A subspace H of a Leibniz algebra L is called a quasi-ideal if [H,K]+[K,H]⊆H+K for every subspace K of L. It includes ideals and subalgebras of codimension one in L. Quasi-ideals of Lie algebras were classified in two remarkable papers by Amayo. The objective here is to extend those results to the larger class of Leibniz algebras, and to classify those Leibniz algebras in which every subalgebra is

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial

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

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

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

We investigate derivations in the semiring of skew Ore polynomials over an idempotent semiring. We show that multiplying each polynomial by x on left is a derivation and construct commutative idempotent semiring consisting of derivations of a skew polynomial semiring. We introduce generalized hereditary derivations as derivations acting only over the coefficients of the polynomial and construct an

In this article, we introduce the notion of M-coidempotent elements of a ring and investigate their connections with fully coidempotent modules, fully copure modules and vn-regular modules where M is a module. We prove that if M is a finitely cogenerated module, then M is fully copure if and only if M is semisimple. We prove that if M is a Noetherian module or M is a finitely cogenerated module, then

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

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

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

Let R be an Artin ring and Θ={Θ(1),Θ(2),…,Θ(n)} be a family of objects in an Artin extriangulated R-category (C,E,s) such that E(Θ(j),Θ(i))=0 for all j≥i. In this article, we show that the class P(Θ) of the Θ-projective objects is a precovering class and the class I(Θ) of the Θ-injective objects is a pre-enveloping one in C. Furthermore, if C has enough projectives and enough injectives, we show that

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.

Let FG denote the group algebra of the group G over the field F with char(F)≠2. Given both a homomorphism σ:G→{±1} and a group involution ∗:G→G, an oriented involution of FG is defined by α=Σαgg↦α⊛=Σαgσ(g)g∗. In this article, we determine the conditions under which the group algebra FG is normal, that is, conditions under which FG satisfies the ⊛-identity αα⊛=α⊛α. We prove that FG is normal if and

Abstract Given a polynomial f with coefficients in a field of prime characteristic p, it is known that there exists a differential operator that raises 1/f to its pth power. We first discuss a relation between the “level” of this differential operator and the notion of “stratification” in the case of hyperelliptic curves. Next, we extend the notion of level to that of a pair of polynomials. We prove

Let G be a group and H≤G. The permutizer of H in G is PG(H)=〈x|x∈G,〈x〉H=H〈x〉〉. H is said to be strongly permuteral in G if PU(H)=U whenever H≤U≤G. Moreover, let PGP(H)=〈x|x is a primary element of G,〈x〉H=H〈x〉〉. H is said to be strongly P-permuteral in G if PUP(H)=U whenever H≤U≤G. In this article, we study the structure of a group G in which every Sylow subgroup and its maximal subgroups are strongly

Let F be a field, charF=2,−1∈F*. Let further D1, D2 be biquaternion algebras over F with Albert forms φ1,φ2. We prove that the dimension of anisotropic part of the form φ1⊗φ2 is at most 24, provided there exists a common splitting field of degree 8m+4 for D1 and D2. We also investigate the symbol length of φ1⊗φ2∈I4/I5(F), i.e., the minimal number n such that φ1⊗φ2∈I4/I5(F) is the sum of n 4-fold Pfister

The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen, Nasseh, Sather-Wagstaff, and Şega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of the vanishing of Tor. This has

Fusion rings are a class of table algebras that generalize group rings with basis the group and character rings of a finite group with basis the irreducible characters. When considering the character ring of a group as a fusion ring, the usual degree of a character coincides with the degree map. Hence classifying fusion rings based on the degree set is a generalization of classifying groups based on

The Frobenius complexity of a local ring R measures asymptotically the abundance of Frobenius operators of order e on the injective hull of the residue field of R. It is known that, for Stanley–Reisner rings, the Frobenius complexity is either −∞ or 0. This invariant is determined by the complexity sequence {ce}e of the ring of Frobenius operators on the injective hull of the residue field. We will

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

We define a cup-product in Hom-Leibniz cohomology and show that the cup-product satisfies the graded Hom-Zinbiel relation.

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

Let g be a simple complex Lie algebra with root system R of rank r. Larsen and Lubotzky observed that the number of irreducible representations of g of degree ≤ n grows roughly like n r|R+|. This paper makes a closer investigation of this phenomenon for algebras of type A, B, C, and D, giving explicit upper and lower bounds whose ratio depends only on r.

Let A be a cellular algebra over a field F with a decomposition of the identity 1A into orthogonal idempotents ei,i∈I (for some finite set I) satisfying some properties. We present a technique to describe the radical and the head of each cell module of the algebra A by using the cell modules of the algebra eiAei for each i. Moreover, we also study the block theory of A by using this decomposition.

We give examples showing that the usual Artin approximation theorems valid for convergent series over a field are no longer true for convergent series over a commutative Banach algebra. In particular, we construct an example of a commutative integral Banach algebra R such that the ring of formal power series over R is not flat over the ring of convergent power series over R.

All groups are finite with Φ(G) denoting the Frattini subgroup of a group G. If G is nilpotent with subgroups H and K where H≤K, then Φ(H)≤Φ(G) and Φ(H)≤Φ(K). However, as demonstrated by the symmetric group S3, there are non-nilpotent groups that also satisfy these two properties. In 1965 H. Bechtell introduced a class of groups that satisfy the property that Φ(H)≤Φ(G) for all subgroups H of a group

For a Poisson algebra A, by studying its universal enveloping algebra Ape, we prove a duality theorem between Poisson homology and cohomology of A.

Abstract A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among other results that if R is a domain such that the conductor (R:R̂) vanishes, then Λ(R) is finite;

We describe equivalence classes of indecomposable exact module categories over a finite graded tensor category. When applied to a pointed fusion category, our results coincide with the ones obtained in [11 Natale, S. (2017). On the equivalence of module categories over a group-theoretical fusion category, SIGMA, Symmetry Integrability Geom. Methods Appl. 13 Paper 042:9. p. DOI: 10.3842/SIGMA.2017.042

This article is devoted to n-dimensional nilpotent finite-dimensional evolution algebras E with maximal index of nilpotency. We are able to find full invariants of these algebras which allows us to classify them. Furthermore, we explicitly describe its associative enveloping algebras. Moreover, we give an affirmative answer to the question: is it possible to extend a given derivation of the nilpotent

Let n⩾1 and k⩾0 be fixed integers and R be a ring. We show that R is (n, k)-SG with G‐gldim(R)=k if and only if R[x1,x2,…,xm] is (n,k+m)-SG with G‐gldim(R[x1,x2,…,xm])=k+m. We also give an example of a 2-SG-Dedekind domain which is not an SG-Dedekind domain.

In this paper, we prove that the minimum size of a hypergroup of type U on the right which contains a proper subsemihypergroup of type U on the right of size 6 and a subgroup of size ≥2 is 18. We study some properties of hypergroups of type U on the right of size 2p that have a cyclic subgroup of size p. Moreover, we give a more general answer to a problem left open in [11 Fasino, D., Freni, D. (2007)

In the present work, a procedure for determining idempotents of a commutative ring having a sequence of ideals with certain properties is presented. As an application of this procedure, idempotent elements of various commutative rings are determined. Several examples are included illustrating the main results.

Throughout this article, all groups are finite and G is a group. Let σ={σi|i∈I} be some partition of the set of all primes P. Then σ(G)={σi|σi∩π(G)≠∅}; σ+(G)={σi|G has a chief factor H/K such that σ(H/K)={σi}}. The group G is said to be: σ-primary if G is a σi-group for some i; σ-soluble if every chief factor of G is σ-primary. The symbol Rσ(G) denotes the product of all normal σ-soluble subgroups

In this article, a new semistar operation, called the q-operation, on a commutative ring R is introduced in terms of the ring Q0(R) of finite fractions. It is defined as the map q:Fq(R)→Fq(R) by A↦Aq:={x∈Q0(R)| there exists some finitely generated semiregular ideal J of R such that Jx⊆A} for any A∈Fq(R), where Fq(R) denotes the set of nonzero R-submodules of Q0(R). The main superiority of this semistar

Let ℓ be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an ℓ-Sylow subgroup. The classification assists in describing the ℓ-Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation

First, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined

In this article, we investigate the representation rings (or Green rings) of the Drinfeld doubles of the Taft algebras. It is shown that these Green rings are commutative rings generated by infinitely many elements subject to certain relations. The generators together with the subjecting relations are given. The stable Green rings of of the Drinfeld doubles of the Taft algebras are also described.

Matlis showed that an injective module over a commutative Noetherian ring R can be completely decomposed as a direct sum of indecomposable injective submodules. In this paper, we prove the Matlis’ Theorem for almost Dedekind domains. Then we characterize the secondary modules and classify the indecomposable secondary modules over almost Dedekind domains. Also we prove every P-secondary module over

Let A be an n-Gorenstein ring. Employing the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we introduce the concept of Gorenstein tilting pair. Moreover, we give a simple characterization on Gorenstein tilting pair, which shows that Gorenstein cotilting and tilting modules are special examples of Gorenstein tilting pair.

In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1 − ba (resp., ba) being pseudo core invertible while 1 − ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1 − ba (resp., ba) is given in terms of 1 − ab

In this article, we define the ϕ-Krull dimension as a generalization of Krull dimension. This property is treated here as a part of a study on some constractions of rings such as direct products, amalgamation of rings, and trivial ring extensions.

There are only 10 Euclidean forms, that is flat closed three-dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of n-fold coverings over orientable Euclidean manifolds G2 and G4, and calculate the numbers of nonequivalent coverings of each type. We classify subgroups in the fundamental groups π1(G2) and π1(G4) up to isomorphism and

Dickson’s commutative semifields are an important class of finite division algebras. We generalize Dickson’s construction over any base field by doubling not just finite field extensions (which would correspond to the classical setup), but also central simple algebras. The latter case yields division algebras which are no longer commutative nor associative. We compute their automorphisms.

Let ϕ:G→G be an automorphism of a group which is a free product of finitely many groups each of which is freely indecomposable and two of the factors contain proper finite index characteristic subgroups. We show that G has infinitely many ϕ-twisted conjugacy classes. As an application, we show that if G is the fundamental group of a three-manifold that is not irreducible, then G has property R∞, that

A subgroup H of a finite group G is said to be nearly s-embedded in G if G has an s-permutable subgroup T and an s-semipermutable subgroup HssG contained in H such that HsG=HT and H∩T≤HssG, where HsG is the intersection of all s-permutable subgroups of G containing H. In this article, we investigate the influence of some nearly s-embedded subgroups of Sylow p-subgroups on the p-nilpotency of finite

In this article, we give a generalization of O’Halloran’s theorem that gives rise to families of Weyl modules with simple radicals. This generalization is then applied to simply connected, semisimple algebraic groups of types B, C, and D. We also construct families of Weyl modules with formal characters of length 4, and compute the extensions and cohomology of simple modules associated with these Weyl

There has been some interest in understanding the relationship between the number of cyclic subgroups of a group and its order. This relationship is controlled in many cases by the size of the set of non-squares of the group. We improve upon previously established bounds and classify groups that obtain our new bound.

The pro-étale topology by Bhargav Bhatt and Peter Scholze allows us to reconstruct the derived category of constructible Q¯ℓ-sheaves on a topologically noetherian scheme in the usual derived manner. We extend this result from schemes to algebraic stacks.

We prove that for a right perfect ring R, then the following hold: if A, B are projective right R-modules and there exist epimorphisms A→B and B→A, then A is isomorphic to B. A consequence is the following: if A and B are right R-modules with A, an epimorphic image of B and B, an epimorphic image of A, then the projective cover of A and the projective cover of B are isomorphic.

Let Γ be a graph with vertex set V(Γ). A subset C of V(Γ) is a perfect code of Γ if C is an independent set such that every vertex in V(Γ)∖C is adjacent to exactly one vertex in C. A subset T of V(Γ) is a total perfect code of Γ if every vertex of Γ is adjacent to exactly one vertex in T. Let G be a finite group. The proper reduced power graph of G is the undirected graph whose vertex set consists

In this paper, we provide a complete classification of non-symplectic automorphisms of order 9 of complex K3 surfaces.

We show that when d≥3, the Hilbert scheme HilbdT+1−(d−12)(G(k,n)) has 2 connected components, even though elements in both components have the same cohomology class. Moreover, we show that the Hilbert scheme associated to the Hilbert polynomial (T+mm)−(T+m−dm) in Grassmannian has at most 2 connected components. Communicated by Lawrence Ein

Let K be a field and q∈K,q≠0,1. Let Hn(q,Q) be an Ariki–Koike algebra, where the cyclotomic parameter Q=(Q1,Q2,⋯,Qr)∈Kr with r≥2, Qi=qai,ai∈Z. For a weight one block B of Hn(q,Q), we prove in this article that radB=I, where I is the nilpotent ideal constructed for a symmetric cellular algebra [Radicals of symmetric cellular algebras, Colloq. Math. 133 (2013) 67–83]. We also give some applications of

In this paper, we determine the derivations of affine Lie superalgebras. As the nature of affine Lie superalgebra of type A(ℓ,ℓ)(1) is slightly different from the nature of the other types, we study separately the derivations for type X=A(ℓ,ℓ)(1) and X≠A(ℓ,ℓ)(1). We precisely determine derivations up to inner derivations; in the former type, we get a four-dimensional vector space while in the latter

Let R be commutative ring with 1≠0. A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements a,b,c∈R and abc∈I, then ab∈I or c∈I. It is proved that every primary ideal of R is 1-absorbing primary and every 1-absorbing primary ideal of R is semi-primary (that is ideals with prime radical). However, these three concepts are different. In this paper, we characterize

Let (A, M) be any finite local (nonzero commutative unital) ring. Then the set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A⊂B is a decomposed (minimal ring) extension is in one-to-one correspondence with the collection of sets {I1,I2} of ideals I1andI2 of A such that M=I1⊕I2 (internal direct sum), with any such {I1,I2} corresponding to the class

Recently Gao-Jing-Xia-Zhang defined the structures of quantum N-toroidal algebras uniformally, which are a kind of natural generalizations of the classical quantum toroidal algebras, just like the relation between 2-toroidal Lie algebras and N-toroidal Lie algebras. Based on this work, we construct a level-one vertex representation of quantum N-toroidal algebra for type F4. In particular, we can also

In this paper we explicitly describe the symbolic powers of the ideal defining the curve C(q,m) in P3 parametrized by (xd+2m,xd+mym,xdy2m,yd+2m), where q, m are positive integers, d=2q+1 and gcd(d,m)=1. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal p:=pC(q,m) defining the curve C(q,m) is

Let G be a graph and let I=I(G) be its edge ideal. When G is unicyclic, we give a decomposition of symbolic powers of I in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of I. When G is an odd cycle, we explicitly compute the regularity of I(s) for all s∈N. In doing so, we also give a natural lower bound for the regularity function