We give a geometric classification of complex 4-dimensional nilpotent ℭ𝔇-algebras. The corresponding geometric variety has dimension 18 and decomposes into 2 irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras. In particular, there are no rigid 4-dimensional complex nilpotent ℭ𝔇-algebras.

In this paper, we introduce the concept of Baer (p,q)-sets. Using this notion, we define Rickart, Baer, quasi-Baer and π-Baer (S,R)-bimodules, respectively. We show how these conditions relate to each other. We also develop new properties of the minus binary relation, ≤-, we extend the relation ≤- to (S,R)-bimodules and use it to characterize the aforementioned Rickart, Baer, quasi-Baer, and π-Baer

In this paper, we calculate the combinatorial rank of the positive part uq+(𝔤) of the multiparameter version of the small Lusztig quantum group, where 𝔤 is a simple Lie algebra of type G2. Supposing that the main parameter of quantization q has multiplicative order t, where t is finite, t>3, we prove that the combinatorial rank equals 3.

Let 𝔽 denote a field with char 𝔽≠2. The Racah algebra ℜ is the unital associative 𝔽-algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert that [A,B]=[B,C]=[C,A]=2D and each of the elements α=[A,D]+AC−BA,β=[B,D]+BA−CB,γ=[C,D]+CB−AC is central in ℜ. Additionally, the element δ=A+B+C is central in ℜ. We call each element in D2+A2+B2+(δ+2){A

Let R be a polynomial ring in n indeterminates with coefficients in the field K of characteristic p>0 and 𝒟 be the ring of differential operators over R. In this paper, we introduce the notion of generalized Eulerian 𝒟-modules for characteristic p>0 and establish their properties. We show that if 𝒯 is any graded Lyubeznik functor on the category of modules over R, then 𝒯(R) is a generalized Eulerian

The Manturov (3,4)-group G43 is the group generated by four elements a, b, c, d with defining relations a2=b2=c2=d2=(abcd)2=(acdb)2=(adbc)2=1. We apply the algebraic discrete Morse theory to calculate the Anick chain complex for G43, evaluate the Hochschild cohomology groups of the group algebra 𝕂G43 with coefficients in all 1-dimensional bimodules over a field 𝕂 of characteristic zero, and derive

A Kuribayashi quartic curve 𝒞a:X4+Y4+Z4+a(X2Y2+Y2Z2+Z2X2)=0,a∈ℂ∖{−1,±2}, carries total sextactic points if and only if a=14 or a is a zero of P(a)=a3+68a2−91a+98, cf. [1]. In [2], the authors describe the subgroup generated by the total sextactic points in the Jacobian of a Kuribayashi quartic curve when a is a zero of P(a). In this paper, we describe this group when a=14.

The aim of this paper is to introduce and study Yetter–Drinfeld category over a weak monoidal Hom–Hopf algebra H. We first show that the category HH𝒲ℳℋ𝒴𝒟 of Yetter–Drinfeld modules over H with a bijective antipode is a braided monoidal category. Secondly, we discuss some properties on the symmetries of the category HH𝒲ℳℋ𝒴𝒟. Finally, we prove that the representation category of triangular weak

Let ℱ𝒫∞R be the class of all left R-modules M which has a projective resolution by finitely generated projectives. An exact sequence 0→A→B→C→0 of right R-modules is called neat if the sequence 0→A⊗RG→B⊗RG→C⊗RG→0 is exact for any G∈ℱ𝒫∞R. An exact sequence 0→M→N→L→0 of left R-modules is called clean if the sequence 0→HomR(G,M)→HomR(G,N)→HomR(G,L)→0 is exact for any G∈ℱ𝒫∞R. We prove that every R-module

In this paper, we investigate the resolving resolution dimension with respect to the recollements of abelian categories. Let (𝒜,ℬ,𝒞) be a recollement of abelian categories such that ℬ and 𝒞 have enough projective objects and let 𝒳𝒜, 𝒳ℬ, 𝒳𝒞 be resolving subcategories of 𝒜, ℬ and 𝒞, respectively. We establish some upper and lower bounds of 𝒳ℬ-resolution dimension of ℬ in terms of the 𝒳𝒜-resolution

Let k be an algebraically closed field, n≥1 an integer, 𝒯 a k-linear Hom-finite (n+2)-angulated category with n-suspension functor Σn, a Serre functor S, and split idempotents. Let T be a basic n-rigid object and Γ the endomorphism algebra of T. We introduce the notion of relative n-rigid objects, i.e. ΣnT-rigid objects of 𝒯. Then we show that the basic maximal ΣnT-rigid objects in 𝒯 are in bijection

This paper deals with Leibniz superalgebras L=L0⊕L1, whose even part is a simple Lie algebra 𝔰𝔩2. We describe all such Leibniz superalgebras when odd part is an irreducible Leibniz bi-module on 𝔰𝔩2. We show that there exist such Leibniz superalgebras with nontrivial odd part only in case of dimL1=2.

In this paper, we give a largely self-contained proof that the quartic extension 𝔽q4 of the finite field 𝔽q contains a primitive element α such that the element α+α−1 is also a primitive element of 𝔽q4, and Tr𝔽q4|𝔽q(α)=a for any prescribed a∈𝔽q. The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed

The category of internal coalgebras in a cocomplete category 𝒞 with respect to a variety 𝒱 is equivalent to the category of left adjoint functors from 𝒱 to 𝒞. This can be seen best when considering such coalgebras as finite coproduct preserving functors from 𝒯𝒱op, the dual of the Lawvere theory of 𝒱, into 𝒞: coalgebras are restrictions of left adjoints and any such left adjoint is the left

For an integer N≥2 and three subcategories 𝒲,𝒳,𝒴 of modules, we study the notion of (𝒲,𝒳,𝒴)-Gorenstein N-complexes. We show that under certain hypotheses, an N-complex C is (𝒲,𝒳,𝒴)-Gorenstein if and only if Cn is a (𝒲,𝒳,𝒴)-Gorenstein module for each n∈ℤ, HomR(V,C) and HomR(C,U) are N-acyclic for any V∈𝒴̃ and U∈𝒳̃, where 𝒳̃ and 𝒴̃ denote the subcategory of 𝒳N-complexes and 𝒴N-complexes

For a Lie comodule M of a Lie coalgebra C and the Lie algebra 𝔤(C,M) (respectively, the group 𝔊(C,M)) of diagonal coderivations (respectively, Lie coalgebra automorphisms) of the semidirect sum C⋉M, we show that there is a representation of 𝔤(C,M) (respectively, 𝔊(C,M)) on the second cohomology group Hc2(M,C), from which we construct obstruction classes of extensibility of pairs of coderivations

We investigate purity and approximation with respect to a class of morphisms of modules. Let ℐ be a class of left R-module morphisms. An epimorphism ϕ:A→B of left R-modules is called ℐ-pure if for any morphism ψ:X→B in ℐ, there is a morphism 𝜃:X→A such that ϕ𝜃=ψ. An ℐ-pure epimorphism f:M→N is called ℐ-superfluous if every morphism g:A→M with fgℐ-pure is itself ℐ-pure. We get many properties of ℐ-pure

We study the representation theory of the Bershadsky–Polyakov algebra 𝒲k=𝒲k(sl3,f𝜃). In particular, Zhu algebra of 𝒲k is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category 𝒪 for the Bershadsky–Polyakov algebra 𝒲k for k=−5/3,−9/4,−1,0. In the case k=0, we show that the Zhu algebra A(𝒲k) has two-dimensional indecomposable

Assume k is a positive integer and p is an odd prime. Let ν(G) be the number of conjugacy classes of nonnormal subgroups of a finite group G and 𝒩CN(p,k)={ν(G)∈[0,kp]|G is a finite p-group}. The set 𝒩CN(p,k) has been determined for k≤3. In this paper, the set 𝒩CN(p,4) is determined, and it is discovered that there are three gaps in the values that ν(G) can take in the case of finite p-groups.

Let R be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if 𝔞 is an ideal of R which is linked by the ideal I, then cd(𝔞,R)∈{grad𝔞,cd(𝔞,H𝔠grad𝔞(R))+grad𝔞}, where 𝔠:=⋂𝔭∈AssRI−V(𝔞)𝔭. Also, it is shown that for every ideal 𝔟 which is geometrically linked with 𝔞,cd(𝔞,H𝔟grad𝔟(R)) does not depend on 𝔟.

A subgroup H of a finite group G is said to be S-semipermutable in G, if H permutes with all Sylow q-subgroups of G for the primes q not dividing |H|. In this paper, we consider the S-semipermutability of some p-subgroups to investigate the p-supersolubility of a finite group. Some interesting results are obtained which extend some known results.

A class of integer-valued functions defined on the set of ideals of an integral domain R is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the e-operation. We use this class of functions to recover familiar semistar operations

In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type A2n−1,Dn+1,A2n and D4 based on the newly found Moody–Rao–Yokonuma-like presentation.

We study an abstract class of Hamming spaces (known also as measure algebras) that generalizes standard Hamming spaces (ℤ/2ℤ)n. We classify countable locally standard Hamming spaces and show that each of them can be realized as the Boolean algebra of idempotents of a Cartan subalgebra of a locally matrix algebra.

A module M is said to be lifting if, for any submodule X of M, there exists a decomposition M=A⊕B such that A⊆X and X/A is a small submodule of M/A. A lifting module is defined as a dual concept of the extending module. A module M is said to have the finite internal exchange property if, for any direct summand X of M and any finite direct sum decomposition M=M1⊕⋯⊕Mn, there exists a direct summand Ni

We give algebraic and geometric classifications of complex four-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only 18 non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of three rigid algebras and two one-parametric families of algebras.

The purpose of this paper is to introduce a concept of commutator in a multiplicative Lie algebra which will be termed as a Lie commutator. Consequently, we develop the theory of Lie solvability and Lie nilpotency of multiplicative Lie algebras.

Let R be an associative ring with identity. A right R-module MR is said to have Property (A), if each finitely generated ideal I⊆Z(MR) has a nonzero annihilator in MR. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc.155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property (A). We study and construct various classes of modules with Property

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra RcoA̲ with a certain subalgebra of the smash product R#Â. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.

In the present paper, we prove that any finite nontrivial irreducible module over a rank two Lie conformal algebra ℋ is of rank one. We also describe the actions of ℋ on its finite irreducible modules explicitly. Moreover, we show that all finite nontrivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.

We characterize the structure of finite unitary rings R in which every proper subring is commutative and show that R is a commutative ring.

For a composition λ of n we consider the Kazhdan–Lusztig cell in the symmetric group Sn containing the longest element of the standard parabolic subgroup of Sn associated to λ. In this paper, we extend some of the ideas and results in [Beiträge zur Algebra und Geometrie, 59(3) (2018) 523–547]. In particular, by introducing the notion of an ordered k-path, we are able to obtain alternative explicit

Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper, we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also obtain a level-one module of the quantum toroidal algebra for type C as a special case.

We show that except in several cases conjugacy classes of classical Weyl groups W(Bn) and W(Dn) are of type D. We prove that except in three cases Nichols algebras of irreducible Yetter–Drinfeld (YD) modules over the classical Weyl groups are infinite dimensional.

This paper aims to study the concept of ϕ-ideals of a finite-dimensional Lie algebra which is analogous to the concept of c-ideal and c-normal subgroup. We compile some basic properties of ϕ-ideals and consider the influence of this concept on the structure of a finite-dimensional Lie algebra, especially its solvability and supersolvability. We also show that the counterpart of some results for c-ideals

We study Wilf’s conjecture for numerical semigroups S such that the second least generator a2 of S satisfies a2>c(S)+μ(S)3, where c(S) is the conductor and μ(S) the multiplicity of S. In particular, we show that for these semigroups Wilf’s conjecture holds when the multiplicity is bounded by a quadratic function of the embedding dimension.

A group is called a CA-group if the centralizer of every non-central element is abelian. Furthermore, a group is called a minimal non- CA-group if it is not a CA-group itself, but all of its proper subgroups are. In this paper, we give a classification of the finite non-solvable minimal non- CA-groups.

Let R be a ring with nonzero identity. The Idempotent graph of R, denoted by GId(R), has its set of vertices equal to the set of all elements of R; Distinct vertices x and y are adjacent if and only if x+y is an idempotent of R. In this paper, we study some basic properties of GId(R) such as connectedness, diameter and girth.

In this paper, we introduce and investigate three new versions of the Rickart condition for rings. These conditions, as well as, three new corresponding regularities are defined using projection invariance. We show how these conditions relate to each other as well as their connections to the well-known Baer, Rickart, quasi-Baer, p.q.-Baer, regular, and biregular conditions. Applications to polynomial

We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviors.

Symmetric k-varieties are a natural generalization of symmetric spaces to general fields k. We study the action of minimal parabolic k-subgroups on symmetric k-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of k-split groups that depends only on the dimension of a

For the bounded derived category of an abelian category, bounds of the dimension with respect to a complete hereditary cotorsion pair are given. We also characterize levels of DG-modules and study how levels involved in a recollement of derived categories over DG-rings are related.

A staggered linear basis is a specific basis for a polynomial ideal generating the ideal as a linear vector space. There are some algorithms in the literature to compute these bases and we show that they do not work properly. In this paper, due to the strong connection between Gröbner bases and staggered linear bases and by applying a signature-based structure, we present a new and correct algorithm

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout’s theorem, IEEE Trans. Inform. Theory46(2) (2000) 635–641, On (or in) Dick Blahut’s ‘footprint’, Codes, Curves Signals (1998) 3–9] provides us with an upper bound on the number

In this paper, the non-abelian simple groups having at most 20 distinct irreducible character degrees are classified.

Recently, a new type of generalized inverse called the n-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let R be a ring and n be a positive integer, an element x∈R is called the n-strong Drazin inverse of a∈R if it satisfies xax=x, ax=xa and an−ax∈Rnil. The main aim of this paper is to consider some equivalent characterizations for the n-strong Drazin invertibility in

In this paper, first, we study linear deformations of a Lie–Yamaguti algebra and introduce the notion of a Nijenhuis operator. Then we introduce the notion of a product structure on a Lie–Yamaguti algebra, which is a Nijenhuis operator E satisfying E2=Id. There is a product structure on a Lie–Yamaguti algebra if and only if the Lie–Yamaguti algebra is the direct sum of two subalgebras (as vector spaces)

In this paper, we study the generalized right near domains and its relations with sharply 2-transitive groups. We find a necessary and sufficient condition when a generalized right near domain is a field. Also, we have found the isomorphism classes of right near domains in a sharply 2-transitive group.

Let R be a commutative ring with identity. An edge labeled graph is a graph with edges labeled by ideals of R. A generalized spline over an edge labeled graph is a vertex labeling by elements of R, such that the labels of any two adjacent vertices agree modulo the label associated to the edge connecting them. The set of generalized splines forms a subring and module over R. Such a module is called

This paper continues the study of the reduced ring order (rr-order) in reduced rings where a≤rrb if a2=ab. A reduced ring is called rr-good if it is a lower semi-lattice in the order. Examples include weakly Baer rings (wB or PP-rings) but many more. Localizations are examined relating to this order as well as the Pierce sheaf. Liftings of rr-orthogonal sets over surjections of reduced rings are studied

In this paper, we introduce the annihilator graph AG(A) of an MV-algebra A. We show that AG(A) contains the zero-divisor graph ZG(A) as a spanning subgraph. We then prove that AG(A)=ZG(A) if and only if |A|≤4. Moreover, we obtain that the girth gr(AG(A))∈{3,4,∞}.

The purpose of this paper is to use the notion of M-sets (cocliques) introduced by the second author in [S. Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type E6(K) for fields of characteristic two, Beitrag Zur Algebra und Geometry58 (2017) 529–534.] and using Levi components and unipotent radical subgroups of E6(K) to give an elementary and self-contained construction of the stabilizer

Denote by νp(G) the number of Sylow p-subgroups of G. For every subgroup H of G, it is easy to see that νp(H)≤νp(G), but νp(H) does not divide νp(G) in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory21(4) (2018) 695–712], we say that a group G satisfies DivSyl(p) if νp(H) divides νp(G) for every subgroup H of G. In this paper, we show that “almost

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive n-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links

Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first

Let q be a p-power where p is a fixed prime. In this paper, we look at the p-power maps on unitriangular group U(n,q) and triangular group T(n,q). In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the related (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.

Let n≥1 be a squarefree integer, and let M, A be two groups of order n. Using our previous results on the enumeration of Hopf–Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group M and additive group A. As an application, we enumerate skew braces whose order is the product of three distinct primes

In this paper, we study the category of finitely generated modules over a class of right 4-Nakayama artin algebras. This class of algebras appear naturally in the study of representation-finite artin algebras. First, we give a characterization of right 4-Nakayama artin algebras. Then, we classify finitely generated indecomposable right modules over right 4-Nakayama artin algebras. We also compute almost

We study knowledge bases as a certain algebraic and model-theoretical structure. Algebraic model of a knowledge base allows to distinguish some algebraic and logical invariants of knowledge bases. In particular, our interest is to find out how these invariants are related to the informational equivalence of knowledge bases.