Showing papers in "Communications in Algebra in 2006"
TL;DR: Weak Armendariz rings as mentioned in this paper are a generalization of semicommutative rings and are shown to be weak armendariz if and only if for any n, the n-by-n upper triangular matrix ring T n (R) is weak ARM.
Abstract: We introduce weak Armendariz rings which are a generalization of semicommutative rings and Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak Armendariz if and only if for any n, the n-by-n upper triangular matrix ring T n (R) is weak Armendariz. If R is semicommutative, then it is proven that the polynomial ring R[x] over R and the ring R[x]/(x n ), where (x n ) is the ideal generated by x n and n is a positive integer, are weak Armendariz.
92 citations
TL;DR: In this article, it was shown that strongly clean rings are additive analogs of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute.
Abstract: A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.
76 citations
TL;DR: In this article, the authors define the notions of normal, prime, maximal, and Jacobson radical of a hyperring and by considering these notions, they obtain some results arising from a study of hyperrings.
Abstract: The purpose of this article is to present certain results arising from a study of theory of hyperrings. By a hyperring we mean a Krasner hyperring, that is, a triple (R, +,·) is such that (R, +) is a canonical hypergroup, (R, ·) is a semigroup with a zero 0 where 0 is the scalar identity of (R, +) and · is distributive over + . In this article, we define the notions of normal, prime, maximal, and Jacobson radical of a hyperring and by considering these notions we obtain some results. We define hyperring of fractions and hyper-valuation on a hyperring. For this, as in the classical case, we need a mapping from R onto an ordered group G. Finally, we shall state and prove the Chinese Remainder Theorem for the case of hyperrings.
69 citations
TL;DR: In this article, it was shown that for a semiprime power-serieswise Armendariz ring R with a.c.c on annihilator ideals, the power series ring with an indeterminate x over R has finitely many minimal prime ideals, such that B 1,…,B m ǫ = 0 and B i = A i of R for all i, where A 1, […],A m are all minimal prime ideal ideals of R.
Abstract: In this note we continue to study zero divisors in power series rings and polynomial rings over general noncommutative rings. We first construct Armendariz rings which are not power-serieswise Armendariz, and find various properties of (power-serieswise) Armendariz rings. We show that for a semiprime power-serieswise Armendariz (so reduced) ring R with a.c.c. on annihilator ideals, R[[x]] (the power series ring with an indeterminate x over R) has finitely many minimal prime ideals, say B 1,…,B m , such that B 1… B m = 0 and B i = A i [[x]] for some minimal prime ideal A i of R for all i, where A 1,…,A m are all minimal prime ideals of R. We also prove that the power-serieswise Armendarizness is preserved by the polynomial ring extension as the Armendarizness, and construct various types of (power-serieswise) Armendariz rings.
64 citations
TL;DR: In this article, the ideal structure of the Kauffman monoid is described in terms of the Jones monoid and a purely combinatorial numerical function with linearly ordered ideals.
Abstract: The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kauffman. Borisavljevic, Dosen, and Petric gave a complete proof of its abstract presentation by generators and relations, and suggested the name “Kauffman monoid”. We bring the theory of semigroups to the study of a certain finite homomorphic image of the Kauffman monoid. We show the homomorphic image (the Jones monoid) to be a combinatorial and regular *-semigroup with linearly ordered ideals. The Kauffman monoid is explicitly described in terms of the Jones monoid and a purely combinatorial numerical function. We use this to describe the ideal structure of the Kauffman monoid and two other of its homomorphic images.
62 citations
TL;DR: In this article, the derivation algebra Der and the automorphism group Aut of the twisted Heisenberg-Virasoro algebra ℒ have been given, as well as the corresponding automorphisms.
Abstract: In this article, we give the derivation algebra Der ℒ and the automorphism group Aut ℒ of the twisted Heisenberg–Virasoro algebra ℒ.
61 citations
TL;DR: In this paper, the zero-divisor graph of R with respect to I, denoted by Γ I (R), is the graph whose vertices are the set {x ∊ R\I|xy∊ I for some y∊ R/I} with distinct vertices x and y adjacent if and only if xy ∊ I.
Abstract: Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ I (R), is the graph whose vertices are the set {x ∊ R\I | xy ∊ I for some y ∊ R\I} with distinct vertices x and y adjacent if and only if xy ∊ I. In the case I = 0, Γ0(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ I (R) ≅ Γ J (S) and Γ(R/I) ≅ Γ(S/J). We also discuss when Γ I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ I (R).
58 citations
TL;DR: In this article, a poset A is defined as a pomonoid with a right S-action (a,s) that is monotone in both arguments and satisfies the conditions a(st) = (as)t and a(1) = 1.
Abstract: Let S be a partially ordered monoid, or briefly, pomonoid. A right S-poset (often denoted A S ) is a poset A together with a right S-action (a,s)↝ as that is monotone in both arguments and that satisfies the conditions a(st) = (as)t and a1 = 1 for all a ∊ A, s,t ∊ S. Left S-posets S B are defined analogously, and the left or right S-posets form categories, S-POS and POS-S, whose morphisms are the monotone maps that preserve the S-action. In these categories, as in the category POS of posets, the monomorphisms and epimorphisms are the injective and surjective morphisms, respectively, but the embeddings and quotient maps have stronger properties; in particular, an embedding is a monomorphism that is also an order embedding. A tensor product A S ⊗ S B exists (a poset) that has the customary universal property with respect to balanced, bi-monotone maps from A × B into posets. Various flatness properties of A S can be defined in terms of the functor A S ⊗ − from S-POS into POS. More specifically, a...
54 citations
TL;DR: In this paper, the authors define a group as strongly bounded if every isometric action on a metric space has bounded orbits, which is equivalent to the so-called uncountable strong cofinality.
Abstract: We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently initiated by Bergman. Our main result is that G I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that ω1-existentially closed groups are strongly bounded.
53 citations
TL;DR: In this article, it was shown that Hopf group coalgebras and group-algebraic groups can be classified into a symmetric monoidal category, which is called the Turaev category.
Abstract: We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter–Drinfeld modules over Hopf group-coalgebras using the center construction.
52 citations
TL;DR: In this paper, a necessary and sufficient condition for the pair (α, δ) such that the polynomial ring A[x] has the Poisson bracket for all a, b ∊ ǫ ∊ A and construct a class of poisson algebras including the coordinate rings of Poisson 2-×-2-matrices and Poisson symplectic 4-space.
Abstract: Let A be a Poisson algebra with Poisson bracket {·, ·} A and let α,δ be linear maps from A into itself. Here we find a necessary and sufficient condition for the pair (α,δ) such that the polynomial ring A[x] has the Poisson bracket for all a, b ∊ A and construct a class of Poisson algebras including the coordinate rings of Poisson 2 × 2-matrices and Poisson symplectic 4-space.
TL;DR: Bovdi et al. as discussed by the authors showed that the symmetric elements (RG) of an R-linear extension of an involution defined on a commutative ring form a group ring.
Abstract: Let R be a commutative ring, G a group, and RG its group ring. Let ϕ: RG → RG denote the R-linear extension of an involution ϕ defined on G. An element x in RG is said to be symmetric if ϕ (x) = x. A characterization is given of when the symmetric elements (RG)ϕ of RG form a ring. For many domains R it is also shown that (RG)ϕ is a ring if and only if the symmetric units form a group. The results obtained extend earlier work of Bovdi (2001), Bovdi et al. (1996), Bovdi and Parmenter (1997), Broche Cristo (2003, to appear), Giambruno and Sehgal (1993), and Lee (1999), who dealt with the case that ϕ is the involution * mapping g ∈ G onto g−1.
TL;DR: In this paper, the set of equivalence classes of topological Lie algebras over topological fields of characteristic zero is described as a disjoint union of affine spaces with translation group H 2 (, ())[S], where [S] denotes the equivalence class of the continuous outer action S : i.e.
Abstract: In this article we extend and adapt several results on extensions of Lie algebras to topological Lie algebras over topological fields of characteristic zero. In particular, we describe the set of equivalence classes of extensions of the Lie algebra by the Lie algebra as a disjoint union of affine spaces with translation group H 2(, ())[S], where [S] denotes the equivalence class of the continuous outer action S : → der sp;. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to a crossed module whose obstruction class in H 3(, ()) S is the characteristic class of the corresponding crossed module. The correspondence between crossed modules and extensions further leads to a description of -extensions of in terms of certain ()-extensions of a Lie algebra which is an extension of by /(). We discuss several types of examples, describe applications to Lie algebras of vect...
TL;DR: In this paper, Bulacu et al. showed that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic, and they proved that the category of finite-dimensional left Yetter Drinfeld modules is rigid, and then they computed explicitly the canonical isomorphisms in.
Abstract: We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra H are isomorphic. We prove also that the category of finite dimensional left Yetter-Drinfeld modules is rigid, and then we compute explicitly the canonical isomorphisms in . Finally, we show that certain duals of H 0, the braided Hopf algebra (introduced in Bulacu and Nauwelaerts, 2002; Bulacu et al., 2000) are isomorphic as braided Hopf algebras if H is a finite dimensional triangular quasi-Hopf algebra. Communicated by M. Takeuchi.
TL;DR: For weakly Laskerian modules, it was shown in this paper that the set of associated primes of the local cohomology module is finite for all i ≤ 0.
Abstract: The notion of weakly Laskerian modules was introduced recently by the authors. Let R be a commutative Noetherian ring with identity, 𝔞 an ideal of R, and M a weakly Laskerian module. It is shown that if 𝔞 is principal, then the set of associated primes of the local cohomology module is finite for all i ≥ 0. We also prove that when R is local, then is finite for all i ≥ 0 in the following cases: (1) dim R ≤ 3, (2) dim R/𝔞 ≤ 1, (3) M is Cohen-Macaulay, and for any ideal 𝔟, with l = grade(𝔟, M), is weakly Laskerian.
TL;DR: In this article, the Miyashita-Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable extensions, which yields new characterizations of separable and H-separable extensions.
Abstract: A ring extension A ‖ B is depth two if its tensor-square satisfies a projectivity condition w.r.t. the bimodules A A B and B A A . In this case the structures (A ⊗ B A) B and End B A B are bialgebroids over the centralizer C A (B) and there is a certain Galois theory associated to the extension and its endomorphism ring. We specialize the notion of depth two to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse, we observe that normal Hopf subalgebras over a field are depth two extensions. A generalized Miyashita–Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable extensions, which yields new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius.
TL;DR: In this article, a commutative unital ring and E a unital R-module are shown to give minimal ring extensions of R which are not isomorphic as R-algebras.
Abstract: Let R be a commutative unital ring and E a unital R-module. Then the canonical injective ring homomorphism from R into the idealization R(+) E is a minimal ring homomorphism if and only if E is a simple R-module. For E nonzero, R(+)E is not (R-algebra isomorphic to) an overring of R. If E 1 and E 2 are nonisomorphic simple R-modules, then R(+) E 1 and R(+) E 2 give minimal ring extensions of R which are not isomorphic as R-algebras. The ring of dual numbers over R is a minimal ring extension of R ⇔ R × R is a minimal ring extension of R ⇔ R is a field.
TL;DR: In this paper, it was shown that there exists a unique pair (E ,) where E is an Archimedean vector lattice and ǫ is a symmetric lattice s-morphism.
Abstract: Let s ∊ {2.3,…} and E be an Archimedean vector lattice. We prove that there exists a unique pair (E ,), where E is an Archimedean vector lattice and :E× ··· ×E (s times) → E is a symmetric lattice s-morphism, such that for every Archimedean vector lattice F and every symmetric lattice s-morphism T:E × ··· × E (s times) → F, there exists a unique lattice homomorphism T :E → F such that T = T ○. We give two approaches to construct (E ,) based on f-algebras and functional calculus, respectively, provided that E is also uniformly complete.
TL;DR: In this paper, it was shown that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity of the identity.
Abstract: We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity.
TL;DR: In this article, the authors revisited finite racks and quandles using a perspective based on permutations, which can aid in the understanding of the structure and recover old results and prove new ones.
Abstract: We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples. Communicated by M. Dixon.
TL;DR: In this paper, the equations and components of the jet schemes of a monomial subscheme of affine space were explicitly computed from an algebraic perspective, and they were shown to be polynomially equivalent.
Abstract: We explicitly compute the equations and components of the jet schemes of a monomial subscheme of affine space from an algebraic perspective.
TL;DR: In this paper, the degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ∊cd (G).
Abstract: Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(G) is the graph whose set of vertices is the set of primes that divide degrees in cd (G), with an edge between p and q if pq divides a for some degree a ∊ cd (G). We determine the graph Δ(G) for the finite simple groups of types A l(q) and 2 A l (q 2), that is, for the simple linear and unitary groups.
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TL;DR: The Galois Comodules as discussed by the authors generalize the notion of Galois corings and have common properties with tilting (co)modules and have been shown to be generators.
Abstract: Generalizing the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring 𝒞 for which P A is finitely generated and projective and the evaluation map μ𝒞:Hom 𝒞 (P, 𝒞) ⊗ S P → 𝒞 is an isomorphism (of corings) where S = End 𝒞 (P). It has been observed that for such comodules the functors − ⊗ A 𝒞 and Hom A (P, −) ⊗ S P from the category of right A-modules to the category of right 𝒞-comodules are isomorphic. In this note we use this isomorphism related to a comodule P to define Galois comodules without requiring P A to be finitely generated and projective. This generalises the old notion with this name but we show that essential properties and relationships are maintained. Galois comodules are close to being generators and have common properties with tilting (co)modules. Some of our results also apply to generalised Hopf Galois (coalgebra Galois) extensions.
TL;DR: In this paper, a crossed module representing the cocycle was constructed for a simple complex Lie algebra, and the goal was to generate H 3( ∆; ∆, ∆) for the problem.
Abstract: The goal of this article is to construct a crossed module representing the cocycle 〈[,],〉 generating H 3(;ℂ) for a simple complex Lie algebra .
TL;DR: Weak-injective R-modules as discussed by the authors are an envelope class over any domain, giving a partial answer to the existence of envelope classes in the hierarchy of injective and divisible modules.
Abstract: Pure-injective and RD-injective R-modules over domains R have been investigated by many authors. We introduce another class of R-modules, called weak-injective modules, which turn out to be useful in addressing several unanswered questions between the two classes of modules. We also find that this class is an envelope class over any domain, giving a partial answer to the existence of envelope classes in the hierarchy of injective and divisible modules. Communicated by I. Swanson.
TL;DR: The description of Galois points with respect to a Hermitian curve is given in this article, which suggests that Yoshihara's theory needs modifying if the characteristic of the ground field is positive.
Abstract: The description of Galois points with respect to a Hermitian curve is given, which suggests that Yoshihara's theory of Galois points needs modifying if the characteristic of the ground field is positive.
TL;DR: In this paper, the structure of a finite group G under the assumption that certain subgroups of prime power orders belong to the subgroup K of the group G to be investigated.
Abstract: A subgroup K of a finite group G is called an ℋ-subgroup of G if the following condition is satisfied: The set of all ℋ-subgroups of a finite group G will be denoted by ℋ(G). In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of prime power orders belong to ℋ(G). Communicated by M. Dixon
TL;DR: In this paper, it was shown that an associative ring R is Iwanaga-Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero is orthogonal complement of each other with respect to the extended functor.
Abstract: The main purpose of this article is to present some applications of the notion of Gorenstein injective dimension of complexes over an associative ring. Among the applications, we give some new characterizations of Iwanaga–Gorenstein rings. In particular, we show that an associative ring R is Iwanaga–Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero and the class of complexes of finite projective dimension are orthogonal complement of each other with respect to the ‘Ext’ functor.
TL;DR: In particular, it was shown in this article that the inner derivation superalgebra of the odd contact Lie superalgebras of KO(n, n−+−1, t ) is Abelian.
Abstract: The finite-dimensional odd contact Lie superalgebras KO(n, n + 1, t ) over a field of prime characteristic are studied, where n is a positive integer and t is an n-tuple of non-negative integers. In particular, it is proven that KO(n, n + 1, t ) is simple and has no non-singular associative bilinear forms. Moreover, an explicit description of the derivation superalgebra of KO(n, n + 1, t ) is given, and as a consequence it is shown that the outer derivation superalgebra of KO(n, n + 1, t ) is Abelian of dimension . Communicated by K. Misra.
TL;DR: In this paper, the p-nilpotency of a group for which every maximal subgroup of its Sylow p-subgroups is s-semipermutable for some prime p was investigated.
Abstract: A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with (|H|,|K|) = 1 H is said to be s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p,|H|) = 1 In this article, we investigate the p-nilpotency of a group for which every maximal subgroup of its Sylow p-subgroups is s-semipermutable for some prime p We generalize some recent theorems in Guo and Shum (2003) Communicated by M Dixon