A unit-picker is a map 𝒢 that associates to every ring R a well-defined set 𝒢(R) of central units in R which contains 1R and is invariant under isomorphisms of rings and closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let 𝒢 be a unit-picker. A ring R is called (strongly) nil 𝒢-clean if for each a∈R, a=ve+b where

A Hom-structure on a Lie algebra (𝔤,[⋅,⋅]) is a linear map σ:𝔤→𝔤 which satisfies the Hom–Jacobi identity [[x,y],σ(z)]+[[z,x],σ(y)]+[[y,z],σ(x)]=0 for all x,y,z∈𝔤. A Hom-structure is called regular if σ is also a Lie algebra isomorphism. Let 𝒩 be the Lie algebra consisting of all strictly upper triangular (n+1)×(n+1) matrices over a field 𝔽. In this paper, we prove that if n≥4, any regular Hom-structure

Let R be a ring, Q a small k-preadditive category, and let Q,RMod be the category of k-linear functors from Q to the left R-module category RMod. Given a cotorsion pair (𝒜,ℬ) in RMod, we construct four cotorsion pairs (Γ(𝒜),Γ(𝒜)⊥), (⊥Γ(ℬ),Γ(ℬ)), (Θ(𝒜),Θ(𝒜)⊥) and (⊥ξ(ℬ),ξ(ℬ)) in Q,RMod and investigate when these cotorsion pairs are hereditary and complete. Moreover, under few assumptions, we show

Let 𝔤=𝔤0̄⊕𝔤1̄ be a finite-dimensional simple Lie superalgebra of type D(2,1;α), G(3) or F(4) over ℂ. Let G be the simply connected semisimple algebraic group over ℂ such that Lie(G)=𝔤0̄. Suppose e∈𝔤0̄ is nilpotent. We describe the centralizer 𝔤e of e in 𝔤 and its center 𝔷(𝔤e) especially. We also determine the labeled Dynkin diagram for e. We prove theorems relating the dimension of 𝔷(𝔤e)Ge

First Grigorchuk group 𝒢 and Grigorchuk’s overgroup 𝒢̃, introduced in 1980, are self-similar branch groups with intermediate growth. In 1984, 𝒢 was used to construct the family of generalized Grigorchuk groups {Gω|ω∈{0,1,2}ℕ}, which has many remarkable properties. Following this construction, we generalize the Grigorchuk’s overgroup 𝒢̃ to the family {G̃ω|ω∈{0,1,2}ℕ} of generalized Grigorchuk’s

For q=3r (r∈ℕ), denote by 𝔽q the finite field of order q and for a positive integer m≥2, let 𝔽qm be its extension field of degree m. We establish a sufficient condition for existence of a primitive normal element α such that f(α) is a primitive element, where f(x)=ax2+bx+c, with a,b,c∈𝔽qm satisfying b2≠ac in 𝔽qm.

This paper considers a new alphabet set, which is a ring that we call 𝔽4R, to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize 𝔽4R-skew cyclic codes. Interesting connections between the image

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every 4-dimensional (4D) real unital algebra 𝒜 into an 8D real unital algebra denoted by FD(𝒜). We also find the conditions on 𝒜 under which FD(𝒜) is a division algebra. This covers the most classes of known 4D real division algebras. The second process allows us to

We investigate the structure and properties of an Artinian monomial complete intersection quotient A(n,d)=𝕂[x1,…,xn]/(x1d,…,xnd). We construct explicit homogeneous bases of A(n,d) that are compatible with the Sn-module structure for n=3, all exponents d≥3 and all homogeneous degrees j≥0. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible

Let (A,𝔪) be a Noetherian local ring of dimension d>0 and I an 𝔪-primary ideal of A. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring A to be passed onto the associated graded ring of filtration. Let ℐ denote an I-good filtration. We prove that if A is Buchsbaum and the 𝕀 -invariant, 𝕀(A) and 𝕀(G(ℐ)), coincide then the associated graded ring G(ℐ) is Buchsbaum

Let Γ(R), ΓE(R), ΓAnn(R) denote the zero-divisor graph, compressed zero-divisor graph and annihilating ideal graph of a commutative ring R, respectively. In this paper, we prove that ΓE(R)≅ΓAnn(R) for a semisimple commutative ring R and represent Γ(R) as a generalized join of a finite set of graphs. Further, we study the zero-divisor graph of a semisimple group-ring 𝔽qCn and proved several structural

We prove an analogue of the Kac–Wakimoto conjecture for the periplectic Lie superalgebra 𝔭(n), stating that any simple module lying in a block of non-maximal atypicality has superdimension zero.

In this paper, we consider semisimple group algebras 𝔽qG of split metacyclic groups over finite fields. We construct left codes in 𝔽qG in the case when the order G is pmℓn, where p and ℓ are different primes such that gcd(q,p,ℓ)=1 extend the construction described in a previous paper, determine their dual codes and find some good codes.

We consider the Graver basis, the universal Gröbner basis, a Markov basis and the set of the circuits of a toric ideal. Let A,B be any two of these bases such that A⊉B in general, we prove that there is no polynomial on the size or on the maximal degree of the elements of A which bounds the size or the maximal degree of the elements of B correspondingly.

The notion of maximal non-ϕ-pseudo-valuation ring is introduced which generalizes the concept of maximal non-pseudo-valuation domain. The equivalence of maximal non-ϕ-PVR and maximal non-local ring is established under some conditions. The concept of maximal non-PVR is also introduced and it is shown that a maximal non-PVR is either a ϕ-PVR or a maximal non-ϕ-PVR. Examples are also given to show that

Let D be an integral domain with quotient field K, R be an overring of D, X be an indeterminate over R and E be a subset of K. We consider the ring of D-valuedR-polynomials onE, denoted by IntR(E,D), formed by the polynomials f∈R[X] such that f(a)∈D for each a∈E, that is, IntR(E,D):={f∈R[X]:f(E)⊆D}. In this paper, we study localization properties, local freeness and faithful flatness of IntR(E,D) over

In this paper, we study biprojectivity of generalized module extension Banach algebras and second dual of Banach algebras. As a main result, we prove that if I is a contractible closed ideal of A such that A/I is biprojective, then A is biprojective. As a consequence, we give some results on biprojectivity of generalized module extension Banach algebras. Indeed, this is a generalization of results

The main purpose of this paper is to answer a question which was left open in N. Jarboui and S. Aljubran [Maximal non-integrally closed subrings of an integral domain, Ric. Mat. (2020), https://doi.org/10.1007/s11587-020-00500-0] asking for a characterization of maximal non-integrally closed subrings of arbitrary rings.

In this paper, we consider fusion categories generated by a self-dual simple object of FP-dimension 2. We use the classification results of indecomposable generalized Cartan matrices to determine possible types of these fusion categories. It turns out that there are only five types for such categories. Moreover, the fusion rules of all types of these fusion categories are explicitly described according

In this paper, we study non-central almost subnormal subgroups of the multiplicative group of a division ring satisfying a nonzero generalized rational identity. The main result generalizes Chiba’s theorem on subnormal subgroups. As an application, we get a theorem on almost subnormal subgroups satisfying a generalized algebraic rational identity. The last theorem has several corollaries which generalize

The goal of this paper is to supply an explicit description of the universal factorization algebra of the generic polynomial of degree n into the product of two monic polynomials, one of degree r, as a representation of Lie algebras of n×n matrices with polynomial entries. This is related with the bosonic vertex representation of the Lie algebra gl∞ due to Date, Jimbo, Kashiwara and Miwa.

We prove that the profinite completion construction is a covariant functor from the category of (universal) algebras of a given type into the category of the corresponding Stone algebras. A Grothendieck problem for finitely presented MV-algebras is also formulated and solved. Finally, we characterize finitely presented MV-algebras for which profinite completions and MacNeille completions coincide.

We prove that a valued field of positive characteristic p that has only finitely many distinct Artin–Schreier extensions (which is a property of infinite NTP2 fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has p-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open

An R-module M is called strongly D2 if M(n) is a D2 (equivalently, direct projective) module for every positive integer n. In this paper, we consider the class of quasi-projective R-modules, the class of strongly D2R-modules and the class of D2R-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations

Let G be a finite group. We say that a conjugacy class of g in G is vanishing if there exists some irreducible character χ of G such that χ(g)=0. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence we obtain that every central simple algebra of degree 16 is split by a 2-extension of degree at most 216.

Let G be a finite group. A sequence over G means a finite sequence of terms from G, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over G (with the concatenation of sequences as the operation) is a finitely generated C-monoid

In this paper, the s.Baer module concept and some of its generalizations (e.g. quasi-s.Baer, π-s.Baer and p.q.-s.Baer) are developed. To this end, we characterize the class of rings for which every module is quasi-s.Baer as the class of rings which are finite direct sums of simple rings. Connections are made between the s.Baer (quasi-s.Baer, π-s.Baer) and the extending (FI-extending, π-extending) properties

In this paper we enumerate the skew braces of non-abelian type of size p2q for p,q primes with q>2 by the classification of regular subgroups of the holomorph of the non-abelian groups of the same order. Since Crespo dealt with the case q=2, this paper completes the enumeration of skew braces of size p2q started in a previous work by the authors. In some cases, we provide also a structural description

Let G be a finite group and ψ(G)=∑g∈Go(g), where o(g) denotes the order of g. The function ψ′′(G)=ψ(G)/|G|2 was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9], some lower bounds for ψ″(G) are determined such that if ψ″(G) is greater than each of them, then

The Grigorchuk and Gupta–Sidki groups play a fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction

Let R be a ring. By the notation [R,R] we denote the additive subgroup of R generated by all [a,b]=ab−ba in R. In this work, we partially generalize a result due to Herstein [I. N. Herstein, Topics in Ring Theory (University of Chicago Press, 1969)] showing that if 1∈[R,R], then the subring generated by [R,R] is equal to R. This result implies that [R,R] cannot be a proper subring of R.

In this paper, we classify all finite solvable groups having at most two imprimitive irreducible characters.

In this note, we prove that vector bundles which satisfy the point property over a very general principally polarized Jacobian, Prym and abelian variety are indecomposable. We also compare two known constructions of vector bundles satisfying the point property over a very general principally polarized abelian surface.

An overring Ro of an integral domain R is said to be comparable if Ro≠R, Ro≠qf(R), and each overring of R is comparable to Ro under inclusion. We do provide necessary and sufficient conditions for which R has a comparable overring. Several consequences are derived, specially for minimal overrings, or in the case where the integral closure R¯ of R is a comparable overring, or also when each chain of

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right ℝ-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump’s correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of

In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on C(X) to characterize spaces X when every irreducible ideal of C(X) is pseudoprime. We

Let N be a nontrivial normal subgroup of G and ∅≠π⊆π(N). In this paper, we show that if 2∉π and for every π-element x∈N−Z(N) with |π(x)|≤2, |clG(x)|=m, for some integer m, then N has nilpotent Hall π-subgroups. Further, we show that if |π(N/Z(N))∩π|≥2, then N has abelian Hall π-subgroups.

It is well known that injective objects play a fundamental role in many branches of mathematics. The question whether a given category has enough injective objects has been investigated for many categories. Also, quasi-injective modules and acts have been studied by many categorists. In this paper, we study quasi-injectivity in the category of actions of an ordered monoid on ordered sets (Pos-S) with

Let G be a group and φ be an automorphism of G. Two elements x,y∈G are said to be φ-twisted conjugate if y=gxφ(g)−1 for some g∈G. We say that a group G has the R∞-property if the number of φ-twisted conjugacy classes is infinite for every automorphism φ of G. In this paper, we prove that twisted Chevalley groups over a field k of characteristic zero have the R∞-property as well as the S∞-property if

To each symmetric graded Frobenius superalgebra we associate a W-algebra. We then define a linear isomorphism between the trace of the Frobenius Heisenberg category and a central reduction of this W-algebra. We conjecture that this is an isomorphism of graded superalgebras.

In this paper, we prove that if R is an Archimedean reduced ring and satisfy ACC on annihilators, then R[[x]] is also an Archimedean reduced ring. More generally, we prove that if R is a right Archimedean ring satisfying the ACC on annihilators and α is a rigid automorphism of R, then the skew power series ring R[[x;α]] is right Archimedean reduced ring. We also provide some examples to justify the

Let R be a commutative ring with identity. Let R[x] and R[[x]] be the collection of polynomials and, respectively, of power series with coefficients in R. There are a lot of multiplications in R[x] and R[[x]] such that together with the usual addition, R[x] and R[[x]] become rings that contain R as a subring. These multiplications are from a class of sequences λ={λn}n=0∞ of positive integers. The trivial

As introduced by Cǎlugǎreanu and Lam in [G. Cǎlugǎreanu and T. Y. Lam, Fine rings: a new class of simple rings, J. Algebra Appl.15(9) (2016) 1650173, 18 pp.], a fine ring is a ring whose every nonzero element is the sum of a unit and a nilpotent. As a natural generalization of fine rings, a ring is called a generalized fine ring if every element not in the Jacobson radical is the sum of a unit and

The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a residuated poset. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly

The Hilbert curve of a complex polarized manifold (X,L) is the complex affine plane curve of degree dim(X) defined by the Hilbert-like polynomial χ(xKX+yL), where KX is the canonical bundle of X and x and y are regarded as complex variables. A natural expectation is that this curve encodes several properties of the pair (X,L). In particular, the existence of a fibration of X over a variety of smaller

We consider Anderson t-motives M of dimension 2 and rank 4 defined by some simple explicit equations parameterized by 2×2 matrices. We use methods of explicit calculation of h1(M) — the dimension of their cohomology group H1(M) (=the dimension of the lattice of their dual t-motive M′) developed in our earlier paper. We calculate h1(M) for M defined by all matrices of the form 0a12a210, and by some

We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous q-Hermite polynomials and the star product

We give the classification of 5- and 6-dimensional complex one-generated nilpotent assosymmetric algebras.

A proper ideal I of a commutative ring R is called lifting whenever idempotents of R/I lift to idempotents of R. In this paper, many of the basic properties of lifting ideals are studied and we prove and extend some well-known results concerning lifting ideals and lifting idempotents by a new approach. Furthermore, we give a necessary and sufficient condition for every proper ideal of a commutative

Let M be a finitely generated torsion-free module over a generalized Dedekind domain D. It is shown that if M is a projective D-module, then it is a generalized Dedekind module and v-multiplication module. In case D is Noetherian it is shown that M is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module M[x] is a generalized Dedekind D[x]-module (a Krull D[x]-module)

We develop categorical and number-theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank 8. In particular we find three distinct families of prime categories in rank 8 in contrast to the lower rank cases for which there is only one such family.

We prove that a uniform pro-p group with no non-abelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche–Wendt groups.

We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial n-Kummerian extension is equivalent to either a radical or an I-radical

Let Λ be a radical square zero Nakayama algebra with n simple modules and let Γ be the Auslander algebra of Λ. Then every indecomposable direct summand of a tilting Γ-module is either simple or projective. Moreover, if Λ is self-injective, then the number of tilting Γ-modules is 2n; otherwise, the number of tilting Γ-modules is 2n−1.

In this paper, we study the word map w:SU(2)×SU(2)→SU(2), where w is a word in the free group F of rank 2. We give a necessary and sufficient condition for the surjectivity of the word map by using the trace polynomials. As applications of our method, we give example of word w∉F(2) for which the word map is not surjective. For families of words belonging to F(2) of the form [[a,b],[a,bn]] and [[a,b]

Let Cn denote the cyclic group of order n, and let Hol(Cn) denote the holomorph of Cn. In this paper, for any odd integer m≥3, we find necessary and sufficient conditions on an integer A, with |A|≥3, such that 𝔉m,A(x)=x2m+Axm+1 is irreducible over ℚ. When m=q≥3 is prime and 𝔉q,A(x) is irreducible, we show that the Galois group over ℚ of 𝔉q,A(x) is isomorphic to either Hol(Cq) or Hol(C2q), depending

Let R be a commutative ring, we say that 𝒜⊆Spec(R) has prime avoidance property, if I⊆⋃P∈𝒜P for an ideal I of R, then there exists P∈𝒜 such that I⊆P. We exactly determine when 𝒜⊆Spec(R) has prime avoidance property. In particular, if 𝒜 has prime avoidance property, then 𝒜 is compact. For certain classical rings we show the converse holds (such as Bezout rings, QR-domains, zero-dimensional rings

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of

Let A be a residually finite dimensional algebra (not necessarily associative) over a field k. Suppose first that k is algebraically closed. We show that if A satisfies a homogeneous almost identity Q, then A has an ideal of finite codimension satisfying the identity Q. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra L over k is almost d-Engel