Abstract In this paper, we introduce deficient complete bipartite graphs and deduce a formula for graded Betti numbers of their edge rings in terms of associated combinatorial data. As a consequence, we obtain the graded Betti numbers of edge rings of complete bipartite graphs K m , n and the n-crown graph C n , n . Also, the projective dimension and the regularity of edge rings of these graphs are

(2020). Statement of retraction: Eight-dimensional octonion-like but associative normed division algebra. Communications in Algebra. Ahead of Print.

Abstract Assume that a basic algebra A over an algebraically closed field k with a basic set A 0 of primitive idempotents has the property that eAe = k for all e ∈ A 0 . Let n be a nonzero integer, and ϕ and ψ two automorphisms of the repetitive category A ̂ of A with jump n (namely, they send A [ 0 ] to A [ n ] , where A [ i ] is the i-th copy of A in A ̂ for all i ∈ ℤ ). If ϕ and ψ coincide on the

Abstract A group G is said to have an exact factorization if there exist proper subgroups Ai for i = 1 , 2 , … , n such that G = A 1 A 2 … A n and | G | = | A 1 | . | A 2 | … | A n | . The number n is called length of this factorization. An exact factorization of length n is called minimal length if there is no exact factorization of length less than n. In this paper, up to conjugacy, we determine

Abstract A ring R is called right (or left) socle-injective if every R-homomorphism from the right (or left) socle of R into R extends to R. In this article, we show that any semiprime ring R with socle S, is socle-injective if and only if End R ( S ) ≅ Q ′ / I , where Q′ is a suitable subring of maximal right ring of quotients of R and I = { r ∈ Q ′ | r S = 0 } is an ideal of Q′. Furthermore, an explicit

Abstract We study minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups. As a result, we find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes.

Abstract Let G be a finite group. A classical theorem of Burnside shows that a group with order divisible by at most two primes is solvable. The aim of this article is studying finite groups each of whose vanishing class sizes has at most two prime divisors, where a conjugacy class of x in G is a vanishing class if there exists some irreducible character χ of G such that χ ( x ) = 0 .

Abstract Let R be a commutative ring with nonzero identity and, S ⊆ R be a multiplicatively closed subset. An ideal P of R with P ∩ S = ∅ is called an S-prime ideal if there exists an (fixed) s ∈ S and whenver a b ∈ P for a , b ∈ R then either s a ∈ P or s b ∈ P . In this article, we construct a topology on the set SpecS (R) of all S-prime ideals of R which is generalization of prime spectrum of R

Abstract A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to Prüfer domains. In the present paper, we investigate some analogs of these concepts for modules over group rings.

Abstract We study so called regular Lie algebras, i.e., Lie algebras in which each nonzero element is regular. We make a connection with an open problem whether any element of reduced trace zero in a simple associative algebra is a commutator.

Abstract It is well known that Brauer graph algebras coincide with symmetric special biserial algebras and there has been a lot of work on Brauer graph algebras and their representation theory. Given a Brauer graph algebra A associated with a Brauer graph G, we denote by gr(A) the graded algebra associated with the radical filtration of A. We give a criterion for gr(A) to be representation-finite in

Abstract We define a new class of non-solvable groups S p * containing every group G whose every chief factor A/B satisfies one of the following conditions: (1) A/B is a p-group; (2) A/B is a p ′ -group; (3) | A / B | p = p . We call a subgroup H is a C A P S p * -subgroup of a finite group G if for any pd-chief factor A/B of G, we have either HA = HB or | H ∩ A : H ∩ B | p ≤ p . In this article, some

Abstract Let R be an associative ring. In this paper, we give a positive answer to the question of M.P. Drazin, “whether a ∈ R has a unique core-quasi-nilpotent decomposition implies that it must be quasi-invertible”.

Abstract Some conditions for the Galois map to be injective are given in the groupoid acting on a noncommutative ring context. In the particular case in which the Galois extension is a central Galois algebra, it is given a complete characterization of that kind of extension with Galois map bijective.

Abstract We say that a Puiseux monoid is exponential provided that it is generated by some of the powers of a rational number. Here we study the atomic properties of exponential Puiseux monoids and semirings. First, we characterize atomic exponential Puiseux monoids, and we prove that the finite factorization property, the bounded factorization property, and the ACCP coincide in this context. Then

Abstract For an affine toric variety Spec ( A ) , we give a convex geometric interpretation of the Gerstenhaber product HH 2 ( A ) × HH 2 ( A ) → HH 3 ( A ) between the Hochschild cohomology groups. In particular, our construction also generalizes the known results about the cup product T A 1 × T A 1 → T A 2 .

Abstract We construct an infinite-dimensional Heisenberg algebra in the derived Hall algebra for the category of nilpotent representations of Jordan quiver. Using this Heisenberg algebra, we reconstruct the theory of symmetric functions, focusing on Hall-Littlewood symmetric functions and various operators acting on them.

Abstract We produce a family of complexes called trimming complexes and explore its applications. We demonstrate how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by certain subsets of a generating set for an arbitrary ideal I. In particular, we compute the Betti table of the ideal obtained by removing an arbitrary generator from the

Abstract Given a prime number p we consider C p the topological completion of the algebraic closure of the field of p-adic numbers with respect to the usual p-adic absolute value. In this paper, we point out new connections between zeros and singularities of the trace functions associated with arbitrary elements of C p .

Abstract Let M be the backward identity matrix. A complex matrix G is perplectic if G is an isometry of the symmetric scalar product ( x , y ) → x T M y , that is, ( x , y ) = ( G x , G y ) for all column vectors x and y. The set of perplectic matrices P(n) forms a noncompact and nonconnected group. We show an Iwasawa-like decomposition for P(n), that is, we write every element G as a product KAN,

Abstract In ‘A Hermitian analogue of the Morita theorems’ by Dasgupta and Dasgupta, we proved a Hermitian Morita II. There is an error in the proof of Proposition 3.1 which nullifies the paper.

Abstract For the cyclic group Z p (p is a prime) and a positive integer k, we study the representations of the orbifold vertex operator algebra L s l 2 ̂ ( k , 0 ) Z p . All the irreducible modules for L s l 2 ̂ ( k , 0 ) Z p are classified and constructed explicitly. Quantum dimensions and fusion rules for L s l 2 ̂ ( k , 0 ) Z p are completely determined.

Abstract In this paper we give the classification of 5-dimensional complex nilpotent associative algebras. We use χ ( A ) = ( dim A , dim A 2 , dim A 3 , … , dim A n ) - the isomorphism invariant. We prove that there are only two possible isomorphism invariants χ ( A ) = ( 5 , 2 , 1 , 0 , 0 ) or χ ( A ) = ( 5 , 3 , 1 , 0 , 0 ) for any 5-dimensional complex nilpotent associative algebra A with the condition

Abstract Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that [ g ( x m ) x n − x s h ( x t ) , x r ] k = 0 for all x ∈ I , where m , n , s , t , r , k are fixed positive integers. Then, one of the following conditions holds: there exist α , β ∈ C such that g

Abstract In the present work, we consider differential rings of the form ( B , D ) where B is a Danielewski surface and D is a locally nilpotent derivation on B . Influenced by several recent works, we describe the isotropy group of a locally nilpotent derivation, D, on Danielewski surfaces, in the cases x y = φ ( z ) , x n y = φ ( Z ) , and f ( x ) x = φ ( z ) .

Let g be a hyperbolic Kac-Moody algebra of rank 2. We give a necessary and sufficient condition for the crystal graph of the Lakshmibai-Seshadri path crystal B ( λ ) to be connected for an arbitrary integral weight λ.

We improve a result of Isaacs et al. with respect to character degrees and class sizes of finite groups [1, Corollary D].

Motivated from understanding higher-dimensional supersymmetric conformal field theory, we study various inhomogeneous oscillator representations of s l ( n + 1 | m ) and o s p ( m | n + 1 ) on supersymmetric polynomial algebras and on spaces of supersymmetric exponential–polynomial functions, analogous to the projective oscillator representations of s l ( n + 1 ) and s p ( 2 m + 2 ) . In particular

For several classes of special p-groups G, of exponent p, p > 2, we show that the near-ring, C 0 ( G ) , of congruence preserving functions on G is a ring if and only if G is a 1-affine complete group.

We previously have developed two methods (Key–Moori Methods 1 and 2) for constructing codes and designs from finite groups (mostly simple finite groups). In this article, we introduce a new method (Method 3) for constructing codes and designs from fixed points of elements of finite transitive groups. We first discuss background material and results required from finite groups, permutation groups and

We first obtain that NI rings satisfy a property that if ab is central for elements a, b, then ( a b ) n = ( b a ) n for some n ≥ 1 , by applying a property of reduced rings. We prove next the following: Let R be a ring and I be the ideal of R generated by the subset { a b − b a | a , b ∈ R   such   that a b   is   central   in   R } . (i) Suppose that ab is central for a , b ∈ R and ab – ba is a nonzero

We describe a linear equivariant isomorphism K from the enveloping algebra U(gl(n)) to the algebra C[Mn,n]≅Sym(gl(n)) of polynomials in the entries of a “generic” square matrix of order n. The isomorphism K maps any Capelli bitableau [S|T] in U(gl(n)) to the (determinantal) bitableau (S|T) in C[Mn,n] and any Capelli *-bitableau [S|T]* in U(gl(n)) to the (permanental) *-bitableau (S|T)* in C[Mn,n].

Let σ={σi|i∈I} be some partition of the set P of all primes, that is, P=∪i∈Iσi and σi∩σj=∅ for all i≠j. Let G be a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G and H contains exactly one Hall σi-subgroup of G for every σi∈σ(G). G is said to be a σ-group if it possesses a complete Hall σ-set. A σ-group G

Let P be the set of all primes. A subgroup H of G is called P-subnormal in G, if either H = G, or there exists a chain H=H0≤H1≤…≤Hn=G,|Hi:Hi−1|∈P,∀i. Recall that G is w-supersoluble, if every Sylow subgroup of G is P-subnormal in G. The structure of w-supersoluble residual of G = AB with P-subnormal w-supersoluble subgroups A and B is obtained.

Let R be a commutative ring with identity. Denote by FPR(R) the set of all R-modules admitting a finite projective resolution consisting of finitely generated projective modules. Then the small finitistic dimension of R is defined as fPD(R)=sup{pdRM|M∈FPR(R)}. Cahen et al. posed an open problem as follows: Let R be a Prüfer ring. Is fPD(R)⩽1? In this paper, we show that the answer to this problem is

In this article, some properties of the Schur multiplier and residual nilpotency of Lie rings are studied. In the case of nilpotent Lie rings and given special homomorphism between rings, some isomorphisms will be obtained. Furthermore, we will prove that the residual nilpotent of a Lie ring L is independent of the choice of the free presentation of L. Finally, we give a method of constructing residually

This is one of a series of papers which aim toward determining the automorphism groups of Frobenius groups. This paper solves the problem in the case where the Frobenius kernels are elementary abelian and Frobenius complements are cyclic.

We construct quasi-particle bases of principal subspaces of standard modules L(Λ), where Λ=k0Λ0+kjΛj, and Λj denotes the fundamental weight of affine Lie algebras of type Bl(1),Cl(1),F4(1) or G2(1) of level one. From the given bases we find characters of principal subspaces.

For a Khovanov-Lauda-Rouquier algebra K defined by an unoriented graph of type An, we prove that K is derived equivalent to a matrix-like algebra T. Under this equivalence, the higher K-groups and global dimension of K have been described.

The purpose of this article is to study the construction of 3-Bihom-Lie algebras. We give some ways of constructing 3-Bihom-Lie algebras from 3-Bihom-Lie algebras and 3-totally Bihom-associative algebras. Furthermore, we introduce Tθ-extensions and Tθ*-extensions of 3-Bihom-Lie algebras and prove the necessary and sufficient conditions for a 2n-dimensional quadratic 3-Bihom-Lie algebra to be isomorphic

Let G be a finite group, ψ(G)=∑g∈Go(g) and ψ″(G)=ψ(G)/|G|2, where o(g) denotes the order of g. In this paper, we prove that if p is a prime number and ψ″(G)>(p2+p+1)/(4p2), then G=Op(G)×Op′(G), where Op(G) is cyclic. In addition, G is supersolvable and G″≤Z(G).

Let R be an alternative ring containing a nontrivial idempotent and D be a multiplicative Lie-type derivation from R into itself. Under certain assumptions on R, we prove that D is almost additive. Let pn(x1,x2,…,xn) be the (n−1)-th commutator defined by n indeterminates x1,…,xn. If R is a unital alternative ring with a nontrivial idempotent and is {2,3,n−1,n−3}-torsion free, it is shown under certain

We first prove that a left Novikov algebra N is right nilpotent if and only if it is solvable. Then we show that, every Novikov algebra that can be represented as the sum of two solvable subalgebras is itself solvable, moreover, if the two solvable subalgebras are abelian, then the whole algebra is metabelian. Finally, we show that for every n≥2, every n-generated non-abelian free solvable (or non-abelian

Let R be a commutative Noetherian ring and I be an ideal of R. Let M be an arbitrary R-module. In this paper we establish some results concerning the cofiniteness properties of modules. It is shown that, M is I-cominimax if and only if there is an ideal J⊇I with dimR/J≤0 such that M is (I,V(J))-cofinite and the R-module ExtRi(R/J,M) is finitely generated, for each i≥0. Moreover, it is shown that M

Analogous to the types A, B, and C cases, we address the computation of the index of seaweed subalgebras in the type-D case. Formulas for the algebra’s index can be computed by counting the connected components of its associated meander. We focus on a set of distinguished vertices of the meander, called the tail of the meander, and using the tail, we provide comprehensive combinatorial formulas for

For the field F and F-vector space A, let G be a subgroup of GL(F, A), the group of invertible F-linear transformations of A. A subspace B of A is G-core-free if ∩g∈GgB=0. In this article, we discuss groups G and vector spaces A such that every subspace of A is either G-invariant or G-core-free.

In a semigroup S with fixed c∈S, one can construct a new semigroup (S,·c) called a variant by defining x·cy:=xcy. Elements a,b∈S are primarily conjugate if there exist x,y∈S1 such that a=xy,b=yx. This coincides with the usual conjugacy in groups, but is not transitive in general semigroups. Araújo et al. proved that transitivity holds in a variety W of epigroups containing all completely regular semigroups

Let x be an element of a finite group G and denote the order of x by ord(x). We consider a finite group G such that gcd(ord(x),ord(y))⩽2 for any two vanishing elements x and y contained in distinct conjugacy classes. We show that such a group G is solvable. When G with the property above is supersolvable, we show that G has a normal metabelian 2-complement.

A finite group P is said to be primary if |P|=pa for some prime p. We say a primary subgroup P of a finite group G satisfies the Frobenius condition in G if NG(P)/CG(P) is a p-group provided P is p-group. In this article, we determine the structure of a finite group G in which every non-subnormal primary subgroup satisfies the Frobenius condition. In particular, we prove that if every non-normal primary

We classify all n-dimensional reduced Cohen-Macaulay modular quotient varieties AFn/C2p and study their singularities, where p is a prime number and C2p denotes the cyclic group of order 2p. In particular, we present an example that demonstrates that the problem proposed by Yasuda has a negative answer if the condition that “G is a small subgroup” was dropped.

Let G be a finite group, σ={σi|i∈I} a partition of the set of all primes P and σ(G)={σi|σi∩π(|G|)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i∈I and H contains exactly one Hall σi-subgroup of G for every σi∈σ(G). G is said to be σ-full if G possesses a complete Hall σ-set. We say a subgroup H of G is sσ-quasinormal

Let (A,m) be an alternative algebra with maximal ideal m which is complete and separated for the m-adic topology. Assuming that A/m:=k is a perfect field of positive characteristic and that the associated graded algebra is a k-algebra, we show that the reduction map W(k)→k from the Witt ring W(k) lifts canonically to a morphism W(k)→A thereby giving A the structure of a unital W(k)-algebra.

Let B be the one-point extension algebra of A by an A-module X. We proved that every support τ-tilting A-module can be extended to be a support τ-tilting B-module by two different ways. As a consequence, it is shown that there is an inequality | s τ ‐ tilt    B | ⩾ 2 | s τ ‐ tilt  A | .

A ring is nil clean if each of its elements is the sum of an idempotent and a nilpotent. In Sahinkaya et al. [13 Sahinkaya, S. , Tang, G. , Zhou, Y. (2017). Nil-clean group rings. J. Algebra Appl . 16(07):1750135. DOI: 10.1142/S0219498817501353.[Crossref] , [Google Scholar]], it was shown that, for a ring R and a symmetric group S 3, the group ring RS 3 is nil clean iff R and M 2 ( R ) are nil clean

A group G is called mixed identity-free if for every n ∈ ℕ and every w ∈ G ∗ F n , there exists a homomorphism φ : G ∗ F n → G such that φ is the identity on G and φ ( w ) is nontrivial. In this paper, we make a modification to the construction of elementary amenable lacunary hyperbolic groups provided by Ol’shanskii et al. to produce finitely generated elementary amenable groups which are mixed identity-free

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Particularly, Grothendieck-type theorem is obtained which states that there is a canonical correspondence between the idempotents of a ring and the clopens of its pure spectrum. It is also proved that a given ring is a Gelfand ring iff its maximal spectrum equipped with the

The paper is devoted to give a full classification of all finite-dimensional nilpotent Lie algebras L of class 4 with dim  L 2 = 3. Moreover, we specify the capable ones.

Let k be a number field and Ok its ring of integers. Let Γ be a finite group. Let M be a maximal Ok -order in the semi-simple algebra k [ Γ ] containing O k [ Γ ] . Let C l ( O k [ Γ ] ) (resp. C l ( M ) ) be the locally free classgroup of O k [ Γ ] (resp. M ). We denote by R ( D − 1 , O k [ Γ ] ) (resp. R ( D − 1 , M ) ) the set of classes c in C l ( O k [ Γ ] ) (resp. C l ( M ) ) such that there

Let g be a hyperbolic Kac-Moody algebra of rank 2, and set λ = Λ 1 − Λ 2 , where Λ 1 , Λ 2 are the fundamental weights. Denote by V ( λ ) the extremal weight module of extremal weight λ with v λ the extremal weight vector, and by B ( λ ) the crystal basis of V ( λ ) with u λ the element corresponding to v λ . We prove that (i) B ( λ ) is connected, (ii) the subset B ( λ ) μ of elements of weight μ

Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic different from 2. Let E ( r 1 , … , r n ) ( v 1 , … , v n ) be a n-induced ( R , + ) -grading on E and let S ( r 1 , … , r n ) ( v 1 , … , v n ) be its support. In this paper, we study the properties of the set S ( r 1 , … , r n ) ( v 1 , … , v n ) as a subset of R . Assuming that this set is bounded, we describe