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In this paper, some methods are given to construct dipolarizations in Kac–Moody algebras as well as their some subalgebras, and an illustrative example is provided. The dipolarizations found in this paper include symmetric and non-symmetric ones. [ABSTRACT FROM AUTHOR]

In this paper, by using the Anick resolution and Gröbner-Shirshov basis for quantized enveloping algebra of type B2, we compute the minimal projective resolution of the trivial module of U q + (B 2) , and as an application we compute the global dimension of U q + (B 2). [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, Q be its maximal right ring of quotients, and C be its extended centroid. Suppose that f (x 1 , ... , x n) is a non-central multilinear polynomial over C, 0   ≠   p   ∈   R , and F, G are two b-generalized derivations of R. In this paper we describe all possible forms of F and G in the case p G (F (f (r)) f (r)) = 0 for all r = (r 1 , ... , r n) in Rn. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char . If R admits a generalized derivation associated with a derivation such that for all , then for all or for all . Moreover, a related result is also obtained. [ABSTRACT FROM AUTHOR]

In an earlier paper, we defined a degree stable Lie algebra, and determined the Lie algebra automorphism group AutLie(S+(2)) of the Lie algebra S+(2). In this paper, we determine the Lie algebra automorphism group AutLie(S(1,0,2)) of the Lie algebra S(1,0,2). [ABSTRACT FROM AUTHOR]

Let 픽q be the finite field with q elements, where q is a power of some prime p. In the classical paper [2], Dickson defined the polynomial $[e_1,e_2,\dots,e_n]:=\det (x_j^{q^{e_i}})_{n\times n} \in {\Bbb F}_q[x_1,x_2,\dots,x_n]$, where e1,..., en are non-negative integers. He observed that any [e1,..., en] is divisible by [0,1,...,n-1], and the quotient is GLn(픽q)-invariant. He then went on to show that one can pick n of these invariants to generate the entire invariant subring of GLn(픽q). In this paper, we answer the following two natural questions: How to express any given [e1,..., en]/[0,1,...,n-1] in terms of the fundamental generators? In general, when does [e1,..., en] divide [f1,..., fn]? The answers are Theorems 1.2 and 1.3, respectively. [ABSTRACT FROM AUTHOR]

It is well known that the notion of ∗-modules plays an important role in the generalization of Morita equivalences. Recently, Wei et al. introduced the concept of ∗n-modules, which induces an equivalence of particular subcategories. In this paper, the dual situation of ∗n-modules, called co-∗n-modules, is considered, and their characterizations and properties are derived. We also present a characterization in terms of category duality, which can be regarded as a generalization of Morita duality. In particular, it is found that the co-selfsmall n-cotilting modules are co-∗n-modules. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of finite groups G under the assumption that certain abelian subgroups of largest possible exponent of prime power order lie in the 픉-hypercenter of G. We improve and extend some results of Ezzat, Shaalan, Elashiry, Asaad, Ramadan, and Buckley. [ABSTRACT FROM AUTHOR]

Let S = {x1,..., xn} be a set of n distinct positive integers and e ≥ 1 an integer. Then we call the n × n matrix (Se) = ((xi, xj)e) having the eth power of the greatest common divisor (xi, xj) of xi and xj as its (i, j)-entry the power greatest common divisor (GCD) matrix on S. A well-known formula for det(Se) when S = {1, ..., n} was given by H.J.S. Smith in 1875/76. In this paper, we consider the power GCD matrices for a UFD along the direction of their structure, determinant and non-singularity, and consequently, we extend the classical result of Smith as well as the results of Beslin and Kassar on UFD given in 1989. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous work [10]. By GAERS-1, we denote the class of generalized abelian exchange rings with stable range 1. In this paper, we first prove that for any ring R ∈ GAERS-1 and any ideal I of R, K0(R/I) is an archimedean ℓ-group, which is a natural generalization of [10, Theorem 5.3]. As applications, we establish explicit characterizations for the K0-simplicity of such rings in the sense of [3], and investigate the norm completeness of their K0-groups. Finally, we characterize the primitive idempotents in R by K0(R) with ordered structure, from which we can further determine completely the structure of K0(R). [ABSTRACT FROM AUTHOR]

Let R be a ring and g(x) a polynomial in C[x], where C=C(R) denotes the center of R. Camillo and Simón called the ring g(x)-clean if every element of R can be written as the sum of a unit and a root of g(x). In this paper, we prove that for a, b ∈ C, the ring R is clean and b-a is invertible in R if and only if R is g1(x)-clean, where g1(x)=(x-a)(x-b). This implies that in some sense the notion of g(x)-clean rings in the Nicholson–Zhou Theorem and in the Camillo–Simón Theorem is indeed equivalent to the notion of clean rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that heightMI > 0. In this paper, there is a definition of the integral closure Na for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences AssR M/(InN)a and AssR (InM)a/(InN)a(n=1,2,...) of associated prime ideals are increasing and ultimately constant for large n. [ABSTRACT FROM AUTHOR]

In this paper, δ-lifting and δ-supplemented modules are defined as generalizations of lifting and supplemented modules. Several properties of these modules are proved. New characterizations of δ-semiperect and δ-perfect rings studied in [9] are obtained using δ-lifting and δ-supplemented modules. [ABSTRACT FROM AUTHOR]

A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that an is regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S. [ABSTRACT FROM AUTHOR]

In this paper, we provide an elementwise comparison between two realizations of affine Lie algebras of type Ãn-1, given respectively by Kac and Moody via central extensions of loop algebras and by Peng and Xiao via root categories of nilpotent representations of cyclic quivers. In particular, we give an explicit description of Heisenberg generators in the realization of Peng and Xiao. [ABSTRACT FROM AUTHOR]

This paper describes briefly a theory of Cartesian decompositions of a set Ω that are preserved by a permutation group G on Ω. If G has a transitive minimal normal subgroup M, then there is a one-to-one correspondence between the Cartesian decompositions preserved by (3 and certain families of subgroups of M, called Cartesian systems. The various types of Cartesian decompositions can be categorised by various types of Cartesian systems. These types arc described. and examples of each are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a pseudo-metric on the space of all m × s matrices with entries from the finite ring Zq, generalizing the classical Lee metric and the array RT metric, and name this pseudo-metric as the generalized Lee-RT pseudo-metric (or the GLRTP-metric). We also obtain some bounds on the parameters of array codes equipped with the GLRTP-metric. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the sheaf 픗픤 of germs of non-commutative holomorphic functions in elements of a finite-dimensional nilpotent Lie algebra 픤, which is a sheaf of non-commutative Fréchet algebras over the character space of 픤. We prove that 픗픤(D) is a localization over the universal enveloping algebra ${\mathcal U}({\mathfrak g})$ whenever D is a polydisk, which in turn allows to describe the Taylor spectrum of a supernilpotent Lie algebra of operators in terms of the transversality. [ABSTRACT FROM AUTHOR]

In this paper we study flexible algebras (possibly infinite-dimensional) satisfying the polynomial identity x(yz) = y(zx). We prove that in these algebras, products of five elements are associative and commutative. As a consequence of this, we get that when such an algebra is a nil-algebra of bounded nil-index, it is nilpotent. Furthermore, we obtain optimal bounds for the index of nilpotency. Another consequence that we get is that these algebras are associative when they are semiprime. [ABSTRACT FROM AUTHOR]

Let R be a ring and α,β be endomorphisms of R. An additive mapping F: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d: R → R such that F(xy)=F(x) α(y) + β(x)d(y) holds for all x, y ∈ R. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (α,β)-derivation F satisfying any one of the properties: (i) [F(x),x]α,β=0, (ii) F([x,y])=0, (iii) F(x ◦ y)=0, (iv) F([x,y])=[x,y]α,β, (v) F(x ◦ y)=(x ◦ y)α,β, (vi) F(xy)- α(xy) ∈ Z(R), (vii) F(x)F(y)- α(xy) ∈ Z(R) for all x, y in an appropriate subset of R. [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]

It is well known that there are several non-equivalent types of prime near-rings which are all equivalent in the case of associative rings. In this paper we introduce various characterizations of prime modules in a zero-symmetric near-ring R. The connection of a prime R-ideal P of a module M and the ideal (P:M) of the near-ring R is also investigated. [ABSTRACT FROM AUTHOR]

Several authors investigate Lie algebras and Lie superalgebras graded by finite root systems. In this paper, we study Leibniz algebras graded by finite root systems and obtain some results in simply-laced cases. [ABSTRACT FROM AUTHOR]

In this paper, we mainly construct the quantum double D(H) of a weak T-coalgebra H and then prove that it is also a weak Hopf group coalgebra. Furthermore, we prove that the category of modules over D(H) is isomorphic to the category of weak Yetter–Drinfeld group modules. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional algebra over an arbitrary field k. Assume that a bounded above derived category D-(ModA) admits a recollement relative to bounded above derived categories of two finite dimensional k-algebras B and C: $$D^{-}(\mbox{Mod\,}B)\ {\scriptsize\begin{array}{c} \;i^{\star}\\[-1.5mm] \longleftarrow\\[-1.4mm] \;i_{\star}\\[-1.5mm] \longrightarrow\\[-1.mm] \,i^{!}\\[-1.7mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}A)\ {\scriptsize\begin{array}{c} \,j_{!}\\[-1.5mm] \longleftarrow\\[-1.2mm] \;j^{\star}\\[-1.5mm] \longrightarrow\\[-1.5mm] \;j_{\star}\\[-1.5mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}C).$$ In this paper, we prove that if there exist M ∈ modA and N ∈ modB such that i⋆(N)=M, then the bounded above derived category D-(ModA[M]) admits a recollement relative to bounded above derived categories of two finite dimensional k-algebras B[N] and C: $$D^{-}(\mbox{Mod\,}B[N])\ {\scriptsize\begin{array}{c} \longleftarrow\\[-1.5mm] \longrightarrow\\[-1.5mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}A[M])\ {\scriptsize\begin{array}{c} \longleftarrow\\[-1.5mm] \longrightarrow\\[-1.5mm] \longleftarrow \end{array}}\ D^{-}(\mbox{Mod\,}C),$$ where A[M] and B[N] are the one-point extensions of A by M and of B by N, respectively. As a consequence, we obtain the main result of Barot and Lenzing [1]. [ABSTRACT FROM AUTHOR]

In this paper we prove that a finite p-solvable group G is solvable if its every conjugacy class size of p′-elements with prime power order equals either 1 or m for a fixed integer m. In particular, G is 2-nilpotent if 4 does not divide every conjugacy class size of 2′-elements with prime power order. [ABSTRACT FROM AUTHOR]

Let (W,S) be a Weyl group of type Bn. In this paper, we find certain nontrivial distinguished involutions in the two-sided cell Ωt of W with a-value $\frac12(n^2-n+4t^2+2t)$ for 1≤ 2t ≤ n and n even (resp., $\frac12(n^2-n+4t^2-2t)$ for 1≤ 2t < n and n odd). Besides, for 1≤ i1< ⋯ < it≤ n-1, let Li1⋯ it be a left cell of W with a-value t2 and R(Li1⋯ it) ={si1,...,sit}. There are only two involutions y(i1,i1)⋯ y(it-1,it-1) sit and y(i1,i1)⋯ y(it,it) in the left cell Li1⋯ it. We prove that the former (resp., latter) is a distinguished involution in the left cell Li1⋯ it for t odd (resp., even). [ABSTRACT FROM AUTHOR]

Linton in 1978 proved that a λ-large subgroup of an abelian p-group A is totally projective if and only if A is totally projective. In this paper it will be shown that a special λ-large subgroup of a CΩ-p-group A is summable if and only if A is summable. [ABSTRACT FROM AUTHOR]

This paper considers Lie bialgebra structures on Lie algebras of Block type and proves that all such Lie bialgebras are triangular coboundary. [ABSTRACT FROM AUTHOR]

Table Algebras (A, B) with distinct non-trivial basis elements a, b ∈ B satisfying either $ab = m \overline{a} + n \overline{b}$ or $ab = m \overline{a} + nb$ for some m, n ∈ ℝ+ are studied in this paper. The case $ab = m \overline{a} + nb$ was partially classified by Arad and Blau, and the case $ab = m \overline{a} + n \overline{b}$ was partially classified by Arad and Cohen. [ABSTRACT FROM AUTHOR]

In the category of ℓ-groups, we introduce the concepts of Hopfian and generalized Hopfian ℓ-groups. An ℓ-group G is called Hopfian if every surjective ℓ-homomorphism f : G → G is an ℓ-isomorphism, and G is called generalized Hopfian if every surjective ℓ-homomorphism f : G → G has a small kernel in G. By an example we show that the class of generalized Hopfian ℓ-groups is a proper subclass of Hopfian ℓ-groups. In this paper, we establish some characterizations for them, which generalize some results in [9]. [ABSTRACT FROM AUTHOR]

The classification of nilpotent Lie algebras of maximal rank was solved by Santharoubane. In the present paper, we prove that the classification of non-degenerate solvable Lie algebras of maximal rank can be obtained from the work of Santharoubane. [ABSTRACT FROM AUTHOR]

Let (A,B,f) be a C-algebra. For any subset N of B, its kernel kerfN is a subset of $\hat {\bf B}_f$. The kernel kerf has been used to obtain important results for C-algebras (see [2] and [4]). In this paper, we study further properties of kerf. In particular, for a transitional C-algebra (A,B,f) and quotient subsets M and N such that N ⊆ M and M is also a C-subset, we prove that $\widehat{({\bf M}/{\bf N})}_{\tilde f_{\bf N}|_{\langle {\bf M}/{\bf N}\rangle}} \cong_x \ker_f {\bf N}/ \ker_f {\bf M}$. In order to show this result, exact sequences in C-algebras are studied. Finally, we present a few examples and applications. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of finite p-groups all of whose subgroups of index p2 are abelian. [ABSTRACT FROM AUTHOR]

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra $\overline{WP_{m,n,s}}$ and its restricted subalgebra $\overline{WP_{m,n,s}}_r$ are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra $\overline{WP_{A_{\cdot n}g_n,A_{\cdot m},A_{\cdot s}}}_B$ and the semi-Lie algebra $\overline{WP_{A_{\cdot n}g_n,A_{\cdot m},A_{\cdot s}}}_{B_{[\,,\,]}}$, where $B=\{1,\partial_1,\partial_2,\partial_{12},\partial_{1}^2,\partial_{2}^2\}$. We prove that the algebra is simple and find all its non-associative algebra derivations. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of dimension filtration of an Artinian module and study a class of Artinian modules, called sequentially co-Cohen–Macaulay modules, which contains strictly all co-Cohen–Macaulay modules. Some characterizations of co-Cohen–Macaulayness in terms of the Matlis duality and of local homology are also given. [ABSTRACT FROM AUTHOR]

In this paper, Kaplansky-type theorems are given to characterize GCD-monoids and valuation monoids. Also, (unique) r-factorable monoids are defined and it is shown that S is a Krull monoid if and only if S is a unique t-factorable (resp., w-factorable) monoid if and only if S is a t-factorable (resp., w-factorable) t-Prüfer monoid. [ABSTRACT FROM AUTHOR]

In this paper, we consider an analogue of the level two homogeneous construction of the affine Kac–Moody algebra $A^{(1)}_{1}$ by vertex operators. We construct modules for the toroidal Lie algebra and the extended toroidal Lie algebra of type A1. We also prove that the module is completely reducible for the extended toroidal Lie algebra. [ABSTRACT FROM AUTHOR]

Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on $so_n\widetilde{({\Bbb C}_Q)}$ are determined. [ABSTRACT FROM AUTHOR]

In this paper, we classify the fine representations and R-groups of quasi-split simple groups, and describe the set of good roots with the help of Yan's method. Part of the results are not new, but the method used here is new and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between each Jacobson-type near-ring and the corresponding matrix near-ring. [ABSTRACT FROM AUTHOR]

The notion of xst-rings was introduced by García and Marín in 1999. In this paper, we characterize Morita-like equivalences for right xst-rings, obtain the universal theory of Morita equivalences, and prove that two right xst-rings R and T are Morita-like equivalent if and only if there is a Morita context between R and T. We also prove that Morita-like equivalences can be realized by the covariant functors Hom and ⊗ for these rings. [ABSTRACT FROM AUTHOR]

The Held group He discovered by Held [10] is a sporadic simple group of order 4030387200 = 210.33.52.73.17. The group He has 11 conjugacy classes of maximal subgroups as determined by Butler [5] and listed in the 픸핋핃픸핊. Held himself determined much of the local structure of He as well as the conjugacy classes of its elements. Thompson calculated the character table of He. In the present paper, we determine the Fischer–Clifford matrices and hence compute the character table of the non-split extension 3·S7, which is a maximal subgroups of He of index 226560 using the technique of Fischer–Clifford matrices. Most of the computations were carried out with the aid of the computer algebra system 픾픸ℙ. [ABSTRACT FROM AUTHOR]

Hypersubstitutions were introduced in [3] as a way of making precise the concepts of hyperidentity and M-hyperidentity. The monoid of hypersubstitutions has been widely studied by many authors. Knowledge of the monoid of hypersubstitutions can be applied to the concept of M-hyperidentities. In this paper, we show that the order of hypersubstitutions of type τ = (3) is 1, 2, 3 or infinite. [ABSTRACT FROM AUTHOR]

Let H be a coalgebra. In this paper, we show that H is a dual quasi-bialgebra if and only if the category ${\cal M}^{H}$ of comodules is a tensor category; and H is a braided dual quasi-bialgebra if and only if ${\cal M}^{H}$ is a braided tensor category. If H is a braided dual quasi-Hopf algebra, it is shown that the antipode of H is inner, i.e., s2(h) = ∑ τ (h1)h2τ-1(h3). [ABSTRACT FROM AUTHOR]

In this paper, we classify metacyclic p-groups of nilpotency class at least three up to isomorphism by giving a canonical presentation for each isomorphism class. [ABSTRACT FROM AUTHOR]

Let K be a complete field with respect to a real non-trivial valuation v, and $\bar{v}$ be the extension of v to an algebraic closure $\overline{K}$ of K. A well-known result of Ostrowski asserts that the limit of a Cauchy sequence of elements of $\overline{K}$ does not always belong to $\overline{K}$ unless $\overline{K}$ is a finite extension of K. In this paper, it is shown that when a Cauchy sequence { bn } of elements of $\overline{K}$ is such that the sequence { [K(bn) : K] } of degrees of the extensions K(bn) / K does not tend to infinity as n approaches infinity, then { bn } has a limit in $\overline{K}$. We also give a characterization of those Cauchy sequences { bn } of elements of $\overline{K}$ whose limit is not in $\overline{K}$, which generalizes a result of Alexandru, Popescu and Zaharescu. [ABSTRACT FROM AUTHOR]

If G is a finite group and ${\cal A}$ is a group of automorphisms of G, then it is known that the matrix near-ring ${\Bbb M}_m(M_{\cal A}(G); G)$ is a subnear-ring of the centralizer near-ring $M_{\cal A}(G^m)$ for every m ≥ 2. Conditions are known under which ${\Bbb M}_m(M_{\cal A}(G);G)$ is a proper subnear-ring of $M_{\cal A} (G^m)$, and if ${\cal A}$ and G are abelian, then conditions are known which imply the equality ${\Bbb M}_m(M_{\cal A}(G); G) =M_{\cal A}(G^m)$. In this paper, we characterize the groups ${\cal A}$ of automorphisms of a cyclic p-group G for which this equality holds. We also show that for every group ${\cal A}$ of automorphisms of a cyclic p-group G, either all the nonzero orbits of G are of unique type or none of the orbits of G is of unique type if p is odd, and there is a third possibility if p=2, namely precisely one of the orbits of G is of unique type. [ABSTRACT FROM AUTHOR]

In this paper, we show that the coset diagram for the action of a linear-fractional group generated by $u:z\rightarrow \frac{z-1}{z}$ and $v:z\rightarrow \frac{-1}{z+1}$ on the rational projective line is transitive and the coset diagram for the action is connected. Using the coset diagram, we show that $u^{3}=v^{3}=1$ are defining relations for the group. [ABSTRACT FROM AUTHOR]

Let v be a henselian valuation of a field K, and $\bar{v}$ be the (unique) extension of v to a fixed algebraic closure $\overline{K}$ of K. For an element $\theta\in \overline{K} \backslash K$, a chain $\theta=\theta_{0},\theta_{1},\dots, \theta_{m}$ of elements of $\overline {K}$ such that θi is of minimal degree over K with the property that $\bar{v}(\theta_{i-1}-\theta_{i}) =\sup\{\bar{v}(\theta_{i-1}-\beta) \mid [K(\beta):K] < [K(\theta_{i-1}):K]\}$ and θm ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each $\theta \in \overline{K} \backslash K$ when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements $\theta\in \overline {K} \backslash K$ for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ. [ABSTRACT FROM AUTHOR]