Journal of Algebra and Its Applications

The aim of this work is to prove a technical result that had been stated by Boyallian et al. [Classification of finite irreducible modules over the Lie conformal superalgebra $C{K}_6$, Comm. Math. Phys. 317 (2013) 503–546], on the degree of singular vectors of finite Verma modules over the exceptional Lie superalgebra $E(1,6)$ that is isomorphic to the annihilation superalgebra associated with the conformal superalgebra $C{K}_6$.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be a BNA-subgroup of $G$ if either $H^x=H$ or $x\in 〈H,H^x〉$ for all $x\in G$. A subgroup $H$ of $G$ is said to be a weakly BNA-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T$ is a BNA-subgroup of $G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that every minimal subgroup of $G$ not having a supersolvable supplement in $G$ is a weakly BNA-subgroup of $G$.

We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking.

In this paper, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by Fonseca and Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution.

Let $R$ be a semiperfect commutative Noetherian ring with identity and $C,S$ be semidualizing $R$-modules. We study the theory of linkage with respect to $S$ for modules of finite $G_C$-dimension. For a module which is horizontally linked with respect to $S$, the connections of its invariants reduced grade with respect to $C$, $G_C$-dimension and depth are discussed. Along the way, we provide a characterization of horizontally linked modules with respect to $S$ of $G_C$-dimension zero.

We contribute to the theory of G-dimension relative to a semidualizing module $C$, in connection to the properties of a module being totally $C$-reflexive, $C$-$k$-torsionless, and $C$-$k$-syzygy, where $k\ge 0$ is an integer. We extend several known results, and for an integer $q\ge 0$ we introduce the class of $C$-$q$-Gorenstein rings. We also consider $C$-duals and initiate the study of $C$-valued derivation modules over a local ring $R$ of low depth; if $C=R$ and $R$ is a three-dimensional Gorenstein domain, we find a bound for the number of generators and we propose a question when $C$ is general.

Let $G$ be a finite group in which every non-nilpotent maximal subgroup has prime index and $p$ the largest prime divisor of $\left|G\right|$, without using the solvability of $G$ we prove that either the Sylow $p$-subgroup of $G$ is normal or $G$ has a normal $p$-complement.

A proper ideal $I$ of a commutative ring $R$ is said to be a strongly quasi-primary ideal if, whenever $a,b\in R$ with $ab\in I$, then $a^2\in I$ or $b\in \sqrt{I}$ (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of $R$, called the class of quasi-$n$-ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of $R$ is a maximal or a minimal ideal of $R$. Many examples are given to illustrate the obtained results.

In this paper, we study gyro-groups associated to groups, group extensions admitting gyro-sections, and corresponding co-homologies. We also describe the obstructions in terms of co-homomology. The notion of gyro-Schur multiplier and that of gyro-Milnor $K_2$ group are introduced.

We find a short equational basis for the variety of $3$-supernilpotent loops. We also present a conceptually simple proof that $k$-nilpotence and $k$-supernilpotence are equivalent for groups. Connections between $3$-supernilpotent loops, Moufang loops, code loops, automorphic loops and AIM loops are explored.

This paper is a contribution to the globalization problem for partial group actions on non-associative algebras. We principally focus on partial group actions on Lie algebras, Jordan algebras and Malcev algebras. We give sufficient conditions for the existence and uniqueness of a globalization for partial group actions on the algebras already mentioned. As an application of this result, we show that in characteristic zero every partial group action on a semisimple Malcev algebra admits a globalization, unique up to isomorphism. We give a criterion for the existence and uniqueness of a globalization for a partial group action on a unital Jordan algebra in characteristic different from two, and on a sympathetic Lie algebra (a perfect Lie algebra without center and outer derivations).

We first recall the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then, we construct a corresponding non-coassociative permutative (NAP) coproduct on the subspecies of finite connected $T_0$ topological spaces, i.e. finite connected posets, and we prove that the vector space generated by isomorphism classes of finite posets is a free pre-Lie algebra and also a cofree NAP coalgebra. Furthermore, we give an explicit duality between the non-associative permutative product and the proposed NAP coproduct. Finally, we prove that the results in this paper remain true for finite connected topological spaces.

In this paper, some properties of vertex algebras acted by a Hopf algebra will be given. We prove that $V\#H$ is an ${𝒮}$-local vertex algebra, where $V$ is an $H$-module vertex algebra. In addition, we extend the ${𝒮}$-locality to the quantum vertex algebras.

In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of ${\mathrm{\text{SL}}}_3(k)$ and ${\mathrm{\text{SL}}}_4(k)$ where $k$ is a finite field of odd characteristic. We provide some generalized results for ${\mathrm{\text{SL}}}_n(k)$.

Let $R$ be a commutative ring with a nonzero identity and $M$ an $R$-module. Set $\varphi \text{-tor }(M)=\{x\in M|sx=0\text{ for some }s\in R\backslash \mathrm{\text{Nil}}(R)\}$, if $\varphi$-$\mathrm{\text{tor}}(M)=M$ then $M$ is called a $\varphi$-torsion module. An $R$-module $M$ is said to be $\varphi$-flat, if $0\to A{\otimes }_{R}M\to B{\otimes }_{R}M\to C{\otimes }_{R}M\to 0$ is an exact $R$-sequence, for any exact sequence of $R$-modules $0\to A\to B\to C\to 0$, where $C$ is $\varphi$-torsion. In this paper, we study some new properties of $\varphi$-flat modules. Then we introduce and study the class of $\varphi$-$P$-flat modules which is a generalization of $\varphi$-flat modules and $P$-flat modules. Finally, we give some new characterizations of the $\varphi$-von Neumann regular ring and its transfer to various contexts of constructions such as the amalgamation of rings along an ideal and trivial ring extension.

We verify a conjecture proposed by Qian on character codegrees in the case of symmetric and alternating groups.

Assume that we are given a sequence $F$ of $k$ homogeneous polynomials in $n$ variables of degree at most $d$ and the ideal ${ℐ}$ generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity $d^{O(n)}$, whether $F$ is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if $F$ is regular then we can transform ${ℐ}$ into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity $d^{O(n^2)}$. Finally, under the same assumption, we establish the new upper bound $2{(d^k/2)}^{2^{n-k-1}}$ for the maximum degree of the elements of any reduced Gröbner basis of ${ℐ}$ in the case that $n>k$.

This paper analyzes some of the fundamental results of the unit groups of the commutative modular group algebras. We introduce a more visible and concise form of a number of these results. We consider with a corresponding justification some essentially inexact and unproved results in some papers and for certain of them either we mark the corrections or we point out the corrections, which are founded in another papers.

For a monic polynomial $f$ over a commutative, unitary ring $A$ the splitting algebra $A_f$ is the universal $A$-algebra such that $f$ splits in $A_f$. The symmetric group acts on the splitting algebra by permuting the roots of $f$. It is known that if the intersection of the annihilators of the elements $2$ and $D_f$ (where $D_f$ depends on $f$) in $A$ is zero, then the invariants under the group action are exactly equal to $A$. We show that the converse holds.

Let $P$ be a primary subgroup of a finite group $G$. The structure of $G$ is studied by the local permutability of $P$.

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension.

In this paper, we investigate inclusion properties in a ∗-ring. It is proven that for two $\{1,3\}$-invertible elements $a$ and $b$ in a ∗-ring, $a=b$ if and only if $a\{1,3\}=b\{1,3\}$. For two $\{1,2,3\}$-invertible elements $a$ and $b$, $a\{1,2,3\}=b\{1,2,3\}$ if and only if $(a-b)Ra=(a-b)Rb=0$. Moreover, we give a characterization of $\{3\}$-inverses, and the absorption law for $\{3\}$-inverses is studied.

Let $R$ be a ring, $\sigma$ be an automorphism of $R$ and $\delta$ be a $\sigma$-derivation of $R$. We use $\mathrm{\Omega }\subseteq \text{End}(R,+)$ to denote the set of all words composed of $\sigma$, ${\sigma }^{-1}$ and $\delta$. A $(\sigma ,{\sigma }^{-1},\delta )$-ideal $P$ of $R$ is $\mathrm{\Omega }$-prime if whenever $a,b\in R$ are such that $aR\omega (b)\subseteq P$ for any $\omega \in \mathrm{\Omega }$, we have $a\in P$ or $b\in P$. In this paper, we first introduce the $\mathrm{\Omega }$-prime ideal and the $\mathrm{\Omega }$-prime radical of a ring $R$, to obtain connections between the prime radical of the Ore extension $R[x;\sigma ,\delta ]$ and the $\mathrm{\Omega }$-prime radical of the base ring $R$. Based on these results, we next give definitions of the $\mathrm{\Omega }$-LS-prime ideal, the $\mathrm{\Omega }$-strongly prime ideal and the $\mathrm{\Omega }$-uniformly strongly prime ideal of a ring $R$ to provide formulas for the LS-prime radical, the strongly prime radical and the uniformly strongly prime radical of the Ore extension.