Abstract Let H be a transfer Krull monoid over a subset G 0 of an abelian group G with finite exponent. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · … · u k . The set L ( a ) of all possible factorization lengths k is called the set of lengths of a, and H is said to be half-factorial if | L ( a ) | = 1 for all a ∈ H . We show that, if a ∈ H is a non-unit and

If a finite quasisimple group G with simple quotient S is embedded into a suitable classical group X through the smallest degree of a projective representation of S, then N X ( G ) is a maximal subgroup of X, up to two series of exceptions where S is a Ree group, and four exceptions where S is sporadic.

Let G be a finite group. A subgroup H of G is called S-permutable in G if H permutes with all Sylow subgroups of G. A subgroup H of G is called SS-supplemented in G if there exists a subgroup K of G such that HK = G and H ∩ K is S-permutable in K. In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of a Sylow subgroup of G are SS-supplemented

The aim of this paper is twofold. Firstly, to give short proofs of the celebrated results of Rudd (1970), De Marco (1972), Brookshear (1977), Neville (1990), and Vechtomov (1996) with the help of some well-known results in ring theory. Secondly, to establish new algebraic characterizations for when C(X) is von Neumann regular, semihereditary, or a Bézout ring. The notions of the greatest common divisor

Quasi-Frobenius rings are precisely rings over which any right module is a homomorphic image of an injective module. We investigate the structure of rings whose proper cyclic right modules are homomorphic image of injectives. The class of such rings properly contains that of right self-injective rings. We obtain some structure theorems for rings satisfying the said property and apply them to the Artin

Let R be a commutative ring with identity and X r 1 ( R ) the set of regular height one prime ideals of R. We will show that R is a Krull ring if and only if each regular prime ideal of R contains a t-invertible prime ideal, if and only if (1) R = ∩ P ∈ X r 1 ( R ) R [ P ] and it has finite character and (2) ( R [ P ] , [ P ] R [ P ] ) is a rank one DVR for each P ∈ X r 1 ( R ) . It is also shown that

Let D be an integral domain. Then D is an almost valuation (AV)-domain if for a , b ∈ D ∖ { 0 } there exists a natural number n with a n | b n or b n | a n . AV-domains are closely related to valuation domains, for example, D is an AV-domain if and only if the integral closure D ¯ is a valuation domain and D ⊆ D ¯ is a root extension. In this note, we explore various generalizations of DVRs (which

In this note we prove the inductive blockwise Alperin weight condition for PSL ( 3 , 4 ) by establishing an equivariant bijection between 3-irreducible Brauer character and 3-weights of the 3 ′ -cover of PSL ( 3 , 4 ) . Thus the inductive blockwise Alperin weight condition holds for the simple groups PSL ( 3 , q ) for every prime dividing its order.

In this article, we study modules for twisted full toroidal lie algebras (TFTLA). We define a category of bounded modules for TFTLA and classify all the irreducible modules in that category. A class of irreducible bounded modules is explicitly constructed by Michael Lau and Yuly Billig using Vertex Operator Algebras. We show that they exhaust all the irreducible bounded modules. We prove that the top

In this article, we study homotopes of finite-dimensional algebras (not necessarily, associative). In the case of associative algebras, we study homotopes by the methods of Category theory and give description of the so-called well-tempered elements of finite-dimensional associative algebra in algebraic terms.

In this article, we prove that if H is a non-normal subgroup of a finite nilpotent group G non-isomorphic to the dihedral group D 8 of order 8, then the number of isomorphism classes of normalized right transversals (NRTs) of H in G is greater than 16, where the isomorphism classes are formed with respect to the induced right-quasigroup structure (induced by the binary operation of G) on each NRT of

In this work, we use probability groups, introduced by Harrison in 1979, as a tool to study a semisimple Hopf algebra H with a commutative character ring and prove that the algebra generalized by the dual probability group is the center Z ( H ) of H and the product of two class sums is an integral combination up to a factor of dim ( H ) − 1 of the class sums of H. We classify all the 2-integral probability

We study the classes of free and plus-one generated hyperplane arrangements. Specifically, we describe how to compute the associated prime ideals of the Jacobian ideal of such an arrangement from its lattice of intersection. Moreover, we prove that the localization of a plus-one generated arrangement is free or plus-one generated.

Let A be an abelian group. If for any x , y ∈ A we define x ⊳ y = 2 x − y , then ( A , ⊳ ) is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by R ( A ) . Indeed, we characterize the subquandles of T(A) and show that R ( A ) is the lattice of cosets of A if and only if A is an odd abelian

Let p be a prime. Let G be a finite group, and let χ be an irreducible character of G. Suppose F is a finite extension of Q p , the field of p-adic numbers. If all the values of χ are in F, then, associated with χ, is a specific element of Br(F) the Brauer group of F, and, since Br(F) is canonically isomorphic to Q / Z , χ has, uniquely associated with it, an element of Q / Z . Some recent conjectures

We study iterated differential polynomial rings over a locally nilpotent ring and show that a large class of such rings are Behrens radical. This extends results of Chebotar and Chen et al.

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.

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 ∈ G g B = 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.

Let G be a reductive split p-adic group and let U be the unipotent radical of a Borel subgroup. We study the cohomology with trivial Zp-coefficients of the profinite nilpotent group N=U(OF) and its Lie algebra n, by extending a classical result of Kostant to our integral p-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.

We introduce the notion of strongly G-regular rings, and we observe that most rings that were known to be G-regular are in fact strongly G-regular. This property allows us to describe totally acyclic complexes over tensor product rings in which one of the factors is strongly G-regular and the other factor is Gorenstein.

We study the structure of bounded simple weight sl(∞)-, o(∞)-, sp(∞)-modules, which have been recently classified by D. Grantcharov and I. Penkov. Given a splitting parabolic subalgebra p of sl(∞),o(∞),sp(∞), we introduce the concepts of p-aligned and pseudo p-aligned sl(∞)-, o(∞)-, sp(∞)-modules, and give necessary and sufficient conditions for bounded simple weight modules to be p-aligned or pseudo

Let Ĝ be the finite simply connected version of an exceptional Chevalley group, and let V be a nontrivial irreducible module, of minimal dimension, for Ĝ over its field of definition. We explore the overgroup structure of Ĝ in GL(V), and the submodule structure of the exterior square (and sometimes the third Lie power) of V. When Ĝ is defined over a field of odd prime order p, this allows us to

We give an algebraic classification of complex 5-dimensional one-generated nilpotent terminal algebras.

A quasi-equigenerated monomial ideal I in the polynomial ring R=k[x1,…,xn] is a Freiman ideal if μ(I2)=l(I)μ(I)−(l(I)2) where l(I) is the analytic spread of I and μ(I) is the number of minimal generators of I. Freiman ideals are special since there exists an exact formula computing the minimal number of generators of any of their powers. In this work, we address the question of characterizing which

A global analog of Fontaine’s category of triples is introduced. While the latter is antiequivalent to the category of formal group laws over the ring of Witt vectors over a finite field of characteristic p, the introduced category is antiequivalent to the category of formal group laws over ℤ. Explicit constructions of kernels and cokernels in this category are given, thus proving that it is pre-abelian

Our aim in this work is to study exact Osserman limit linear series on curves of compact type X with three irreducible components. This case is quite different from the case of two irreducible components studied by Osserman. For instance, for curves of compact type with two irreducible components, every refined Eisenbud-Harris limit linear series has a unique exact extension. But, for the case of three

In this paper, we extend the notion of the Brauer–Clifford group to the case of an (S,G,H)-comodule algebra, where H is a commutative Hopf algebra, G is a Lie algebra in the symmetric monoidal category of right H-comodules, and S is a commutative algebra which is an H-comodule algebra, a G-module algebra and the H-coaction is compatible with the G-action. This Brauer–Clifford group turns out to be

The Hirzebruch–Milnor class is given by the difference between the homology Hirzebruch characteristic class and the virtual one. It is known that the Hirzebruch–Milnor class for a certain singular hypersurface can be calculated by using the Steenbrink spectrum of the local defining function at a point in each stratum of singular locus. Thus far there is no explicit calculation of this invariant for

In this article, we continue the study of strongly m-clean ring which we introduced in the paper “On m-clean and strongly m-clean rings” (Purkait et al. 2020). Mainly, we characterize strongly m-clean ring in terms of m-semiperfect ring. For that, we introduce the notion of m-semiperfect ring and establish some results on this ring. We prove that under certain conditions a ring is m-semiperfect if

The minus partial order is already known for sets of matrices over a field and bounded linear operators on arbitrary Hilbert spaces. Recently, this partial order has been studied on Rickart rings. In this paper, we extend the concept of the minus relation to the module theoretic setting and prove that this relation is a partial order when the module is regular. Moreover, various characterizations of

(2020). Correction to: Characterization of multiplication commutative rings with finitely many minimal prime ideals. Communications in Algebra: Vol. 48, No. 10, pp. 4554-4554.

Let G be an extra-special p-group of order p 2 n + 1 for n ≥ 2. The aim of this paper is to investigate what possible combinations of degrees (counting multiplicities) of irreducible α-characters of G can occur for [ α ] in the Schur multiplier M ( G ) , and then to determine the numbers of [ α ] ∈ M ( G ) that give the same combinations. Solutions are given when n = 2 and in general for p = 2. Writing

A ring R is called left AIP if R modulo the left annihilator of any ideal is flat. In this paper, we characterize a module MR for which the endomorphism ring E n d R ( M ) is left AIP. We say a module MR is endo-AIP (resp. endo-APP) if M has the property that “the left annihilator in E n d R ( M ) of every fully invariant submodule of M (resp. E n d R ( M ) m , for every m ∈ M ) is pure as a left ideal

Let G be a finite group, V a faithful finite-dimensional representation of G over the complex field C , and C ( V ) G be the corresponding invariant field. The Bogomolov multiplier B 0 ( G ) of G is canonically isomorphic to the unramified cohomological group H nr 2 ( C ( V ) G , ℚ / Z ) , which has been used by Saltman in 1984 and Bogomolov in 1988 to provide counterexamples to the rationality problem

In this article, we present the theory of standard bases for modules over polynomial subalgebras based on arbitrary gradings (induced by an additive monoid Γ) which we called SΓ-bases. This paper deals with the problem of construction of SΓ-bases for submodules of free module over polynomial subalgebras. For this purpose, we present an algorithm based on SΓ-normal form and discuss it for some special

We show that there exist mathematical 4-instanton bundles F on the projective 3-space such that F(2) is globally generated (by four global sections). This is equivalent to the existence of elliptic space curves of degree 8 defined by quartic equations. There is a (possibly incomplete) intersection theoretic argument for the existence of such curves in D’Almeida [Bull. Soc. Math. France 128 (2000),

Abstract Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let I⊂K[x1,…,xn] be a monomial ideal and let G(I) denote the (unique) minimal monomial generating set of I. How small can |G(Ii)| be in terms of |G(I)|? Until recently, it was widely expected that |G(I2)|≥|G(I)| would always hold. The first counterexamples emerged in 2018 for n = 2. In this article we show

Let G be a finite group, σ = { σ i | i ∈ I } a partition of the set of all primes P and σ ( G ) = { σ i | σ i ∩ 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 σ i ∈ σ ( G ) . G is said to be σ-full if G possesses a complete Hall

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.

In this article, we provide several descriptions of w-coherent rings in terms of modules. We show that a ring R is w-coherent if and only if every direct product of flat modules is w-flat. To do this, we introduce the class w ‐ F ‐ ML of all w-Mittag-Leffler modules with respect to all flat modules and obtain that R is w-coherent if and only if every (finitely generated) ideal is in w ‐ F ‐ ML . We

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.

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.

A corner of a ring A is a subring eAe, where e is an idempotent. Radical and semi-simple classes which are hereditary for corners and cases where the radical of a ring contains all radical corners are studied.

Generalized Weyl algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite order automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings

For a division ring F, the polynomials f∈F[z] can be evaluated “on the left” and “on the right” giving rise to left and right Lagrange interpolation problems. The problems containing interpolation conditions of the same type were considered in [7 Lam, T. Y. (1986). A general theory of Vandermonde matrices. Exposition Math. 4(3):193–215. [Google Scholar]] where the solvability criterion was given in

(2020). Correction to: Non-absolutely irreducible elements in the ring of integer-valued polynomials. Communications in Algebra: Vol. 48, No. 9, pp. 4100-4101.

It is well known that when a ring R satisfies ACC on right annihilators of elements, then the right singular ideal of R is nil, in this case, we say R is right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial

We will show that every nilpotent element N in M n ( D ) , the ring of square matrices over a division ring, can be presented as a single commutator, that is, N = A B − B A for some matrices A, B in M n ( D ) . We will also construct an example illustrating that there exists a prime ring with unity over which some nilpotent matrices cannot be presented as commutators.

We present an eight-dimensional even sub-algebra of the 2 4 = 16 -dimensional associative Clifford algebra C l 4 , 0 and show that its eight-dimensional elements denoted as X and Y respect the norm relation ‖ XY ‖ = ‖ X ‖   ‖ Y ‖ , thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar

A finite group is called a CZ(n)-group if the number of the orders of its quasikernels is n. In this note, we obtain a possible upper bound of derived length of the solvable CZ(n)-groups. Also, we give a necessary and sufficient condition for a group to be a CZ(2)-group.

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 T θ -extensions and T θ * -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 ∈ G o ( 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 ) > ( p 2 + p + 1 ) / ( 4 p 2 ) , then G = O p ( G ) × O p ′ ( G ) , where O p ( G ) is cyclic. In addition, G is supersolvable and G ″ ≤ Z ( G ) .

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

Let R be a commutative ring with identity. Denote by F P R ( 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 { pd R M | M ∈ F P R ( 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

Let G be a reductive split p-adic group and let U be the unipotent radical of a Borel subgroup. We study the cohomology with trivial Zp-coefficients of the profinite nilpotent group N=U(OF) and its Lie algebra n, by extending a classical result of Kostant to our integral p-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.

We classify all n-dimensional reduced Cohen-Macaulay modular quotient varieties A F n / C 2 p and study their singularities, where p is a prime number and C 2 p 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 σ = { σ i | i ∈ I } be some partition of the set P of all primes, that is, P = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ 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 σ i ∈ σ ( G ) . G is said to be a σ-group if it possesses a

In a semigroup S with fixed c ∈ S , one can construct a new semigroup ( S , · c ) called a variant by defining x · c y : = xcy . Elements a , b ∈ S are primarily conjugate if there exist x , y ∈ S 1 such that a = x y , b = y x . 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

We construct quasi-particle bases of principal subspaces of standard modules L ( Λ ) , where Λ = k 0 Λ 0 + k j Λ j , and Λ j denotes the fundamental weight of affine Lie algebras of type B l ( 1 ) , C l ( 1 ) , F 4 ( 1 ) or G 2 ( 1 ) of level one. From the given bases we find characters of principal subspaces.

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 p n ( x 1 , x 2 , … , x n ) be the ( n − 1 ) -th commutator defined by n indeterminates x 1 , … , x n . If R is a unital alternative ring with a nontrivial idempotent and is { 2 , 3 , n − 1 , n − 3