present the results of Artin–Markov on braid groups by using the Grobner–Shirshov basis. As a consequence, we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.
TL;DR: In this paper, the authors examined the finite dimensional representations of the Euclidean algebra 𝔢(2) that are obtained by embedding embeddings into the Lie algebra of traceless 3 × 3 matrices.
Abstract: In this article, we examine the finite dimensional representations of the Euclidean algebra 𝔢(2) that are obtained by embedding 𝔢(2) into 𝔰𝔩3, the Lie algebra of traceless 3 × 3 matrices. We show that the finite dimensional, irreducible representations of 𝔰𝔩3 restricted to 𝔢(2) are indecomposable and, when possible, we give graphical descriptions of these 𝔢(2) representations.
TL;DR: A contragredient Lie superalgebra has finite-growth if the dimensions of the graded components (in the natural grading) depend polynomially on the degree as mentioned in this paper.
Abstract: A contragredient Lie superalgebra is a superalgebra defined by a Cartan matrix. A contragredient Lie superalgebra has finite-growth if the dimensions of the graded components (in the natural grading) depend polynomially on the degree. In this article we classify finite-growth contragredient Lie superalgebras. Previously, such a classification was known only for the symmetrizable case.
TL;DR: In this paper, it was shown that f-depth(I+Ann(M), N) is the least integer r such that the generalized local cohomology module is not Artinian.
Abstract: Let R be a commutative Noetherian local ring, I a proper ideal of R, M, and N finitely generated R-modules It is proved that f-depth(I + Ann(M), N) is the least integer r such that the generalized local cohomology module is not Artinian Let r ≥ 0 be an integer We also discuss the property that is Artinian for all i ≥ r
TL;DR: In this paper, it was shown that for a variety R ℳ of semimodules over an IBN-semiring R (an IBN is a semiring analog of a ring with IBN), all automorphisms of Aut(Rℳ0) are semi-inner.
Abstract: In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ0) of automorphisms of the category Θ0 of finitely generated free algebras of Θ is of great importance. In this article, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety Rℳ of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut(Rℳ0) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in Plotkin (2002); in particular, for Artinian (Noetherian, PI-) rings R, or a division semiring R, all automorphisms of Aut(Rℳ0) are semi-inner.
TL;DR: In this article, partial skew polynomial rings are defined as natural subrings of the partial skew group ring R ⋆α-G on a cyclic infinite group.
Abstract: In this article, we consider rings R with a partial action α of a cyclic infinite group G on R. We define partial skew polynomial rings as natural subrings of the partial skew group ring R ⋆α G. We study prime and maximal ideals of a partial skew polynomial ring when the given partial action α has an enveloping action.
TL;DR: In this article, a zero-divisor graph of a commutative ring is defined for integral domains, and these graphs are used to characterize certain classes of domains, including UFDs.
Abstract: In Beck (1988), the author introduces the idea of a zero-divisor graph of a commutative ring. We generalize this idea to study factorization in integral domains and define irreducible divisor graphs. We use these irreducible divisor graphs to characterize certain classes of domains, including UFDs.
TL;DR: In this paper, it was shown that the class of C 11-modules is closed under direct sums but not under direct summands, and that all essential extensions of a module M satisfying C 11 are essential extensions from M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M.
Abstract: A module M is said to satisfy the C 11 condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C 1 condition implies the C 11 condition and that the class of C 11-modules is closed under direct sums but not under direct summands. We show that if M = M 1 ⊕ M 2, where M has C 11 and M 1 is a fully invariant submodule of M, then both M 1 and M 2 are C 11-modules. Moreover, the C 11 condition is shown to be closed under formation of the ring of column finite matrices of size Γ, the ring of m-by-m upper triangular matrices and right essential overrings. For a module M, we also show that all essential extensions of M satisfying C 11 are essential extensions of C 11-modules constructed from M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M. Finally, we prove that if M is a C 11-module, then so is its rational hull. Examples are provided to illustrate and delimit the theory.
TL;DR: In this paper, it was shown that a GCD domain D is an antimatter domain if and only if P−1 = D for each maximal t-ideal P of D.
Abstract: An integral domain without irreducible elements is called an antimatter domain. We give some monoid domain constructions of antimatter domains. Among other things, we show that if D is a GCD domain with quotient field K that is algebraically closed, real closed, or perfect of characteristic p > 0, then the monoid domain D[X; ℚ+] is an antimatter GCD domain. We also show that a GCD domain D is antimatter if and only if P−1 = D for each maximal t-ideal P of D.
TL;DR: A ring R is called left generalized morphic if for every element a in R, there exists b ∈ R such that l(a) ≥ R/Rb, where l denotes the left annihilator of a element in R as mentioned in this paper.
Abstract: A ring R is called “left generalized morphic” if for every element a in R, there exists b ∈ R such that l(a)≅ R/Rb, where l(a) denotes the left annihilator of a in R. The aim of this article is to investigate these rings. Several examples are given. They include left morphic rings and left p.p. rings. As applications, some homological dimensions over these rings are defined and studied.
TL;DR: In this paper, the authors studied a class of integral domains characterized by the property that every nonzero finite intersection of principal ideals is a directed union of invertible ideals, and they proved that every directed union is a union of the principal ideals.
Abstract: We study a class of integral domains characterized by the property that every nonzero finite intersection of principal ideals is a directed union of invertible ideals.