Search

An abelian group A is called a T I -group if every associative ring with the additive group A is filial. The filiality of a ring R means that the ring R is associative and all ideals of any ideal of R are ideals in R. In this paper, torsion-free T I -groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition T I implies the indecomposability and homogeneity. The paper contains constructions of 2 ℵ 0 such groups of any rank from 2 to 2 ℵ 0 which are pairwise non-isomorphic. [ABSTRACT FROM AUTHOR]

The enhanced power graph P E (S) of a semigroup S is a simple graph whose vertex set is S and two vertices x , y ∈ S are adjacent if and only if x , y ∈ ⟨ z ⟩ for some z ∈ S , where ⟨ z ⟩ is the subsemigroup generated by z. In this paper, we first describe the structure of P E (S) for an arbitrary semigroup S , and then discuss the connectedness of P E (S). Further, we characterize the semigroup S in the cases when P E (S) is separately a complete, bipartite, regular, tree and null graph. The planarity, together with the minimum degree and independence number, of P E (S) is also investigated. The chromatic number of a spanning subgraph, i.e., the cyclic graph, of P E (S) is proved to be countable. In the final part of this paper, we construct an example of a semigroup S such that the chromatic number of P E (S) need not be countable. [ABSTRACT FROM AUTHOR]

For a nilpotent group G without π -torsion, and x , y ∈ G , if x n = y n for a π -number n , then x = y ; if x m y n = y n x m for π -numbers m , n , then x y = y x. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let m be a given positive integer and A a complex square matrix satisfying that (i) all eigenvalues of A are nonnegative, and (ii) rank ⁡ A 2 = rank ⁡ A ; then A has a unique m -th root X with rank ⁡ X 2 = rank ⁡ X , all eigenvalues of X are nonnegative, and moreover there is a polynomial f (λ) with X = f (A). In addition, let A and B be complex n × n matrices with all eigenvalues nonnegative, and rank ⁡ A 2 = rank ⁡ A , rank ⁡ B 2 = rank ⁡ B ; then (i) A = B when A r = B r for some positive integer r , and (ii) A B = B A when A s B t = B t A s for two positive integers s and t. [ABSTRACT FROM AUTHOR]

A graph G is called triangle-free if G does not contain any triangle as its induced subgraph. Let G n be the set of triangle-free graphs of order n each of which has three positive eigenvalues. In this paper, we find 20 specific graphs in G n , each of which has nullity no more than 2, and we show that in terms of three graph transformations all the other graphs of G n can be constructed from these 20 specific graphs. Hence, we completely characterize the triangle-free graphs with exactly three positive eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient condition under which an involution monoid generates a variety with continuum many subvarieties. According to this result, several involution J -trivial monoids are shown to generate varieties with continuum many subvarieties. These examples include Rees quotients of free involution monoids, Lee monoids with involution, and Straubing monoids with involution. [ABSTRACT FROM AUTHOR]

Let R be a ring (not necessarily commutative) with identity 1 and let CL (R) be its clean graph. In this paper, we investigate the genus number of the compact Riemann surface in which CL (R) can be embedded and explicitly determine all commutative rings R (up to isomorphism) such that CL (R) has genus at most two. It is shown that for any Artinian ring R , CL (R) is a projective graph if and only if R is isomorphic to F 4. Furthermore, we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two. [ABSTRACT FROM AUTHOR]

A finite group G is said to be a B (n , k) group if for any n -element subset { a 1 , ... , a n } of G , | { a i a j   |   1 ≤ i , j ≤ n } | ≤ k. It is of interest to characterize the structure of B (n , k) groups for n (n + 1) / 2 ≤ k ≤ n 2 − 1. The B (5 , k) groups for 15 ≤ k ≤ 19 have been investigated by several authors. In this paper, we give a complete characterization of B (5 , 20) 2-groups by showing there are five classes of such groups which are nontrivial and nonabelian. [ABSTRACT FROM AUTHOR]

In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence. [ABSTRACT FROM AUTHOR]

Suppose that L is a Lie superalgebra over a field F of characteristic different from 2 and 3. In this paper, the so-called 5-sequence of cohomology for a central extension of L is constructed and is proved to be exact. Moreover, the multiplier of L is proved to be isomorphic to the second cohomology group with coefficients in the trivial module of L. Finally, an upper bound of the superdimension of the second cohomology group is given in the situation when L is nilpotent and finite-dimensional. [ABSTRACT FROM AUTHOR]

A signed graph G ˙ = (G , σ) is a graph G = (V (G) , E (G)) with vertex set V (G) and edge set E (G) , together with a function σ : E → { + 1 , − 1 } assigning a positive or negative sign to each edge. In this paper, we present a more elementary proof for the matrix-tree theorem of signed graphs, which is based on the relations between the incidence matrices and the Laplcians of signed graphs. As an application, we also obtain the results of Monfared and Mallik about the matrix-tree theorem of graphs for signless Laplacians. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and n be a positive integer. In this paper, we introduce and investigate a new subclass of ϕ - n -absorbing primary ideals, which are called ϕ - (n , N) -ideals. Let ϕ : I (R) → I (R) ∪ { ∅ } be a function, where I (R) denotes the set of all ideals of R. A proper ideal I of R is called a ϕ - (n , N) -ideal if x 1 ⋯ x n + 1 ∈ I \ ϕ (R) and x 1 ⋯ x n ∉ I imply that the product of x n + 1 with (n − 1) of x 1 , ... , x n is in 0 for all x 1 , ... , x n + 1 ∈ R. In addition to giving many properties of ϕ - (n , N) -ideals, we also use the concept of ϕ - (n , N) -ideals to characterize rings that have only finitely many minimal prime ideals. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the electrical Lie algebra e D 5 is isomorphic to the semidirect product of s p 4 and a 2-step nilpotent Lie algebra. Furthermore, we classify the irreducible highest weight modules for e D 5 . [ABSTRACT FROM AUTHOR]

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective. [ABSTRACT FROM AUTHOR]

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph E A G (R) associated to a commutative ring R , denoted by dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) , respectively. Here we prove the finiteness conditions of dim M (E A G (R)) and dim + (E A G (R)). In addition, we characterize dim M (E A G (R)) , dim + (E A G (R)) and res (E A G (R)) for artinian rings and the direct product of rings. [ABSTRACT FROM AUTHOR]

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph Γ of order 2 p 3 , p a prime, it is well known that Γ is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group Aut (Γ) acting unfaithfully on at least one bipart of Γ , which shows that there is only one infinite family of such graphs with valency p 2. The graphs of this kind have been determined when Aut (Γ) acts faithfully and primitively on at least one bipart of Γ , and thus there is only one remaining case for classifying such graphs of valency p 2 , Aut (Γ) acting faithfully and imprimitively on both biparts of Γ , which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order 2 p 3 with valency p 2. [ABSTRACT FROM AUTHOR]

In this paper, we describe multiplicative derivations on the set of all rank- s matrices of M n (K) over a field K with a relatively small integer s. Concretely, for fixed integers n , s satisfying 1 ≤ s ≤ n / 2 and n ≥ 2 , we prove that if a map δ : M n (K) → M n (K) satisfies δ (x y) = δ (x) y + x δ (y) for any two rank- s matrices x , y ∈ M n (K) , then there exists a derivation D of M n (K) such that δ (x) = D (x) for each rank- k matrix x ∈ M n (K) with 0 ≤ k ≤ s. As an application, we prove that a multiplicative derivation on a special subset of M n (K) must be a derivation. [ABSTRACT FROM AUTHOR]

A finite p -group G is called an A t -group if t is the minimal non-negative integer such that all subgroups of index p t of G are abelian. The finite p -groups G with H ′ = G ′ for all A 2 -subgroups H of G are classified completely in this paper. As an application, a problem proposed by Berkovich is solved. [ABSTRACT FROM AUTHOR]

The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph X of a finite group G , the canonical double covering of X is the direct product X × K 2. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on Z p × Z q and Z 2 n are investigated, where p and q are odd prime numbers. [ABSTRACT FROM AUTHOR]

A set { a 1 , a 2 , ... , a m } of positive integers is called a Diophantine m -tuple if a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m. Let { a , b , c } be the Diophantine triple with c > max { a , b }. In this paper, we find the condition for the reduction of third element c , and using this result, we prove the extendibility of Diophantine pair { F k − 1 F k + 1 , F k − 2 F k + 2 } , where F n is the n -th Fibonacci number. [ABSTRACT FROM AUTHOR]

Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form. [ABSTRACT FROM AUTHOR]

Let p be a prime. In this paper, a complete classification of edge-transitive N -covers of a cubic symmetric graph of order 2 p is given for the case when N is a two-generator 2-group whose derived subgroup is either isomorphic to Z 2 3 or generated by at most two elements. As an application, it is shown that 11 is the smallest value of n for which there exist infinitely many cubic semisymmetric graphs with order of the form 2 n p. [ABSTRACT FROM AUTHOR]

The Lie algebra s l 2 (C) may be regarded in a natural way as a subalgebra of the infinite-dimensional Virasoro Lie algebra, so it is natural to consider connections between the representation theory of the two algebras. In this paper, we explore the restriction to s l 2 (C) of certain induced modules for the Virasoro algebra. Specifically, we consider Virasoro modules induced from so-called polynomial subalgebras, and we show that the restriction of these modules results in twisted versions of familiar modules such as Verma modules and Whittaker modules. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of association schemes by using pairs of subspaces of vector spaces and determine their full automorphism groups. [ABSTRACT FROM AUTHOR]

In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules. Namely, we introduce the right (resp., left) n-trivial extension of a category by a family of endofunctors. Among other results, projective, injective and flat objects of this category are characterized, and two applications are presented at the end of this paper. We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring. [ABSTRACT FROM AUTHOR]

The goal of the present paper is to provide a gallery of small directed strongly regular graphs. For each graph of order n ≤ 12 and valency k < n/2, a diagram is depicted, its relation to other small directed strongly regular graphs is revealed, the full group of automorphisms is described, and some other nice properties are given. To each graph a list of interesting subgraphs is provided as well. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a product structure on a 3-Bihom-Lie algebra, which is a Nijenhuis operator with some conditions. We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras. Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition. Similarly, we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures. Finally, we establish the relation between a complex structure and a product structure. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, M be an R -module, L (M) denote the set of all submodules of M and G ⊆ L (M) \ { 0 M }. For any submodule N of M , we set G V d (N) = { K ∈ G : K ⊆ N } and G ζ d (M) = { G V d (N) : N ∈ L (M) }. Consider χ ⊆ L (R) \ { R } , where L (R) is the set of all ideals of R. We set χ V (I) = { J ∈ χ : I ⊆ J } and χ ζ (R) = { χ V (I) : I ∈ L (R) } for any ideal I of R. In this paper, we investigate when, for arbitrary χ and G as above, χ ζ (R) and G ζ d (M) form a topology and a semimodule, respectively. We investigate the structure of G ζ d (M) in the case that it is a semimodule. [ABSTRACT FROM AUTHOR]

Let (A , B , C) be a recollement of extriangulated categories. In this paper we introduce the global dimension and extension dimension of extriangulated categories, and give some upper bounds of global dimensions and extension dimensions of the categories involved in (A , B , C) , which give a simultaneous generalization of some results in recollements of abelian categories and triangulated categories. [ABSTRACT FROM AUTHOR]

Let G n ([ − 1 ] i) denote the set of all connected graphs on n vertices having distance eigenvalue − 1 of multiplicity i. By using the distribution of the third largest distance eigenvalue and the second least distance eigenvalue of a connected graph, in this paper we completely characterize the graphs in G n ([ − 1 ] i) , where i = n − 1 , n − 2 , n − 3 or n − 4. [ABSTRACT FROM AUTHOR]

In this paper, we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction, silting reduction and τ -tilting reduction. More precisely, if an abelian category A admits a recollement relative to abelian categories A ′ and A ′ ′ , which is denoted by (A ′ , A , A ′ ′ , i ∗ , i ∗ , i ! , j ! , j ∗ , j ∗) , then the assignment C ↦ j ∗ (C) defines a bijection between wide subcategories in A containing i ∗ (A ′) and wide subcategories in A ′ ′. Moreover, a wide subcategory C of A containing i ∗ (A ′) admits a new recollement relative to A ′ and j ∗ (C) which is induced from the original recollement. [ABSTRACT FROM AUTHOR]

Let be an abelian category and M   ∈ A. Then M is called a (D 4) -object if, whenever A and B are subobjects of M with M   =   A   ⊕   B and f : A → B is an epimorphism, Ker ⁡ f is a direct summand of A. In this paper we give several equivalent conditions of (D 4) -objects in an abelian category. Among other results, we prove that any object M in an abelian category is (D 4) if and only if for every subobject K of M such that K is the intersection K 1 ∩ K 2 of perspective direct summands K 1 and K 2 of M with M   =   K 1   +   K 2 , every morphismr φ :   M   →   M / K can be lifted to an endomorphism θ : M → M in E n d A (M). [ABSTRACT FROM AUTHOR]

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature. [ABSTRACT FROM AUTHOR]

The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph X of a finite group G , the canonical double covering of X is the direct product X × K 2. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on Z p × Z q and Z 2 n are investigated, where p and q are odd prime numbers. [ABSTRACT FROM AUTHOR]

A set { a 1 , a 2 , ... , a m } of positive integers is called a Diophantine m -tuple if a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m. Let { a , b , c } be the Diophantine triple with c > max { a , b }. In this paper, we find the condition for the reduction of third element c , and using this result, we prove the extendibility of Diophantine pair { F k − 1 F k + 1 , F k − 2 F k + 2 } , where F n is the n -th Fibonacci number. [ABSTRACT FROM AUTHOR]

Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form. [ABSTRACT FROM AUTHOR]

Herzog, Hibi, Hreindóttir et al. introduced the class of closed graphs, and they proved that the binomial edge ideal JG of a graph G has quadratic Gröbner bases if G is closed. In this paper, we introduce the class of weakly closed graphs as a generalization of the closed graph, and we prove that the quotient ring S/JG of the polynomial ring S   =   K [ x 1 , ... , x n , y 1 , ... , y n ] with K a field and n   =     | V (G) | is F-pure if G is weakly closed. This fact is a generalization of Ohtani's theorem. [ABSTRACT FROM AUTHOR]

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra Λ such that Λ has non-trivial open Frobenius structures. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, I an ideal of R and k ≥ 2 a fixed integer. The ideal-based k -zero-divisor hypergraph H I k (R) of R has vertex set Z I (R , k) , the set of all ideal-based k -zero-divisors of R , and for distinct elements x 1 , x 2 , ... , x k in Z I (R , k) , the set { x 1 , x 2 , ... , x k } is an edge in H I k (R) if and only if x 1 x 2 ⋯ x k ∈ I and the product of the elements of any (k − 1) -subset of { x 1 , x 2 , ... , x k } is not in I. In this paper, we show that H I 3 (R) is connected with diameter at most 4 provided that x 2 ∉ I for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of H I k (R) when R is a product of finite fields. Finally, we find some necessary conditions for a finite ring R and a nonzero ideal I of R to have H I 3 (R) planar. [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group G. In this paper, we investigate the McKay matrix W V of H for tensoring with the 2-dimensional indecomposable H -module V := M (2 , 0). It turns out that the characteristic polynomial, eigenvalues and eigenvectors of W V are related to the character table of the finite group G and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for W V by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of W V and give all eigenvectors of each eigenvalue for W V when G is a dihedral group of order 4 N + 2. [ABSTRACT FROM AUTHOR]

For a ring R (not necessarily commutative) with identity, the comaximal graph of R , denoted by Ω (R) , is a graph whose vertices are all the nonunit elements of R , and two distinct vertices a and b are adjacent if and only if R a + R b = R. In this paper we consider a subgraph Ω 1 (R) of Ω (R) induced by R \ U ℓ (R) , where U ℓ (R) is the set of all left-invertible elements of R. We characterize those rings R for which Ω 1 (R) \ J (R) is a complete graph or a star graph, where J (R) is the Jacobson radical of R. We investigate the clique number and the chromatic number of the graph Ω 1 (R) \ J (R) , and we prove that if every left ideal of R is symmetric, then this graph is connected and its diameter is at most 3. Moreover, we completely characterize the diameter of Ω 1 (R) \ J (R). We also investigate the properties of R when Ω 1 (R) is a split graph. [ABSTRACT FROM AUTHOR]

The inclusion ideal graph I n (S) of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I , J are adjacent if and only if either I ⊂ J or J ⊂ I. The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of I n (S). We investigate the connectedness of I n (S) and show that the diameter of I n (S) is at most 3 if it is connected. We also obtain a necessary and sufficient condition of S such that the clique number of I n (S) is the number of minimal left ideals of S. Further, various graph invariants of I n (S) , viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup S , we investigate properties of I n (S) including its independence number and matching number. Finally, we obtain the automorphism group of I n (S). [ABSTRACT FROM AUTHOR]

Let Δ (φ , ψ) = ( A A N B B M A B ) be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring Δ (φ , ψ) and the algebras A and B. We prove that if Δ (φ , ψ) is a Gorenstein algebra and both M A and A N (resp., both N B and B M) have finite projective dimension, then A (resp., B) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring Δ (φ , ψ) is inherited by the algebras A and B. [ABSTRACT FROM AUTHOR]

Let n , k and l be integers with 1 ≤ k < l ≤ n − 1. The set-inclusion graph G (n , k , l) is the graph whose vertex set consists of all k - and l -subsets of [ n ] = { 1 , 2 , ... , n } , where two distinct vertices are adjacent if one of them is contained in the other. In this paper, we determine the spectrum and automorphism group of G (n , k , l). [ABSTRACT FROM AUTHOR]

The concept of an H X -group is an upgrade of the concept of a group, in which a new operation is defined on the family of non-empty subsets of a group. If this new support set together with the new operation is a group, then we call it an H X -group. On the other hand, a hyperoperation is a mapping having the same codomain as the operation of an H X -group, i.e., the family of non-empty subsets of the initial set, but a different domain — the set itself. This could be (and was indeed) a source of confusion, which is clarified in this paper. Moreover, H X -groups naturally lead to constructions of hypergroups. The links between these two algebraic concepts are presented, with the aim of reviving the old notion of an H X -group in the current research on algebraic hyperstructures. One of such existing links and one newly established link are also discussed. [ABSTRACT FROM AUTHOR]

A ∗ -ring R is called a nil ∗ -clean ring if every element of R is a sum of a projection and a nilpotent. Nil ∗ -clean rings are the ∗ -version of nil-clean rings introduced by Diesl. This paper is about the nil ∗ -clean property of rings with emphasis on matrix rings. We show that a ∗ -ring R is nil ∗ -clean if and only if J (R) is nil and R / J (R) is nil ∗ -clean. For a 2-primal ∗ -ring R , with the induced involution given by (a i j ) ∗ = (a i j ∗) T , the nil ∗ -clean property of M n (R) is completely reduced to that of M n (Z 2). Consequently, M n (R) is not a nil ∗ -clean ring for n = 3 , 4 , and M 2 (R) is a nil ∗ -clean ring if and only if J (R) is nil, R / J (R) is a Boolean ring and a ∗ − a ∈ J (R) for all a ∈ R. [ABSTRACT FROM AUTHOR]

Let W (Γ) be a class of not-finitely graded Lie algebras related to generalized Virasoro algebras with basis { L α , i , C   |   α ∈ Γ ,   i ∈ Z + } , which satisfies relations [ L α , i , L β , j ] = (β - α) L α + β , i + j + (j - i) L α + β , i + j + 1 + δ α + β , 0 δ i + j , 0 α 3 - α 12 C and [ C , L α , i ] = 0. In this paper, W (Γ) -modules of the intermediate series satisfying some conditions are constructed and classified. We also obtain modules of the intermediate series over the related Lie superalgebra. [ABSTRACT FROM AUTHOR]

The distance matrix of a connected graph G , denoted by D (G) , is the matrix whose rows and columns are indexed by the vertex set V (G) such that the (v i , v j) -entry is d (v i , v j) , where v i , v j ∈ V (G). The distance signature sig (D (G)) of G is the inertia of D (G). In this paper, we determine the distance signature of the extended (co-extended) incidence graph of an affine design. Furthermore, we state that an open Graffiti conjecture is true for the extended (co-extended) incidence graphs of affine designs by investigating the lower bound of the matching number. [ABSTRACT FROM AUTHOR]

In this paper we present two new families of 2- (v , k , λ) designs with a flag-transitive and point-primitive automorphism group of product action type. More surprisingly, one of them is still a family of 2- (v , k , λ) designs with a flag-transitive and point-imprimitive automorphism group. [ABSTRACT FROM AUTHOR]

Let L be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic p > 2. In this paper, all the superbiderivations of L are studied. By means of weight space decompositions with respect to a suitable torus and the standard Z -grading structures of L , we show that the supersymmetric superbiderivations of L are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of L are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on L are described. [ABSTRACT FROM AUTHOR]