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An element of a ring R is called strongly J#-clean provided that it can be written as the sum of an idempotent and an element in J#(R) that commute. In this paper, we characterize the strong J#-cleanness of matrices over projective-free rings. This extends many known results on strongly clean matrices over commutative local rings. [ABSTRACT FROM AUTHOR]

In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group Γ=PSL(2,ℤ[i]) on . Graphical interpretation of amalgamation of the components of Γ is also given. Some elements of and their conjugates over ℚ(i) have different signs in the orbits of the biquadratic field when acted upon by Γ. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Γα, and they form a closed path which is the only closed path in the orbit Γα. We also devise a procedure to obtain ambiguous numbers of the form , where b is a positive integer. [ABSTRACT FROM AUTHOR]

Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces (possibly 핊3) branched over (1,1)-knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3-sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper. [ABSTRACT FROM AUTHOR]

In this paper, a new necessary and sufficient condition for the existence of a Hermitian solution as well as a new expression of the general Hermitian solution to the system of matrix equations A1X=C1 and A3XB3=C3 are derived. The max-min ranks and inertias of these Hermitian solutions with some interesting applications are shown. In particular, the max-min ranks and inertias of the Hermitian part of the general solution to this system are presented. [ABSTRACT FROM AUTHOR]

Li and Zhao studied groups with a few conjugacy classes of non-cyclic subgroups. In this paper we study groups with a few non-cyclic subgroups. In fact, among other things, we give some sufficient conditions on the number of non-cyclic subgroups of a finite group to be solvable. [ABSTRACT FROM AUTHOR]

Motivated by Problem 164 proposed by Y. Berkovich and E. Zhmud' in their book 'Characters of Finite Groups', we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character codegree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ℱℐn-envelopes, -envelopes and ℱℐn-covers, where ℱℐn denotes the class of all n-absolute pure modules for an integer n > 0 or n=∞. We prove that -envelopes and ℱℐn-covers exist over an n-coherent ring R, and ℱℐ1-covers and special ℱℐn-preenvelopes exist over any ring R. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of weak operators and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non-strict monoidal category. It is also shown that this category is not trivial, that is to say, it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra. [ABSTRACT FROM AUTHOR]

In this paper we introduce a class of monomial ideals, called k-decomposable ideals. It is shown that the class of k-decomposable ideals is contained in the class of monomial ideals with linear quotients, and when k is large enough, the class of k-decomposable ideals is equal to the class of ideals with linear quotients. In addition, it is shown that a d-dimensional simplicial complex is k-decomposable if and only if the Stanley-Reisner ideal of its Alexander dual is a k-decomposable ideal, where k ≤ d. Moreover, it is shown that every k-decomposable ideal is componentwise k-decomposable. [ABSTRACT FROM AUTHOR]

Train algebras were introduced by Etherington in 1939 as an algebraic framework for treating genetic problems. The aim of this paper is to study the representations and irreducible representations of power-associative train algebras of rank 4. There are three families of such algebras and for two of them we prove that every irreducible representation has dimension one over the ground field. For the third family we give an example of an irreducible representation of dimension three. [ABSTRACT FROM AUTHOR]

In this paper, the concepts of SGQ-projective, quasi-epi -projective and quasi-epi -projective modules are introduced and some properties of these modules are investigated. Hopfian and co-Hopfian modules are investigated via quasi(-epi) -projective and quasi(-epi) -projective modules. [ABSTRACT FROM AUTHOR]

In this paper, we study the complexity of the dual bases of the Gaussian normal bases of type (n, t), for all n and t = 3, 4, 5, 6, of 픽qn over 픽q and provide conditions under which the complexity of the Gaussian normal basis of type (n, t) is equal to the complexity of the dual basis over any finite field. [ABSTRACT FROM AUTHOR]

Let R be a ring with non-zero identity. The unit graph G(R) of R is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and 픪 is a maximal ideal of R such that |R/픪| = 2, then G(R) is a complete bipartite graph if and only if (R, 픪) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessarily commutative), then G(R) is a complete r-partite graph if and only if (R, 픪) is a local ring and r = |R/픪| = 2n for some n ∈ ℕ or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U(R) and the clique number of G(R) is finite, then R is a finite ring. [ABSTRACT FROM AUTHOR]

Let K be a field and G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms defined by g · x(h) = x(gh) for any g, h ∈ G. Denote by K(G) the fixed field K(x(g) : g ∈ G)G. Noether's problem then asks whether K(G) is rational over K. Let p be an odd prime and let G be a p-group of exponent pe. Assume also that (i) K = p > 0, or (ii) K ≠ p and K contains a primitive pe-th root of unity. In this paper we prove that K(G) is rational over K for the following two types of groups: (1) G is a finite p-group with an abelian normal subgroup H of index p such that H is a direct product of normal subgroups of G of the type Cpb × (Cp)c for some b, c with 1 ≤ b and 0 ≤ c; (2) G is any group of order p5 from the isoclinic families with numbers 1, 2, 3, 4, 8 and 9. [ABSTRACT FROM AUTHOR]

Let G be a group. A subset X of G is said to be non-commuting if xy ≠ yx for any x, y ∈ X with x ≠ y. If |X| ≥ |Y| for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, the bound for the cardinality of a maximal non-commuting set in a finite p-group G is determined, where G is a non-abelian p-group given by a central extension as 1 → ℤpm → G → ℤp× ⋯ × ℤp → 1 and its derived subgroup has order p. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the n-simple braid divisible by the generators xi for all 2 ≤ i ≤ n-2 has trivial simple centralizer. Consequently, the commuting graph defined on the set of simple braids is disconnected. We also prove that the graph has one major component. [ABSTRACT FROM AUTHOR]

In this paper, we study Leibniz algebras with a non-degenerate Leibniz-symmetric -invariant bilinear form B, such a pair is called a quadratic Leibniz algebra. Our first result generalizes the notion of double extensions to quadratic Leibniz algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Leibniz algebra to be a quadratic Leibniz algebra by double extension. [ABSTRACT FROM AUTHOR]

In this paper we consider Hibi rings and Rees rings attached to a poset. We classify the ideal lattices of posets whose Hibi relations are indispensable and the ideal lattices of posets whose Hibi relations form a quadratic Gröbner basis with respect to the rank lexicographic order. Similar classifications are obtained for Rees rings of Hibi ideals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the graded path category associated to a quiver. We investigate all n-differentials on such a category, and also study the associated graded Lie algebra. Moreover, a necessary and sufficient condition is given for the graded path category to admit a DG category structure. [ABSTRACT FROM AUTHOR]

One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A ∩ B=0, then A ⊕ B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the spanning simplicial complex Δs(G) associated to a simple finite connected graph G. We characterize all spanning trees of the uni-cyclic graph Un,m. In particular, we give a formula for computing the Hilbert series and h-vector of the Stanley-Reisner ring k[Δs(Un,m)]. Finally, we prove that the spanning simplicial complex Δs(Un,m) is shifted and hence is shellable. [ABSTRACT FROM AUTHOR]

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A. [ABSTRACT FROM AUTHOR]

In this paper, the extended affine Lie algebra is quantized from three different points of view, which produces three non-commutative and non-cocommutative Hopf algebra structures, and yields other three quantizations by an isomorphism of correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra . [ABSTRACT FROM AUTHOR]

This paper interprets mixed multiplicities of good filtrations as Hilbert-Samuel multiplicities, and shows that (휀1,...,휀m)-superficial sequences in [16] (2007) and superficial sequences in [15] (1973) are weak-(FC)-sequences in [19] (2000). As consequences, we not only obtain generalized results for mixed multiplicities of ideals in [15, 16, 19] but also get an improvement of [19, Theorem 3.4] that seems to account well for the essence of [16, Theorem 1.4]. [ABSTRACT FROM AUTHOR]

In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrödinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module Ig(a,b). [ABSTRACT FROM AUTHOR]

In this paper, we describe the finite groups G in which every maximal chain M3 < M2 < M1 < M0 =G contains a proper subnormal subgroup of G. [ABSTRACT FROM AUTHOR]

Let L be a Lie algebra of Block type over ℂ with basis {Lα,i | α,i ∈ ℤ} and brackets [Lα,i,Lβ,j]=(β(i+1)-α(j+1)) Lα+β,i+j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with ℂ[∂]-basis {Lα(w) | α ∈ ℤ} and λ-brackets [Lα(w)λ Lβ(w)]= (α∂+(α+β)λ) Lα+β(w). Finally, we give a classification of free intermediate series B-modules. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Hua-Like theorem in some kind of semirings. This can be regarded as a generalized version of Hua's theorem from rings to semirings. [ABSTRACT FROM AUTHOR]

In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal Gröbner-Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn. [ABSTRACT FROM AUTHOR]

An ideal I of a ring R is strongly separative provided that for all finitely generated projective R-modules A, B with A=AI and B=BI, if 2A ≅ A ⊕ B, then A ≅ B. We prove in this paper that a regular ideal I of a ring R is strongly separative if and only if each a ∈ 1+I satisfying (1-a)R ∝ r(a) is unit-regular, if and only if each a ∈ 1+I satisfying (1-a2)R ∝ r(a2) is unit-regular, if and only if each a ∈ 1+I satisfying R(1-a)R=R r(a) is unit-regular, if and only if each a ∈ 1+I satisfying R(1-a2)R=R r(a2) is unit-regular. [ABSTRACT FROM AUTHOR]

For any complex parameters a, b, let be the Lie algebra with basis {Li,Hi | i ∈ ℤ} and relations [Li,Lj]=(j-i)Li+j, [Li,Hj]=(a+j+bi)Hi+j and [Hi,Hj]=0. In this paper, we construct the conformal algebra for some a, b and its conformal module of rank one. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center , δ: R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if (i) [δ([x,y]),[x,y]]n = 0 for all x, y ∈ R or (ii) for all x ∈ R, except some specific cases. [ABSTRACT FROM AUTHOR]

Given an involution in a group G, it can be extended in various ways to an involution in the group ring RG, where R is a ring, not necessarily commutative. In this paper nonlinear extensions are considered and necessary and sufficient conditions are given on the group G, its involution, the ring R and the extension for the set of skew-symmetric elements to be commutative and for it to be anticommutative. [ABSTRACT FROM AUTHOR]

Let R be a ring, n, d be fixed non-negative integers, the class of (n,d)-injective left R-modules, and the class of (n,d)-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then if and only if , if and only if m+k(M,N) = 0 for all left R-modules M, N and all k ≥ -1, if and only if m-1(M,N) = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then − ⊗ − is right balanced on by , and investigate the global right -dimension of and the global right -dimension of by right derived functors of − ⊗ −. Some known results are obtained as corollaries. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study C-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that C-flat complexes are actually the complexes of C-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a -cover, where is the class of C-flat complexes. We also give a characterization of complexes of modules in that are defined by Sather-Wagstaff, Sharif and White. [ABSTRACT FROM AUTHOR]

In the present paper, the σ-derivations of ℂ[x±1,y±1] are determined, and their applications are considered. [ABSTRACT FROM AUTHOR]

Bertram Huppert conjectured in the late 1990s that the nonabelian simple groups are determined up to an abelian direct factor by the set of their character degrees. Although the conjecture has been established for various simple groups of Lie type and simple sporadic groups, it is expected to be difficult for alternating groups. In [5], Huppert verified the conjecture for the simple alternating groups An of degree up to 11. In this paper, we continue his work and verify the conjecture for the alternating groups of degrees 12 and 13. [ABSTRACT FROM AUTHOR]

In this paper, we study the cover-avoid property of 픉-injectors on chief factors of a group G for a Fitting class 픉 in some universe 픘 with partial solubility. [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]

Let R=⊕n ≥ 0 Rn be a standard graded ring, 픞 ⊇ ⊕n > 0 Rn an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i ≥ 0, the n-th graded component of the i-th generalized local cohomology module of M and N with respect to 픞 vanishes for all n ≫ 0. Some sufficient conditions are proposed to satisfy the equality . Also, some sufficient conditions are proposed for the tameness of such that or i=픞(M,N), where and 픞(M,N) denote the R+-finiteness dimension and the cohomological dimension of M and N with respect to 픞, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of , where j is the first or last non-minimax level of . [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 establish a structure theorem of naturally ordered rpp semigroups with max-idempotents. Results on naturally ordered semigroups with max-idempotents in the literature are extended and amplified. Some previous results obtained by Blyth, McFadden, Guo and Xie on regular and abundant ordered semigroups are extended to ordered rpp semigroups. [ABSTRACT FROM AUTHOR]

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete characterization is established for a finite local ring whose annihilating-ideal graph is a star graph. [ABSTRACT FROM AUTHOR]

In this paper we characterize all the lexsegment ideals which are normally torsion-free. This will provide a large class of normally torsion-free monomial ideals which are not square-free. Our characterization is given in terms of the ends of the lexsegment. We also prove that, for lexsegment ideals, the property being normally torsion-free is equivalent to the property of the depth function being constant. [ABSTRACT FROM AUTHOR]

Let X be a zero-dimensional scheme in ℙ1 × ℙ1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines. [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 M be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Specs(M) of second submodules of M and the spectrum Specc(M) of coprime submodules of M, study several properties of these spaces and investigate their interplay with the algebraic properties of M. [ABSTRACT FROM AUTHOR]

In this paper, we study an infinite-dimensional Lie algebra ℬq, called the q-analog Klein bottle Lie algebra. We show that ℬq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬq are also determined. [ABSTRACT FROM AUTHOR]

In this paper, we extend the concept of Ako and Oka families to submodules, study the behavior of the extended prime submodule principle and use these concepts to give new proofs of some familiar theorems. [ABSTRACT FROM AUTHOR]

In this paper we give a characterization for certain HNN extensions of subgroup separable groups with normal associated subgroups to be weakly potent. We then apply our result to show that certain HNN extensions of finitely generated nilpotent groups with central associated subgroups are weakly potent. [ABSTRACT FROM AUTHOR]