Journal of Algebra and Its Applications

Let $G$ be a group. A subgroup $H$ of $G$ is called an ${\mathscr{H}}$-subgroup in $G$ if $N_G(H)\cap H^x\le H$ for all $x\in G$. Furthermore, a subgroup $H$ of $G$ is called a weakly ${\mathscr{H}}$-subgroup in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is an ${\mathscr{H}}$-subgroup in $G$. In this paper, some new criteria for a group to be $p$-nilpotent and supersolvable are given.

The paper introduces the left and right versions of the large class of Drazin inverses in terms of the left and right annihilators in a ring, which are called left-Drazin and right-Drazin inverses. We characterize some basic properties of these one-sided Drazin inverses, and discuss Jacobson’s lemma for them. In addition, the relation between the Drazin inverses and these two one-sided inverses is given by means of the spectrum and the operator decomposition. As an application, the left-Drazin and right-Drazin invertibilities in the Calkin algebra are investigated.

It is well known that any Lie algebra can be embedded into an associative algebra. We prove that any metabelian Lie algebra can be embedded into an algebra in the subvariety of perm algebras, i.e. associative algebras with the identity $abc-acb=0$. In addition, a technical method to construct the universal enveloping perm algebra for a metabelian Lie algebra is given.

Let $A$ be a commutative noetherian non-positive DG-ring, $\overline{𝔞}$ an ideal of $Ā:={\mathrm{\text{H}}}^0(A)$, and $M\in D(A)$. In this paper, we introduce the notion of cohomological dimension of DG-modules and investigate the interplay between ${\mathrm{\text{cd}}}_A(\overline{𝔞},M):=\mathrm{\text{sup}}\{{\mathrm{\text{cd}}}_{Ā}(\overline{𝔞},{\mathrm{\text{H}}}^n(M))+n|n\in \mathbb{Z}\}$ and $\mathrm{\text{sup}}\mathrm{\text{R}}{\mathrm{\Gamma }}_{\overline{𝔞}}(M)$. It is shown that ${\mathrm{\text{cd}}}_A(\overline{𝔞},M)=\mathrm{\text{sup}}\mathrm{\text{R}}{\mathrm{\Gamma }}_{\overline{𝔞}}(M)$ for any $0\ncong M\in D_{\mathrm{\text{f}}}^{-}(A)$. As an application, we recover a DG-version of Grothendieck’s vanishing and non-vanishing theorems for local cohomology. We also study the cohomological dimension of Koszul DG-modules and get some interesting results.

It is shown that the commuting probability of a finite ring cannot be a fraction with square-free denominator, resolving a conjecture of Buckley and MacHale.

In this paper, we study the connection between the odd cycles of a finite, simple graph $G$ and the symbolic powers of its edge ideal. When the odd cycles of $G$ are disjoint, we give a decomposition of the symbolic powers of the edge ideal based on the number and size of the odd cycles of $G$. This shows that the symbolic powers of the edge ideal are completely dependent on the odd cycles of $G$.

In this paper, we show that some additive mappings of a Banach algebra are derivations. Moreover, we exhibit some commutativity criteria for a prime Banach algebra ${𝒜}$ admitting a derivation satisfying certain algebraic identities.

The objective consists of two parts; one part is to investigate the structure of a right quasi-adequate semigroup $S$ containing a normal medial idempotent which is a left identity. Similarly, a left quasi-adequate semigroup case can be investigated. The other part is to combine the structure of both cases to establish a structure theorem for a quasi-adequate semigroup containing a normal medial idempotent.

In this paper, we take an algebraic approach to the generalized DGHV scheme without using bootstrapping technique. We investigate the homomorphic evaluation algorithm in detail and provide several sufficient conditions for correct homomorphic evaluation of arbitrary polynomial circuits. Compared with the bootstrapping procedure, we show that our approach is much simpler and more efficient. Moreover, we prove that both of the key sizes are actually bounded above and that these upper-bounds depend only on the plaintext space and the security parameter. Hence we need only one pair of secret key and public key for correct homomorphic evaluation of all polynomial circuits, which implies the generalized DGHV scheme is a fully homomorphic encryption in itself. Thus our approach shows that the generalized DGHV scheme provides extremely simple and efficient algorithms.

Let $S=K[x_1,\dots ,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely $t$-spread lexsegment ideals with linear quotients.

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety $X_{\mathrm{\Sigma }}$. The algorithm lends its name from a construction, described by Cox, of $X_{\mathrm{\Sigma }}$ as a GIT quotient $X_{\mathrm{\Sigma }}=({ℂ}^k\setminus Z)//G$ of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ${ℂ}^k$ of $X_{\mathrm{\Sigma }}$ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of $X_{\mathrm{\Sigma }}$. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the $G$-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of $X_{\mathrm{\Sigma }}$. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.

Let $A$ be a finite-dimensional algebra over an algebraically closed field. We study sections in the Auslander–Reiten quiver of the categories of complexes of fixed size and in the Auslander–Reiten quiver of the bounded derived category. One of the aims of this work is to show where the indecomposable $A$-modules can be found in the Auslander–Reiten quiver of the derived category, whenever $A$ is a piecewise hereditary algebra. We also prove that we can obtain a transjective component of the bounded derived category from a section. As an application of the results of sections, we find a bound on the strong global dimension of some piecewise hereditary algebras of tree type. Moreover, we also determine the strong global dimension of some algebras taking into account their ordinary quivers with relations.

We present nodal algebraic hypersurfaces in the complex projective space which are projectively rigid. Defects and Alexander polynomials associated with the hypersurfaces are obtained. There are families of nodal hypersurfaces with nontrivial Alexander polynomials and nodal threefolds with projective rigidity which are potentially infinite.

In this paper, we classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl–Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of $U_q^{+}({𝔰𝔬}_5)$, and a family of algebras which localize to the group algebra of the infinite group with generators $x$ and $y$, subject to the relation $xy=y^{-1}x$.

We parameterize countable locally standard Boolean measure algebras by pairs of a Steinitz number and a real number greater or equal to $1.$ This is an analog of the theorems of Dixmier and Baranov.

In this paper, we prove that every multiplicative anti-derivation map of an arbitrary generalized $n$-matrix ring is additive. Even more, as consequence, we get the same conclusion for the class of triangular $n$-matrix rings, unital prime rings with a nontrivial idempotent, standard operator algebras and factor von Neumann algebras.

This paper is a contribution to the classification of 2-designs admitting a flag-transitive automorphism group. Let ${𝒟}$ be a 2-$(\mathrm{\text{v}},k,\lambda )$ design with $\lambda$ prime and $G$ be a flag-transitive, point-primitive automorphism group of ${𝒟}$. We prove that if the socle of $G$ is an alternating group, then ${𝒟}$ has parameters $(v,k,\lambda )=(15,7,3),(6,3,2),(10,6,5)$ or $(10,4,2)$.

A ring $R$ is said to be i-reversible if for every $a,b$ $\in$$R$, $ab$ is a nonzero idempotent implies $ba$ is an idempotent. It is known that the rings $M_n(R)$ and $T_n(R)$ (the ring of all upper triangular matrices over $R$) are not i-reversible for $n\ge 3$. In this paper, we provide a nontrivial i-reversible subring of $M_n(R)$ when $n\ge 3$ and $R$ has only trivial idempotents. We further provide a maximal i-reversible subring of $T_n(R)$ for each $n\ge 3$, if $R$ is a field. We then give conditions for i-reversibility of trivial, Dorroh and Nagata extensions. Finally, we give some independent sufficient conditions for i-reversibility of polynomial rings, and more generally, of skew polynomial rings.

In this paper, we discuss a few properties of $w$-invertible ideals over infra-Krull domains. Let $w$-${ℐ}(R)$ denote the multiplicative group of $w$-invertible $w$-fractional ideals of an integral domain $R$ and let ${𝒫}(R)$ denote the multiplicative group of principal fractional ideals of an integral domain $R$. We proved that, if $R$ is an infra-Krull domain, then $w$-${ℐ}(R)\cong {\oplus }_{P\in w\mathrm{\text{-}}\mathrm{\text{Max}}(R)}{𝒫}(R_P)$. As an application, we can get one of Neukirch’s results directly.

Let $G$ be a finite group and $H$ a subgroup of $G.$ We say that $H$ is an ${\mathscr{H}}$-subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G;$ $H$ is called weakly ${\mathscr{H}}C$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $H^G=HT$ and $N_T(H)\cap H^g\le H$ for all $g\in G,$ where $H^G$ is the normal closure of $H$ in $G.$ Let $p$ be a prime dividing the order of $G$ and let $P$ be a Sylow $p$-subgroup of $G.$ We fix a $p$-power integer $d$ with $1<d<\left|P\right|$ and study the $p$ -supersolvability criterion of a finite group $G$ under the assumption that each subgroup of $P$ of order $d$ and $pd$ is weakly ${\mathscr{H}}C$-embedded in $G.$ Some recent results about the supersolvability of finite groups are obtained.

We introduce the concept of braided anti-flexible bialgebra and construct cocycle bicrossproduct anti-flexible bialgebras. As an application, we solve the extending problem for anti-flexible bialgebras by using some non-abelian cohomology theory.