Abstract In this paper, we introduce and study a generalization of powerful ideals, in the sense of Badawi and Houston, to the context of integral domains graded by a torsionless monoid.

Abstract The purpose of this article is to provide a common framework for studying various generalizations of Leavitt path algebras. We first define Cohn-Leavitt path algebras of graphs with an additional structure called bi-separated graphs. We then define and study the category BSG of bi-separated graphs with appropriate morphisms so that the functor which associates bi-separated graphs to their

Abstract Given a module T over an F-algebra A , and a basis B for T , the amenability domain of B is the subalgebra of A consisting of all elements r ∈ A with row finite matrix representation [ l r ] B with respect to B . A basis B of T having amenability domain equal to A is said to be an amenable basis. When an amenable basis for T exists, T is said to be an amenable module. A basis having amenability

Abstract In this article, we discuss completeness of non-Lie Leibniz algebras by studying various conditions under which they admit outer derivations. Our study focusses particularly on the class of non-perfect Leibniz algebras whose center is not contained in the Leibniz kernel. We extend to this class of Leibniz algebras several well-known results on derivations of Lie algebras. In particular, we

Abstract A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. It was previously shown by the author that the Hochschild cohomology of a hom-associative algebra A carries a Gerstenhaber structure. In this short paper, we show that this Gerstenhaber structure together with certain operations on the Hochschild homology of A makes a noncommutative differential

Abstract Let G be a finite group and σ 1 ( G ) = 1 | G | ∑ H ≤ G | H | . Answering to a problem posed by M. Tărnăuceanu, we prove that, if σ 1 ( G ) < 117 20 , then G is solvable.

Abstract The aim of this work is to provide a criterion for the rigidity of 2-step nilpotent Lie superalgebras in the variety N ( m | n ) 2 of 2-step nilpotent Lie superalgebras of dimension ( m | n ) . We give several examples of rigid 2-step nilpotent Lie superalgebras of any dimension.

Abstract The notion of L-homologies (of double complexes) as proposed in this paper extends the notion of classical horizontal and vertical homologies along with two other new homologies introduced in the homological diagram lemma called the salamander lemma. We enumerate all L-homologies associated with an object of a double complex and provide new examples of exact sequences. We describe a classification

Abstract Wu-Liu-Ding algebras, that is D ( m , d , ξ ) , are a class of non-pointed affine prime regular Hopf algebras of GK-dimension one. In this paper, we mainly study a class of quotient algebras of D ( m , d , ξ ) , denoted by D ′ ( m , d , ξ ) , which are 2 m 2 d -dimensional non-pointed semisimple Hopf algebras. For a better understanding of the structure of the Hopf algebra D ′ ( m , d , ξ

Abstract Let D be an integral domain and S a multiplicative subset of D. According to the study of Hamed and Hizem, the S-class group of D, denoted by S- C l t ( D ) , is the group of fractional t-invertible t-ideals of D under the t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of D. In this paper we study the case of isomorphism S- C l t ( D ) ≃  S- C l t ( D T ) , where

Abstract We describe an easy way how to find supercharacter theories for a finite group, if its character table is known. Namely, we show how an arbitrary partition of the conjugacy classes or of the irreducible characters can be refined to the coarsest partition that belongs to a supercharacter theory. Our constructions emphasize the duality between superclasses and supercharacters. An algorithm is

Abstract “Unique factorization” was central to the initial development of ideal theory. We update this topic with several new results concerning notions of “unique ideal factorization rings” with zero divisors. Along the way, we obtain new characterizations of several well-known kinds of rings in terms of their ideal factorization properties and examine when monoid rings satisfy various kinds of “unique

Abstract For a commutative algebra A over C , denote g = Der ( A ) . A module over the smash product A # U ( g ) is called a jet g -module, where U ( g ) is the universal enveloping algebra of g . In the present paper, we study jet modules in the case of A = C [ t 1 ± 1 , t 2 ] . We show that A # U ( g ) ≅ D ⊗ U ( L ) , where D is the Weyl algebra C [ t 1 ± 1 , t 2 , ∂ ∂ t 1 , ∂ ∂ t 2 ] , and L is

Abstract Let G be a finitely generated group, and let m n ( G ) denote the number of maximal subgroups of G of index n. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups G (i.e., for lim sup   log   m n ( G )   log   n , the degree of polynomial growth of m n ( G ) ). A condition is given for when this upper bound is attained. Let G = Z k ⋊ A Z ,

Abstract This paper aims to research some basic results on complete Bihom-Lie superalgebras. Particularly, we give some sufficient conditions for a Bihom-Lie superalgebra to be complete. Afterwards, we determine explicitly the derivation superalgebras of centerless perfect Bihom-Lie superalgebras of arbitrary dimension and characteristic over any field are complete. Meanwhile some related properties

Abstract For algebras of global dimension 2 arising from a cut of the quiver with potential associated with a triangulation of an unpunctured surface, Amiot-Grimeland defined integer-valued functions on the first homology groups of the surfaces. Derived equivalences translate to the existence of automorphisms of surfaces preserving these functions. We generalize this to punctured surfaces. Moreover

Abstract Let A be a finite dimensional k-algebra and P a 2-term silting complex in K b ( proj A ) . In this article, we investigate the representation dimension of End D b ( A ) ( P ) by the silting theory. We show that if P is a separating silting complex with certain homological restriction, then, rep.dim A = rep.dim End D b ( A ) ( P ) . This result generalizes the earlier result about tilting modules

Abstract The paper is devoted to the investigation of algebraic hyperstructures. The concept of Menger hyperalgebras, which is a canonical generalization of semihypergroups, is introduced. The emphasis of this paper is on the algebraic nature of such structure concerning subhyperalgebras, homomorphisms and quotient hyperstructures, that allows for a rich algebraic theory. Based on the theory of multiplace

Abstract The aim of this work is to explicitly compute the semi-invariants of low-dimensional Lie algebras by reducing the amount of work, i.e., we can prove that almost every irreducible Lie algebra g , of dimension less than or equal to 5, satisfies the following: It is either a contact Lie algebra or there exists a torus T ⊂ Der ( g ) such that T   ⋉ g is a contact Lie algebra. Therefore, the semi-invariants

Abstract We provide necessary and sufficient conditions for the amalgamation of rings A ⋈ f J to be a divided ring, locally divided ring, going-down ring, and Gaussian ring.

Abstract In the context of factorization in monoid rings with zero divisors, we study associate relations and the resulting notions of irreducibility and factorization length. Building upon these facts, we determine necessary and sufficient conditions for broad classes of monoid rings to satisfy the ascending chain condition on principal ideals or be a bounded factorization ring. Along the way, we

Abstract Let k be a field of characteristic different from 2 and 3. In this paper, we study the maximal k-tori T in the connected simple algebraic groups of type A 2 and G 2. We refer to these tori as unitary tori. We study the cohomology H 1 ( k , T ) and deduce how the presence of maximal k-tori with H 1 ( k , T ) = 0 effects the structure of these groups.

Abstract Given a collection A = { L 1 , … , L n } of intermediate fields in a field extension L/K of finite degree and Λ L , A = L 1 × ⋯ × L n , there is a natural map Ψ L , A : L → Λ L , A given by y ↦ ( Tr L / L 1 ( y ) , … , Tr L / L n ( y ) ) , where Tr L / L i : L → L i denotes the trace. The image set Ψ L , A ( L ) turns out to be contained in a certain subset Λ L , A * of Λ L , A (in fact, a

Abstract This article studies subloops of the Moufang loop of invertible elements in the split Cayley–Dickson algebra over a field K containing a nonassociative subloop formed by elements of the split Cayley–Dickson algebra over a field k such that K is an algebraic extension of k.

Abstract In this current paper, we identify some new properties concerning normal pairs (R, S). Among other interesting results, we show that every prime ideal P of R that is the radical of a finitely generated ideal A such that AS = S is the radical of an ideal with a basis of two elements. Consequently, we derive a significant improvement of Pendleton’s Criterion by producing further characterizations

Abstract We construct a non-Moufang variety of Steiner loops H such that every loop from H satisfies Moufang’s theorem. This example solves Rajah’s problem on the existence of such varieties.

Abstract We investigate the properties of periodic rings R in view of studying general skew polynomials f ( t ) ∈ R [ t ; σ , δ ] . We introduce exponents for these polynomials and give some properties of this notion. We show, in particular, that this notion is right-left symmetric. Using the skew evaluation, we generalize the classical connection between the exponent of a polynomial and the order

Abstract Denote the sum of element orders in a finite group G by ψ ( G ) and let Cn denote the cyclic group of order n. In this paper, we prove that if | G | = n and ψ ( G ) > 13 21 ψ ( C n ) , then G is nilpotent. Moreover, we have ψ ( G ) = 13 21 ψ ( C n ) if and only if n = 6 m with ( 6 , m ) = 1 and G ≅ S 3 × C m . Two interesting consequences of this result are also presented.

Abstract Let μ ( I ) denote the number of generators of a monomial ideal I. It is well known that μ ( I k ) < μ ( I k + 1 ) for k ≫ 0 . In this paper we construct monomial ideals I in F [ x , y ] such that μ ( I k + 1 ) < μ ( I k ) for all k ≤ l , given any positive integer l. Also, we extend some results of Eliahou et al. by constructing monomial ideals in R = F [ x 1 , … , x n ] with μ ( I 2 ) <

Abstract Let S ⊂ R be an arbitrary subset of a unique factorization domain R and K be the field of fractions of R. The ring of integer-valued polynomials over S is the set Int ( S , R ) = { f ∈ K [ x ] : f ( a ) ∈ R ∀ a ∈ S } . This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special

Abstract A finite group G is called a DCI -group if every two isomorphic Cayley digraphs over G are Cayley isomorphic, i.e. their connection sets are conjugate by a group automorphism. We prove that the group C 4 × C p 2 , where p is a prime, is a DCI -group if and only if p ≠ 2 .

Abstract An algebra with trace function has exponentially bounded codimensions if and only if it satisfies an ordinary polynomial identity and what we call an inter-trace permuting identity, namely one of the form ∑ σ ∈ S n α σ t r ( x 1 y σ ( 1 ) ) ⋯ t r ( x n y σ ( n ) ) = 0 in which not all a σ are zero.

Abstract We generalize the concept of the group determinant and prove a necessary and sufficient novel condition for a subset to be a subgroup. This development is based on the group determinant work by Edward Formanek, David Sibley, and Richard Mansfield, where they show that two groups with the same group determinant are isomorphic. The derived condition leads to a generalization of this result.

Abstract We give the necessary and sufficient conditions for an n × n matrix over an integral domain to be a sum of involutions and, respectively, a sum of tripotents. We determine the integral domains over which every n × n matrix is a sum of involutions and, respectively, a sum of tripotents. We further determine the commutative reduced rings over which every n × n matrix is a sum of two tripotents

Abstract In this paper, we present necessary and sufficient conditions for an additive derivation of an incidence algebra of a connected finite partially ordered set X to be inner. These conditions are related to the structure of X as a directed graph and can be applied for finite partially ordered sets that are not connected or even for some that are not finite.

Abstract Let F = F q m ,   m ≥ 7 , n a positive integer, and f = p 1 / p 2 with p 1, p 2 co-prime irreducible polynomials in F [ x ] and deg ( p 1 ) + deg ( p 2 ) = n . We obtain a sufficient condition on (q, m), which guarantees, for any prescribed a, b in E = F q , the existence of primitive pair ( α , f ( α ) ) in F such that Tr F / E ( α ) = a and Tr F / E ( α − 1 ) = b . Further, for every positive

Abstract We introduce enlargements of rings as additive analogues of enlargements of semigroups. For example, a full matrix ring over an idempotent ring is an enlargement of that ring. As our main result we prove that two idempotent rings are Morita equivalent if and only if they have a joint enlargement. We also give a necessary and sufficient condition for a ring with left local units to be Morita

Abstract A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this article, strongly 2-nil-clean rings are studied with an emphasis on their relations with feebly clean rings. This work shows new interesting connections between strongly

Abstract In this article we study solvable Leibniz algebras with abelian nilradicals. The relation between the dimensions of nilradicals and of the corresponding complemented space to the nilradical is estimated. Moreover, we describe ( 2 n − 1 ) -dimensional solvable Leibniz algebras with n-dimensional nilradical. It is know that the nilradical of these algebras is either n-dimensional abelian algebra

Abstract Let a finite group A act on a finite group G via automorphism with ( | A | , | G | ) = 1 and let H be a Hall subgroup of G. We prove that if H is a subgroup of C G ( A ) having a normal complement in C G ( A ) , then H has a normal complement in G.

Abstract The aim of this article is to study quadratic Leibniz superalgebras, which are (left or right) Leibniz superalgebras with an even supersymmetric non-degenerate and associative bilinear forms. In particular, we prove that this class of Leibniz superalgebras are symmetric, which are both a left and right Leibniz superalgebras. We give a characterizations of symmetric Leibniz superalgebras. Moreover

Abstract The enhanced power graph P e ( G ) of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number, and matching number of enhanced power graphs of finite groups. We first study these graph invariants for P e ( G ) when G is any finite group and then determine them when G

Abstract We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products

Abstract An algebraic classification of complex 5-dimensional nilpotent commutative C D -algebras is given. This classification is based on an algebraic classification of complex 5-dimensional nilpotent Jordan algebras.

Abstract For any commutative ring A, we introduce a generalization of S-artinian rings using a hereditary torsion theory σ instead of a multiplicative closed subset S ⊆ A . It is proved that if A is a totally σ-artinian ring, then σ must be of finite type, and A is totally σ-noetherian.

Abstract Recently, in a series of papers “simple” versions of direct-injective and direct-projective modules have been investigated. These modules are termed as “simple-direct-injective” and “simple-direct-projective,” respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if

Abstract We study S 2-orbifolds of N = 1 and N = 2 superconformal vertex algebras. For generic values of the central charge, we determine types of these orbifolds and prove that they are W-algebras. For some special rational values of the central charge, we get new examples of rational N = 1 vertex superalgebras. We also investigate S 2-orbifold of the Heisenberg–Virasoro vertex algebra.

Abstract A module M is said to be generalized extending if for every submodule N ≤ M there exists a direct summand D of M containing N such that D/N is a singular module. In this note we prove that a ring R is right self-injective if and only if the triangular ring T n ( R ) , n ≥ 2 , is right generalized extending. This answers a question which was raised in A. Akalan, G.F. Birkenmeier, A. Tercan

Abstract In this note, we classify finite p-groups all of whose proper subgroups of nilpotency class equal to 2 are metacyclic.

Abstract It is known that the centreless cores of fgc extended affine Lie algebras can be viewed as fixed point subalgebras. In this paper, we study fixed point subalgebras of fgc extended affine Lie algebras. Our main result is that the fixed point subalgebras of untwisted extended affine Lie algebras under a finite abelian group (which is determined by some commuting finite order automorphisms) are

Abstract In this article, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces.

Abstract A Puiseux monoid is an additive submonoid consisting of non-negative rationals. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. Here, we use topological density to understand how much a Puiseux monoid, as well as its set of irreducibles, spread throughout the real line

Abstract Let X ⊂ ℙ r be smooth and irreducible and for k ≥ 0 let ν k ( X ) (resp., δ k ( X ) ) be the k-th contact (resp., the k-th secant) defect of X. For all k ≥ 0 we have the inequality ν k ( X ) ≥ δ k ( X ) and in the case k = 1 we characterize projective varieties X for which equality holds, dim Sing ( X ) ≤ δ 1 ( X ) − 1 and the generic tangential contact locus is reducible.

Abstract We prove that the André–Quillen homology and cohomology modules of F-finite Z ( p ) -algebras are finitely generated.

Abstract We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.

Abstract In this article, we investigate families of connected graphs which do not contain an odd cycle in their complement. Specifically, we consider graphs formed by two complete graphs connected in a particular way. We determine which of these graphs can or cannot occur as the prime character degree graph of a solvable group. An obvious expansion and generalization can also be considered, of which

Abstract We prove that finite categorically equivalent p-rings have isomorphic additive groups (in particular, they have the same cardinality) and that the number of generators is a categorical invariant for finite rings. We also classify rings of size p 3 up to categorical equivalence.

Abstract In 1979, Herzog conjectured that two finite simple groups containing the same number of involutions have the same order. Zarrin, in a 2018 published paper, disproved Herzog’s conjecture with a counterexample. The goal of this article is to prove that there are infinitely many counterexamples to Herzog’s conjecture. In doing so, we obtain an explicit formula for the number of involutions in

Abstract Let R be a noncommutative prime ring with involution ′ * ′ and let Q m s ( R ) be the maximal symmetric ring of quotients of R . In the present paper, we describe the structure of generalized Jordan *-derivations, i.e., additive mappings F : R → R satisfying F ( x 2 ) = F ( x ) x * + x d ( x ) for all x ∈ R , where d is an associated Jordan *-derivation of R . Precisely, we prove that under

Abstract For a sequence of the naturally graded quasi-filiform Leibniz algebra of second type L 5 introduced by Camacho, Gómez, González and Omirov, all the possible right solvable indecomposable extensions over the field C are constructed.