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We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations and over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied. [ABSTRACT FROM AUTHOR]

In this paper, the derivation algebra and the automorphism group of the twisted N=2 superconformal algebra are completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new example of Hopf group coalgebras by considering Radford's biproduct Hopf algebra in the Turaev category. Furthermore, we find some sufficient and necessary conditions for such Radford's biproduct Hopf algebra to admit quasitriangular structures in the sense of Turaev group coalgebras. [ABSTRACT FROM AUTHOR]

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C3, C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ∗-clean. [ABSTRACT FROM AUTHOR]

In the present paper we describe a class of Φ-Auslander-Yoneda algebras over K[x]/(xn) in terms of quivers with relations, and prove that they are actually cellular algebras in the sense of Graham and Lehrer. [ABSTRACT FROM AUTHOR]

In this paper, we obtain (non-commutative) Gröbner-Shirshov bases for rectangular bands, small extensions of semigroups and Bruck-Reilly extensions of monoids in terms of the general product. [ABSTRACT FROM AUTHOR]

Let β be a Lie algebra of Block type with basis {Lɑ, i | ɑ ... Z, i ... Z+} and relations [Lɑ, i, Lβ, j] = (ɑ -- 1)(j + 1) -- (β -- l)(i + l))Lɑ+β,i+j. In the present paper, the derivation algebra and automorphism group of B are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of β is trivial. [ABSTRACT FROM AUTHOR]

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol). [ABSTRACT FROM AUTHOR]

In this paper, we classify and determine the structure of finite groups whose character degree graphs are empty graphs [ABSTRACT FROM AUTHOR]

Bi-Koszul algebras, including two classes of non-Koszul Artin-Schelter regular algebras of global dimension 4, were a class of graded algebras with non-pure resolutions, introduced in [8]. There is a natural question: can we construct bi-Koszul algebras from algebras with pure resolutions? In this paper, we study this question in terms of normal extensions and Ore extensions. More precisely, we attempt to obtain bi-Koszul algebras from algebras with pure resolutions by these two kinds of extensions. Furthermore, some homological properties of bi-Koszul algebras obtained in such ways are discussed. [ABSTRACT FROM AUTHOR]

In this paper we continue to study the lifting of outer actions of groups and introduce the concept of equivalence of two lifting homomorphisms, maximal lifting homomorphisms and maximal split extensions. Some criteria are obtained. [ABSTRACT FROM AUTHOR]

A well known problem in the theory of envelopes is to characterize those rings having the property that every (finitely generated, finitely presented, simple) module has a projective (pre)envelope. In this paper, we introduce the $\mathcal{S}$-projective modules for an arbitrary class $\mathcal{S}$ of finitely generated modules, and characterize in this general setting those rings for which every module in $\mathcal{S}$ has a projective preenvelope. [ABSTRACT FROM AUTHOR]

In this paper, first we show that the categories of semihypergroups and hypergroups have not free objects. Next we give the notion of weak free semihypergroups for the classes of semihypergroups and extension complete semihypergroups (ECS). Also, we give a necessary and sufficient condition for a semihypergroup for being weak free semihypergroups in the category ECS. Finally, the existence of proper weak free objects in ECS is proved. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of n-isoclinism to the class of all pairs of groups (G,M), where M is a normal subgroup of a group G. The details of that notion are studied and some equivalent conditions are given for two pairs of groups to be n-isoclinic. In addition, the concept of subgroup- and quotient-irreducibility for a pair of groups is defined and we derive how to see whether (G,M) has a pair of subgroups or a pair of factor groups n-isoclinic with (G,M). [ABSTRACT FROM AUTHOR]

Let G be a group and G' be its commutator subgroup. An automorphism α of G is called an IA-automorphism if x-1α (x) ∈ G' for each x ∈ G. The set of all IA-automorphisms of G is denoted by IA(G). A group G is called semicomplete if and only if IA(G)=Inn(G), where Inn(G) is the inner automorphism group of G. In this paper we characterize semicomplete finite p-groups of class 2, give some necessary conditions for finite p-groups to be semicomplete, and characterize semicomplete non-abelian groups of orders p4 and p5. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/〈xn〉 is weak quasi-Armendariz, where R[x] is the polynomial ring over R and 〈xn〉 is the ideal of R[x] generated by xn. Various properties of weak quasi-Armendariz rings are also observed. [ABSTRACT FROM AUTHOR]

Let χ be a character of order N=2t(t ⩾ 4) over a finite field 픽q = 픽pf with p ≡ 7, 9 (mod 16). In this paper, we evaluate explicitly the Gauss sums G(χr) related to the group generated by the character χ. [ABSTRACT FROM AUTHOR]

A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra $\cal A$ such that for any variety V of type τ, we have ${\cal A}\in V$ if and only if every identity satisfied by V has the property p. This is equivalent to $\cal A$ being a generator for the variety determined by all identities of type τ which have property p. Płonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of Płonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3. [ABSTRACT FROM AUTHOR]

We take in this paper an arbitrary class $\mathcal{K}$ of groups as a base, and define a radical property 풫 for which every group in $\mathcal{K}$ is 풫-semisimple. This is called the upper radical property determined by the class $\mathcal{K}$. At the same time, we define a radical property 풫 for which every group in $\mathcal{K}$ is a 풫-radical group. This is called the first lower radical property determined by the class $\mathcal{K}$. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one. [ABSTRACT FROM AUTHOR]

In this paper, Lie bialgebra structures on the extended Schrödinger-Virasoro Lie algebra $\mathcal{L}$ are classified. It is obtained that all the Lie bialgebra structures on $\mathcal{L}$ are triangular coboundary. As a by-product, it is derived that the first cohomology group $H^1(\mathcal{L},\mathcal{L}\otimes\mathcal{L})$ is trivial. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate an additive range-inclusive map from a semiprime ring with idempotents into its maximal right quotient ring. As a corollary to our main result we obtain a simple proof of a result by Brešar. Our approach is based on connecting range-inclusive maps with the theories of local derivations and biderivations. [ABSTRACT FROM AUTHOR]

A class of finite-dimensional Cartan-type Lie superalgebras H(n,m) over a field of prime characteristic is studied in this paper. We first determine the derivation superalgebra of H(n,m). Then we obtain that H(n,m) is restrictable and it is an extension of the Lie superalgebra $\widetilde{H}(n)$. Finally, we prove that H(n,m) is isomorphic to a subalgebra of the restricted Hamiltonian Lie superalgebra $H(n,m,\underline{1})$. [ABSTRACT FROM AUTHOR]

There are three standard weight functions on a linear code viz. the Hamming weight, Lee weight and Euclidean weight. The Euclidean weight function is useful in connection with the lattice constructions, where the minimum norm of vectors in the lattice is related to the minimum Euclidean weight of the code. In this paper, we obtain Singleton's bound for Euclidean codes and introduce maximum Euclidean square distance separable (MESDS) codes. [ABSTRACT FROM AUTHOR]

Let G be a group and let w = w(x1, x2,..., xn) be a word in the absolutely free group Fn on free variables x1, x2,..., xn. The set S(n)(G) of all words w such that the equality w(gσ1, gσ2,..., gσn) = w(g1, g2,..., gn) holds for all g1, g2,..., gn∈G and all permutations σ ∈ Sn is a subgroup of Fn, called the subgroup of n-symmetric words for G. In this paper, the groups S(2)(Dp) and S(3)(Dp) for dihedral groups Dp are determined, where p > 3 is a prime. In particular, it turns out that the groups S(3)(Dp) are not abelian. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of almost idempotent semirings as the semirings with semilattice additive reduct satisfying the identity x + x2 = x2, and characterize eight subclasses of the variety $\mathcal{AI}$ of all almost idempotent semirings corresponding to the eight subvarieties of the variety $\mathcal{NB}$ of all normal bands. Every almost idempotent semiring S is a distributive lattice of rectangular almost idempotent semirings. Given a semigroup F, the semiring Pf(F) of all finite non-empty subsets of F is almost idempotent precisely when F is a band, and in this case, Pf(F) is freely generated by the band F in the variety $\mathcal{AI}$. This semiring Pf(F) is free in a subclass of $\mathcal{AI}$ if and only if F is in the corresponding subvariety of $\mathcal{NB}$. [ABSTRACT FROM AUTHOR]

In this paper, we deal with commutative algebras A satisfying the identity 2β{(xy)2 - x2y2} + γ{((xy)x)y + ((xy)y)x - (y2x)x - (x2y)y} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel and Piacentini-Cattaneo [2]. We prove that if the algebra A admits an identity element, then A is associative. We also prove that there exist trace forms on A. Finally, we prove that if A has a non-degenerate trace form, then A satisfies the identity ((yx)x)x = y((xx)x), a generalization of right alternativity. Our results require characteristic ≠ 2, 3. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. The purpose of this paper is to show that if I and J are projectively equivalent ideals with respect to N, then a sequence x := x1,...,xn of elements of R is an asymptotic sequence over I with respect to N if and only if it is an asymptotic sequence over J with respect to N. Also, it is shown that if R is local, then the lengths of all maximal asymptotic sequences over an ideal I with respect to N are the same. As a consequence, we derive a generalization of Rees' theorem [23]. [ABSTRACT FROM AUTHOR]

Let G be a finite group and p a prime. We say that G is quasisupersoluble (resp., p-quasisupersoluble) if for every non-cyclic chief factor H/K of G (resp., for every non-cyclic chief factor H/K of G of order divisible by p), every automorphism of H/K induced by an element of G is inner. In this paper, we study the structure of quasisupersoluble and p-quasisupersoluble finite groups. [ABSTRACT FROM AUTHOR]

Let G be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with $\omega(\mathcal{A}_G)\leq 57$, where 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2,7). We also determine $\omega(\mathcal{A}_G)$ for all finite minimal simple groups G. [ABSTRACT FROM AUTHOR]

In this paper, we define a new type of configurations as two-sided configurations, and investigate which group properties can be characterized by them. It is proved that for polycyclic torsion free groups, having the same finite quotient sets does not imply the (two-sided) configuration equivalence. We show that isomorphisms and configuration equivalences coincide for some free products of groups and a class of nilpotent groups. [ABSTRACT FROM AUTHOR]

Point algebras introduced by Evans are algebraic systems which capture the essence of multiplications (a,b) · (c,d)=(p,q) defined on the set of all ordered pairs of elements of a set S, where p and q are selected from among a,b,c,d by some well-defined rule. In 1961, Jonsson and Tarski gave an interesting example of a variety of algebras of type 〈2,1,1〉 for illustrating the failure of certain free algebra properties. In this paper, we show that this equational class of algebras, called the JT-variety, is a universal variety of point algebras in the sense that every variety generated by a point algebra is a reduct of the JT-variety. [ABSTRACT FROM AUTHOR]

In this paper, we study the class $\widetilde{\mathcal{FI}}$ of complexes of FP-injective left R-modules. It is shown that the pair $(^{\bot}\widetilde{\mathcal{FI}},\widetilde{\mathcal{FI}})$ is a complete cotorsion pair. If R is a left coherent ring, it is proved that every complex has an $\widetilde{\mathcal{FI}}$-cover. We also introduce the FP-injective dimension of complexes. A special attention is paid to the dimension of homologically bounded above R-complexes over a left coherent ring which has a nice functorial description. [ABSTRACT FROM AUTHOR]

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category $^H_H{\cal M}^H_H$ of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category ${\cal YD}^H_H$ of weak Yetter-Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category $^H_H{\cal M}^H_H$ is constructed by the braiding on ${\cal YD}^H_H$, generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category $_H^H{\cal M}^H_H$ are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent. [ABSTRACT FROM AUTHOR]

Given two numerical semigroups S and S', the distance between S and S' is the cardinality of S\S' plus the cardinality of S'\S. In this paper we study those numerical semigroups S for which there is a symmetric numerical semigroup whose distance to S is one. [ABSTRACT FROM AUTHOR]

Let b(G) denote the largest irreducible character degree of a finite group G. In this paper, we prove that if G is a solvable group which does not involve a section isomorphic to the wreath product of two groups of equal prime order p, and if b(G) < pn, then |G:Op(G)|p < pn. [ABSTRACT FROM AUTHOR]