Let S be a commutative ring, x,y∈S a pair of exact zero divisors, and R=S/(x). Let F be a complex of free R-modules. In this paper we explicitly compute cohomological operators of R over S by constructing endomorphisms of F. We consider some properties of these cohomological operators, as well as provide an example in which these cohomological operators act non-trivially.

Let Ḡ be the simple algebraic supergroup SL(m|n) or 𝔬𝔰𝔭(m|2n) over ℂ. Let 𝔤=Lie(Ḡ)=𝔤0̄⊕𝔤1̄ and let G=Ḡ(ℂ), where ℂ is considered as a superalgebra concentrated in even degree. Suppose e∈𝔤0̄ is nilpotent. We describe the centralizer 𝔤e of e in 𝔤 and its center 𝔷(𝔤e). In particular, we give bases for 𝔤e, 𝔷(𝔤e) and (𝔷(𝔤e))Ge. We also determine the labeled Dynkin diagram Δ with respect

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1

We prove a determinantal type formula to compute the characters of a class of finite-dimensional irreducible representations of the general Lie super-algebra 𝔤𝔩(m|n) in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula, originally conjectured by van der Jeugt and Moens, can be regarded as a generalization of the well-known Jacobi–Trudi

In this paper, we construct planar polynomials of the type fA,B(x)=x(xq2+Axq+Bx)∈𝔽q3[x], with A,B∈𝔽q. In particular, we completely classify the pairs (A,B)∈𝔽q2 such that fA,B(x) is planar using connections with algebraic curves over finite fields.

We prove an analog of Deligne’s theorem for finite symmetric tensor categories 𝒞 with the Chevalley property over an algebraically closed field k of characteristic 2. Namely, we prove that every such category 𝒞 admits a symmetric fiber functor to the symmetric tensor category 𝒟 of representations of the triangular Hopf algebra (k[d]/(d2),1⊗1+d⊗d). Equivalently, we prove that there exists a unique

The sum-essential graph 𝒮R(M) of a left R-module M is a graph whose vertices are all nontrivial submodules of M and two distinct submodules are adjacent if and only if their sum is an essential submodule of M. Properties of the graph 𝒮R(M) and its subgraph 𝒫R(M) induced by vertices which are not essential as submodules of M are investigated. The interplay between module properties of M and properties

In this paper, we consider Hom-Lie groups and introduce left invariant almost contact structures on them (almost contact Hom-Lie algebras). On such Hom-Lie groups, we construct the almost contact metrics and the contact forms. We give the notion of normal almost contact Hom-Lie algebras and describe K-contact and Sasakian structures on Hom-Lie algebras. Also, we study some of their properties. In addition

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field 𝔽 can be reduced to the following three classifications, for

For n,m∈ℕ, let Hn,m be the dual of the Radford algebra of dimension n2m. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter–Drinfeld modules over Hn,m. Along the way, we describe the simple objects in Hn,mHn,m𝒴𝒟 and their projective envelopes. Then we determine those simple modules that give rise to finite-dimensional Nichols algebras for the case n=2. There

We investigate the representations of the hyperalgebras associated to the map algebras 𝔤⊗𝒜, where 𝔤 is any finite-dimensional complex simple Lie algebra and 𝒜 is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that

We compute all liftings of braidings of unidentified types 𝔲𝔣𝔬(7) and 𝔲𝔣𝔬(8). Some of these examples had been previously computed by Helbig. We also present a list of examples of liftings of modular type 𝔟𝔯(2,a). Watching a coast as it slips by the ship is like thinking about an enigma. There it is before you, smiling, frowning, inviting, grand, mean, insipid, or savage, and always mute with

Let Fq be a finite field with q elements, n≥2 a positive integer, T(n,q) the semigroup of all n×n upper triangular matrices over Fq under matrix multiplication, T∗(n,q) the group of all invertible matrices in T(n,q), PT∗(n,q) the quotient group of T∗(n,q) by its center. The one-divisor graph of T(n,q), written as 𝕋n(q), is defined to be a directed graph with T(n,q) as vertex set, and there is a directed

Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional algebra which is not self-injective, but such that all simple modules are reflexive (actually, for this example, the simple modules are the only non-projective indecomposable

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra H a Yetter–Drinfeld module braids from the left with H-modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and H-modules. In particular, we will describe

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra Dω(H) defined in [D. Bulacu and F. Panaite, A generalization of the quasi-Hopf algebra Dω(G), Commun. Algebra26 (1998) 4125–4141] and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for

We study the deformation of Courant pairs with a commutative algebra base. We consider the deformation cohomology bi-complex and describe a universal infinitesimal deformation. In a sequel, we formulate an extension of a given deformation of a Courant pair to another with extended base, which leads to describe the obstruction in extending a given deformation. We also discuss the construction of versal

Large order cryptographically suitable quasigroups have important applications in the development of crypto-primitives and cryptographic schemes. These present new perspectives of cryptography and information security. From algebraic point of view polynomial completeness is one of the most important characteristic for cryptographically suitable quasigroups. In this paper, we propose four different

Over an infinite field K, we investigate the minimal free resolution of some configurations of lines. We explicitly describe the minimal free resolution of complete grids of lines and obtain an analogous result about the so-called complete pseudo-grids. Moreover, we characterize the total Betti numbers of configurations that are obtained posing a multiplicity condition on the lines of either a complete

For a finite group G, we associate the quantity β(G)=|L(G)||G|, where L(G) is the subgroup lattice of G. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of β on the class of p-groups of order pn, where n≥3 is an integer. We show that the set containing the quantities β(G), where G is a finite (abelian) group, is dense

A right R-module M is called a U-module if, whenever A and B are submodules of M with A≅B and A∩B=0, there exist two direct summands K and T of M such that A⊆essK, B⊆essT and K⊕T⊆⊕M. The class of U-modules is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper we study the rings whose cyclics are U-modules, extending many of the

It is well known that Frobenius groups can be defined by their complement subgroups. But until now we cannot use a complement subgroup to define a modular Frobenius group. In the present paper, a generalization of Frobenius complements is used as a characterization of a class of modular Frobenius groups. In fact, we build a connection between modular Frobenius groups and Frobenius–Wielandt groups.

A ring R is said to be quasi-central Armendariz if f(x)=∑i=0maixi and g(x)=∑j=0nbjxj∈R[x] satisfy f(x)g(x)=0 then Raibj=aibjR for all i and j. It is proved that if R is a quasi-central Armendariz ring then f(x)g(x)=0 implies that all aibj are in its Wedderburn radical W(R), generalizing and improving the existing result in the literature.

Let R be a unitary ring of finite cardinality pβk, where p is a prime number and p∤k. We show that if the group of units of R has at most one subgroup of order p, then R≅A⊕B, where B is a finite ring of order k and A is a ring of cardinality pβ which is one of the six explicitly described types.

This paper is devoted to study 2-local derivations on W-algebra W(2,2) which is an infinite-dimensional Lie algebra with some outer derivations. We prove that all 2-local derivations on the W-algebra W(2,2) are derivations. We also give a complete classification of the 2-local derivation on the so-called thin Lie algebra and prove that it admits many 2-local derivations which are not derivations.

Let Λ be an algebra with an indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support τ-tilting Λ-modules. In this paper, we will show that it can be restricted to τ-tilting modules.

A notion of n-cotorsion pairs in an extriangulated category with enough projectives and enough injectives is defined in this paper. We show that there exists a one-to-one correspondence between n-cotorsion pairs and (n+1)-cluster tilting subcategories. As an application, this result generalizes the work by Huerta, Mendoza and Pérez in an abelian case. Finally, we give some examples illustrating our

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra130(1) (1990) 17–64.], we introduce the notion of partial matched pair

Suppose that G is a finite group and H is a subgroup of G. H is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G=HB and H permutes with every Sylow subgroup of B. In this note, we fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1<|D|<|P| and study the p-nilpotency of G under the assumption that every subgroup H of P with |H|=|D| is SS-quasinormal

We introduce and study (strongly) π-Rickart objects and their duals in abelian categories, which generalize (strongly) self-Rickart objects and their duals. We establish general properties of such objects, we analyze their behavior with respect to coproducts, and we study classes all of whose objects are (strongly) π-Rickart. We derive consequences for module and comodule categories.

We study the maximum Hamming distance (or rather, the complementary notion of “minimum approximability”) of a general function on a finite group G to either of the sets End(G) and Aff(G), of group endomorphisms of G and affine maps on G, respectively, the latter being a certain generalization of endomorphisms. We give general bounds on these two quantities and discuss an infinite class of extremal

The aim of this paper is twofold. First, to provide a characterization of clopen topologies. We show that a topology is clopen if and only if it is symmetric and Alexandroff. Second, to investigate when some special unitary subrings of a power set define a (clopen) topology.

In this paper, by taking the class of all C3 (or D3) right R-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via D3-covers (or D3-envelopes) and a right V-ring (or a right noetherian V-ring) via C3-covers (or C3-envelopes). By using isosimple-projective preenvelope, we obtained that if R is a semiperfect right FGF ring (or left coherent

The objective of this paper is to determine the finite-dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to those of the infinite dihedral group, we give the algebra the name iD∞.

An element of a ring R is said to be r-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring R are r-precious, then R is called an r-precious ring. We study some basic properties of r-precious rings. We also characterize von Neumann regular elements in M2(R) when R is a Euclidean domain and by this argument

We prove that if G is either a hypercentral-by-finite group or a soluble Baer group and if G has finitely many non-isomorphic factor-groups, then G is a Chernikov group. The converse is also true. Furthermore, we give some information on the structure of a metabelian group with finitely many non-isomorphic factor-groups.

The authors of Berg et al. [J. Algebra348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for ℂℳ, where ℳ is any finite ℛ-trivial monoid. Their method relies on a technical result stating that ℛ-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an ℛ-trivial

Under suitable conditions the skewfield of fractions of a superalgebra which is a noetherian domain is canonically provided with a structure of superalgebra. This gives rise to a notion of rational equivalence in the category of superalgebras. We study from the point of view of this rational equivalence some low dimensional examples of quantum enveloping algebras of Lie superalgebras.

We study actions of pointed Hopf algebras on matrix algebras. Our approach is based on known facts about group gradings of matrix algebras [Y. Bahturin, S. Sehgal and M. Zaicev, Group gradings on associative algebras, J. Algebra241 (2001) 667–698] and other sources.

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane

We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.

Let A=A(α,β) be a graded down-up algebra with (degx,degy)=(1,n) and β≠0, and let ∇A be the Beilinson algebra of A. If n=1, then a description of the Hochschild cohomology group of ∇A is known. In this paper, we calculate the Hochschild cohomology group of ∇A for the case n≥2. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of A is

In this paper, the concept of “Inverse Complemented Matrix Method”, introduced by Eagambaram (2018), has been reestablished with the help of minus partial order and several new properties of complementary matrices and the inverse of complemented matrix are discovered. Class of generalized inverses and outer inverses of given matrix are characterized by identifying appropriate inverse complement. Further

In this paper, we are mainly interested in the two questions “which are the commutative rings on which every finitely presented modules is 1-periodic (respectively, 2-periodic)?”. It is proved that these kinds of rings are particular cases of semiregular rings. So, we call them 1-semiregular and 2-semiregular rings, respectively. We establish characterizations of these rings in terms of various classical

We give a complete classification of pointed fusion categories over ℂ of global dimension p3 for p any odd prime. We proceed to classify the equivalence classes of pointed fusion categories of dimension p3 and we determine which of these equivalence classes have equivalent categories of modules.

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module M over a semiprimary

In this paper, we consider finite 2-groups having the property that every non-abelian subgroup contains its centralizer. Such groups are classified completely up to isomorphism.

Here, we continue to characterize a recently introduced notion, le-modules RM over a commutative ring R with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga.158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec(M) of all prime submodule elements of M. Thus, we extend many results on Zariski

It is well known that if δ:S→S is a derivation in semiring S, then in the semiring Mn(S) of n×n matrices over S, the map δher such that δher(A)=(δ(aij)) for any matrix A=(aij)∈Mn(S) is a derivation. These derivations are used in matrix calculus, differential equations, statistics, physics and engineering and are called hereditary derivations. On the other hand (in sense of [Basic Algebra II (W. H.

In this paper, we find a presentation of a Leavitt inverse semigroup and give normal forms of elements of a Leavitt inverse semigroup.

This paper proposes the matrix representation of formal polynomials over max-plus algebra and obtains the maximum and minimum canonical forms of a polynomial function by standardizing this representation into a canonical form. A necessary and sufficient condition for two formal polynomials corresponding to the same polynomial function is derived. Such a matrix method is constructive and intuitive,

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and

The aim of the paper is to characterize the equation 1+Δ(R)=U(R), where J(R)⊆Δ(R)={x∈R:x+u∈U(R) for all u∈U(R)} that is the largest Jacobson radical subring of R and U(R) is the set of invertible elements of a ring R. We show that this equation is closely related to UJ-rings and rings whose elements can be written as the sum of an idempotent and an element from Δ(R). After presenting several characterizations

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an E-unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying E-unitary cover for inverse semigroups, and as a continuation of Petrich’s works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining

It was shown in [Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq.4 (1997) 121–131.] that each semimodule over an additively regular semiring has an injective envelope. We show the converse statement is true as well; moreover, the result holds even if each finitely generated semimodule has an injective envelope.

We prove that for all non-abelian finite simple groups S, there exists a fake mth Galois action on IBr(X) with respect to X◃X⋊Aut(X), where X is the universal covering group of S and m is any non-negative integer coprime to the order of X. This is one of the two inductive conditions needed to prove an ℓ-modular analogue of the Glauberman–Isaacs correspondence.

An R-module V over a semiring R lacks zero sums (LZS) if x+y=0 implies x=y=0. More generally, a submodule W of V is “summand absorbing” (SA), if, for all x,y∈V, x+y∈W⇒x∈W,y∈W. These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module

Denote the alternating and symmetric groups of degree n by An and Sn, respectively. Consider a permutation σ∈Sn, all of whose nontrivial cycles are of the same length. We find the minimal polynomials of σ in the ordinary irreducible representations of An and Sn.

A ring R is called weakly principally quasi Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left s-unital by left semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. We study the relationship between the weakly p.q.-Baer property of a ring R and those of the skew inverse series rings R((x−1;σ,δ)) and

In this paper, we classify the smooth orientation preserving cyclic p-group actions on the real projective space ℝℙ3 up equivalence, where two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms. We view ℝℙ3 as the lens space L(2,1). We show that any such action on ℝℙ3 is conjugate to a standard action explicitly defined, and we identify the quotient spaces of