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Showing papers in "Communications in Algebra in 2012"


Journal ArticleDOI
TL;DR: In this paper, the connection between supercharacter theories and Schur rings was made explicit, and supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12], Hirasaka and Muzychuk [10], and Tamaschke [20] were constructed.
Abstract: Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12], Hirasaka and Muzychuk [10], and Tamaschke [20].

78 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduce the notion of a partial action of a groupoid on a ring as well as a criteria for the existence of a globalization of it, and construct a Morita context associated to a globalizable partial groupoid action.
Abstract: In this article, we introduce the notion of a partial action of a groupoid on a ring as well as we give a criteria for the existence of a globalization of it. We construct a Morita context associated to a globalizable partial groupoid action, and we introduce the notion of a partial Galois extension, which is related to the strictness of this context.

71 citations


Journal ArticleDOI
TL;DR: In this paper, a prime algebra over a commutative ring K with unity, and a multilinear polynomial over K, not central valued on R, is considered.
Abstract: Let R be a prime algebra over a commutative ring K with unity, and let f(x 1,…, x n ) be a multilinear polynomial over K, not central valued on R. Suppose that d is a nonzero derivation of R and G is a nonzero generalized derivation of R such that for all r 1,…, r n ∈ R. If the characteristic of R is different from 2, then one of the following holds: 1. There exists λ ∈C, the extended centroid of R, such that G(x) = λx, for all x ∈ R; 2. There exist a ∈ U, the Utumi quotient ring of R, and λ ∈C = Z(U) such that G(x) = ax + xa + λx, for all x ∈ R, and f(x 1,…, x n )2 is central valued on R

60 citations


Journal ArticleDOI
TL;DR: In this paper, the notion of Hom-Lie color algebras is introduced and homomorphism relations between hom-lie color algesbras are defined and studied, and a multiplier σ on the abelian group Γ is introduced.
Abstract: The aim of this article is to introduce the notion of Hom-Lie color algebras. This class of algebras is a natural generalization of the Hom-Lie algebras as well as a special case of the quasi-hom-Lie algebras. In the article, homomorphism relations between Hom-Lie color algebras are defined and studied. We present a way to obtain Hom-Lie color algebras from the classical Lie color algebras along with algebra endomorphisms and offer some applications. Also, we introduce a multiplier σ on the abelian group Γ and provide constructions of new Hom-Lie color algebras from old ones by the σ-twists. Finally, we explore some general classes of Hom-Lie color admissible algebras and describe all these classes via G–Hom-associative color algebras, where G is a subgroup of the symmetric group S 3.

50 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give a simpler proof of their results and show that they are generalised to Leibniz algebras by Ayupov and Omirov and in a stronger form by Patsourakos.
Abstract: Engel's Theorem has been generalised to Leibniz algebras by Ayupov and Omirov, and in a stronger form by Patsourakos. I give a simpler proof of their results.

47 citations


Journal ArticleDOI
TL;DR: In this article, the singularity of Richardson varieties is studied in terms of the singular locus of Schubert varieties, which is a generalization of the criterion for a Richardson variety to be smooth in the sense of nonvanishing of certain cohomology classes.
Abstract: Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood–Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easily amenable to inductive geometric constructions. In this article, we study the singularities of each type of generalization. Like Richardson varieties, projection varieties are normal with rational singularities. We also study in detail the singular loci of projection varieties in Type A Grassmannians. We use Kleiman's Transversality Theorem to determine the singular locus of any intersection variety in terms of the singular loci of Schubert varieties. This is a generalization of a criterion for any Richardson variety to be smooth in terms of the nonvanishing of certain cohomology classes which has been known by som...

40 citations


Journal ArticleDOI
TL;DR: In this paper, the authors identify Cech cocycles in nonabelian group cohomology with Maurer-Cartan elements in a suitable L ∞-algebra.
Abstract: We identify Cech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L ∞-algebra. Applications to deformation theory are described.

37 citations


Journal ArticleDOI
TL;DR: In this paper, the authors give an example of a clean ring R and an idempotent e ∈ R, such that the corner ring eRe is not clean.
Abstract: We give an example of a clean ring R and an idempotent e ∈ R, such that the corner ring eRe is not clean.

36 citations


Journal ArticleDOI
TL;DR: In this paper, the authors characterized aspects of the reflexive and one-sided idempotent reflexive properties of a ring and showed that the concept of idemophotonically symmetric rings is not left-right symmetric.
Abstract: Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.

35 citations


Journal ArticleDOI
TL;DR: In this paper, the n-prime Γ-hyperideal and n-semiprime Γ -hyper ideal extensions of a semihypergroup were introduced.
Abstract: In this article, we study Γ-semihypergroups introduced recently. We introduce the hyper versions of Green's relations in Γ-semihypergroups and give some related characterizations. We introduce the n-prime Γ-hyperideal and n-semiprime Γ-hyperideal of a Γ-semihypergroup and show that for any integer n ≥ 2, n-prime Γ-hyperideals are a generalization of prime Γ-ideals. We also give the relationship between n-prime Γ-hyperideals and Γ-hyperideal extensions in Γ-semihypergroups. Then, we deal with prime ideal and introduce the notion of prime radical in a Γ-semihypergroup M, and also, we obtain some results and relations among the prime radicals of M and S, where S is the left operator semihypergroup of M.

33 citations


Journal ArticleDOI
TL;DR: In this article, the cohomological dimension coincides with the dimension of the local ring, and it is conjectured that for any i > 0, the homology annihilator is 0.
Abstract: In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider We examine these annihilators for an arbitrary local ring when i is the cohomological dimension of I. When the cohomological dimension coincides with the dimension of the ring, we provide an explicit description of In general, we examine its dimension over R and conjecture that for i > 0.

Journal ArticleDOI
TL;DR: In this article, it was shown that all modules over a left GF-closed ring have Gorenstein flat covers, which is the same as the one we have shown in this paper.
Abstract: We prove that all modules over a left GF-closed ring have Gorenstein flat covers.

Journal ArticleDOI
TL;DR: In this paper, the existence of free groups and free groups in a division ring with center k and a multiplicative group has been investigated and shown to be a special case of the case when D has an involution.
Abstract: Let D be a division ring with center k, and let D † be its multiplicative group. We investigate the existence of free groups in D †, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D †.

Journal ArticleDOI
TL;DR: In this article, the authors established co-t-structure analogues of Beligiannis and Reiten's corresponding results on compactly generated t-structures, but with some important differences.
Abstract: The idea of a co-t-structure is almost ‘dual’ to that of a t-structure, but with some important differences. This note establishes co-t-structure analogues of Beligiannis and Reiten's corresponding results on compactly generated t-structures.

Journal ArticleDOI
TL;DR: In this paper, the concept of prime submodule defined by Raggi et al. has been studied in the context of prime M-ideal, and it has been shown that if a module M is a progenerator in σ[M] then these concepts are equivalent.
Abstract: We consider the concept of prime submodule defined by Raggi et al. [7]. We find equivalent conditions for a module M progenerator in σ[M], with τ M -Gabriel dimension, to have a one-to-one correspondence between the set of isomorphism classes of indecomposable τ-torsion free injective modules in σ[M] and the set of τ-pure submodules prime in M, where τ is a hereditary torsion theory in σ[M]. Also we give a relation between the concept of prime M-ideal given by Beachy and the concept of prime submodule in M. We obtain that if M is progenerator in σ[M], then these concepts are equivalent.

Journal ArticleDOI
TL;DR: In this article, it was shown that for skew groupoid algebras with commutative principal component, the principal component of a groupoid is maximal commutive if and only if it has the ideal intersection property.
Abstract: We show that, if a groupoid graded ring has a grading satisfying a certain nondegeneracy property, then the commutant of the center of the principal component of the ring has the ideal intersection property, that is it intersects nontrivially every nonzero ideal of the ring. Furthermore, we show that for skew groupoid algebras with commutative principal component, the principal component is maximal commutative if and only if it has the ideal intersection property.

Journal ArticleDOI
TL;DR: In this article, the authors consider the construction of integer-valued polynomials over matrix rings with entries in an integral domain and prove that Int(M n (D) is a ring and investigate its structure and ideals.
Abstract: When D is a commutative integral domain with field of fractions K, the ring Int(D) = {f ∈ K[x] | f(D) ⊆ D} of integer-valued polynomials over D is well-understood. This article considers the construction of integer-valued polynomials over matrix rings with entries in an integral domain. Given an integral domain D with field of fractions K, we define Int(M n (D)): = {f ∈ M n (K)[x] | f(M n (D)) ⊆ M n (D)}. We prove that Int(M n (D)) is a ring and investigate its structure and ideals. We also derive a generating set for Int(M n (ℤ)) and prove that Int(M n (ℤ)) is non-Noetherian.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a proper ideal P of R is (n−− 1, n)-weakly prime (n ≤ 2) if 0 ≤ √ 1, √ n−1 a i−1, a i+1 a n ǫ+1, nǫ∈ P for some i ∈ {1, 1, 2, n, n}.
Abstract: Let R be a commutative ring with identity. We say that a proper ideal P of R is (n − 1, n)-weakly prime (n ≥ 2) if 0 ≠ a 1…a n ∈ P implies a 1…a i−1 a i+1…a n ∈ P for some i ∈ {1,…, n}, where a 1,…, a n ∈ R. In this article, we study (n − 1, n)-weakly prime ideals. A number of results concerning (n − 1, n)-weakly prime ideals and examples of (n − 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n − 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n − 1, n)-weakly prime ideals and (n − 1, n)-prime ideals coincide.

Journal ArticleDOI
TL;DR: In this paper, a mapping from ring R into itself satisfying or for all a, b ∈ R is shown to be additive under some conditions on R. Under these conditions, it is shown that δ is additive.
Abstract: Let δ be a mapping from ring R into itself satisfying or for all a, b ∈ R. Under some conditions on R, we show that δ is additive.

Journal ArticleDOI
TL;DR: In this article, the total graph of R, denoted by τ(R), is studied and some basic graph-theoretical properties, such as the domination number, are determined.
Abstract: Let R be a finite commutative ring with 1 ≠ 0. In this article, we study the total graph of R, denoted by τ(R), determine some of its basic graph-theoretical properties, determine when it is Eulerian, and find some conditions under which this graph is isomorphic to Cay(R, Z(R) \ {0}). We shall also compute the domination number of τ(R).

Journal ArticleDOI
TL;DR: In this paper, a new simultaneous decomposition concerning the general matrix quaternity over an arbitrary division ring is presented, and a practical algorithm for the decomposition of the matrix quaternions is also presented, where necessary and sufficient conditions for the existence of the general solutions to the systems of matrix equations and over ℱ are established.
Abstract: In this article, we give a new simultaneous decomposition concerning the general matrix quaternity over an arbitrary division ring ℱ. A practical algorithm for the decomposition of the matrix quaternity is also presented. As applications, we establish some necessary and sufficient conditions for the existence of the general solutions to the systems of matrix equations and over ℱ. In addition, we give the expressions of the general solutions to the systems when the solvability conditions are satisfied. Numerical examples are also given to illustrate the results of this article. Moreover, we mention that the findings of this article extend the some known results in the literature.

Journal ArticleDOI
TL;DR: In this article, the deformed structures and representations of two-parameter quantum groups U r, s and 𝔤 were studied. And an equivalence of the braided tensor categories was established under the assumption rs −1 ǫ = q 2.
Abstract: This article is the sequel to [11] to study the deformed structures and representations of two-parameter quantum groups U r, s (𝔤) associated to the finite dimensional simple Lie algebras 𝔤. An equivalence of the braided tensor categories 𝒪 r, s and 𝒪 q under the assumption rs −1 = q 2 is explicitly established.

Journal ArticleDOI
TL;DR: In this article, a necessary compatibility condition between action and multiplication of a Hopf quasigroup acting on its quasimodule Hopf Quasigroup for a smash product construction is derived, taking the form of the associativity of the action up to an antipodal operation.
Abstract: Definitions of actions of Hopf quasigroups are discussed in the context of Long dimodules and smash products. In particular, Long dimodules are defined for Hopf quasigroups and coquasigroups, and solutions to Militaru's 𝒟-equation are constructed. A necessary compatibility condition between action and multiplication of a Hopf quasigroup acting on its quasimodule Hopf quasigroup for a smash product construction is derived. This necessary condition takes the form of the associativity of the action up to an antipodal operation. In the case of a Hopf quasigroup with a bijective antipode it is simply the associativity of the action, a condition assumed a priori by Klim and Majid in their original construction of smash products of Hopf quasigroups.

Journal ArticleDOI
TL;DR: In this paper, the authors give an upper bound for the Stanley depth of the intersection of two primary monomial ideals Q, Q′, which is reached if Q, q′ are irreducible, ht(Q+Q′) is odd, and, have no common variable.
Abstract: Let I ⊂ J be monomial ideals of a polynomial algebra S over a field. Then the Stanley depth of J/I is smaller or equal to the Stanley depth of . We give also an upper bound for the Stanley depth of the intersection of two primary monomial ideals Q, Q′, which is reached if Q, Q′ are irreducible, ht(Q + Q′) is odd, and , have no common variable.

Journal ArticleDOI
TL;DR: In this article, it was shown that the endomorphism algebra of a maximal rigid object T of the cluster tube is Gorenstein and of finite representation type, as first shown by Vatne.
Abstract: Given a maximal rigid object T of the cluster tube, we determine the objects finitely presented by T. We then use the method of Keller and Reiten to show that the endomorphism algebra of T is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is not 3. We study how this quiver with potential changes when T is mutated. We also provide a derived equivalence classification for the endomorphism algebras of maximal rigid objects.

Journal ArticleDOI
TL;DR: In this article, the structure of the Yoneda algebra E(A): = ǫ-Ext A (𝕜, &#x 1d 55c;) to E(B) was analyzed.
Abstract: Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦2 property, B will also be 𝒦2. We prove the converse of this result.

Journal ArticleDOI
TL;DR: In this paper, weak McCoy rings and weak Armendariz rings are studied and the weak skew McCoy condition is defined. But weak skewMcCoy rings do not have a property close to the weak McCoy condition.
Abstract: In [41], Nielsen proves that all reversible rings are McCoy and gives an example of a semicommutative ring that is not right McCoy. At the same time, he also shows that semicommutative rings do have a property close to the McCoy condition. In this article we study weak McCoy rings as a common generalization of McCoy rings and weak Armendariz rings. Relations between the weak McCoy property and other standard ring theoretic properties is considered. We also study the weak skew McCoy condition, a generalization of the standard weak McCoy condition from polynomials to skew polynomial rings. We resolve the structure of weak skew McCoy rings and obtain various necessary or sufficient conditions for a ring to be weak skew McCoy, unifying and generalizing a number of known McCoy-like conditions in the special cases. Constructing various examples, we classify how the weak McCoy property behaves under various ring extensions. As a consequence we extend and unify several known results related to McCoy rings and Arm...

Journal ArticleDOI
TL;DR: This work introduces a ‘dual’ notion based upon the replacement of the finite groups used in the definition of algebraic entropy, by subgroups of finite index.
Abstract: The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. In the present work we introduce a ‘dual’ notion based upon the replacement of the finite groups used in the definition of algebraic entropy, by subgroups of finite index. The basic properties of this new entropy are established, and a connection to Hopfian groups is investigated.

Journal ArticleDOI
TL;DR: In this article, the structure of a finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup is weakly ℋ-subgroup in G is investigated.
Abstract: Let G be a finite group. A subgroup H of G is called an ℋ-subgroup in G if N G (H) ∩ H x ≤ H for all x ∈ G. A subgroup H of G is called weakly ℋ-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ℋ-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ℋ-subgroups in G. Some recent results are extended and generalized.

Journal ArticleDOI
TL;DR: In this paper, the Hartshorne-Lichtenbaum Vanishing Theorem is shown to fail for the local cohomology module of a local ring (R, 𝔪).
Abstract: Let I be an ideal of a local ring (R, 𝔪) with d = dim R. For the local cohomology module it is a well-known fact that it vanishes for i > d and is an Artinian R-module for i = d. In the case that the Hartshorne–Lichtenbaum Vanishing Theorem fails, that is we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case R is complete we prove—as a technical tool—that for a certain ideal J ⊂ R. Thus, properties of and its Matlis dual might be described in terms of the local cohomology supported in the maximal ideal.