Journal of Algebra and Its Applications

Let $D$ be a division algebra over an infinite field $K$ such that every element of $D$ is a sum of finitely many algebraic elements. As a generalization of Asano’s theorem, it is proved that every noncentral subspace of $D$ invariant under all inner automorphisms induced by algebraic elements contains $[D,D]$, the additive subgroup of $D$ generated by all additive commutators of $D$. From the viewpoint we study the existence of normal bases of certain subspaces of division algebras. It is proved among other things that $D$ is generated by multiplicative commutators as a vector space over the center of $D$.

In this paper, we introduce the class of quasi-strongly Gorenstein projective modules which is a particular subclass of the class of finitely generated Gorenstein projective modules. We also introduce and characterize quasi-strongly Gorenstein semihereditary rings. We call a quasi-strongly Gorenstein semihereditary domain a quasi-SG-Prüfer domain. A Noetherian quasi-SG-Prüfer domain is called a quasi-strongly Gorenstein Dedekind domain. Let $F$ be a field and $X$ be an indeterminate over $F$. We prove that every ideal of the ring $F+X^{2}F[X]$ is strongly Gorenstein projective. We also show that every ideal of the ring $\mathbb{Z}+3\mathbb{Z}i$ (respectively, $\mathbb{Z}+2\mathbb{Z}i$) is strongly Gorenstein projective. These domains are examples of quasi-strongly Gorenstein Dedekind domains.

In this paper, we prove that the Taketa inequality, namely the derived length of a finite solvable group $G$ is less than or equal to the number of distinct irreducible complex character degrees of $G$, is true under some conditions related to the real and the monolithic characters of $G$.

Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\dots ,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho (I)$, a number for which any $m/r>\rho (I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)}=I^t+J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra $L_K(E)$ has index of nilpotence at most $n$ if and only if no cycle in the graph $E$ has an exit and there is a fixed positive integer $n$ such that the number of distinct paths that end at any given vertex $v$ (including $v$, but not including the entire cycle $c$ in case $v$ lies on $c$) is less than or equal to $n$. Interestingly, the Leavitt path algebras having bounded index of nilpotence turn out to be precisely those that satisfy a polynomial identity. Furthermore, Leavitt path algebras with bounded index of nilpotence are shown to be directly-finite and to be $\mathbb{Z}$-graded $\mathrm{\Sigma }$–$V$ rings. As an application of our results, we answer an open question raised in [S. K. Jain, A. K. Srivastava and A. A. Tuganbaev, Cyclic Modules and the Structure of Rings, Oxford Mathematical Monographs (Oxford University Press, 2012)] whether an exchange $\mathrm{\Sigma }$–$V$ ring has bounded index of nilpotence.

In this work, we give some necessary and/or sufficient conditions for a group algebra of infinite group to satisfy the converse of Schur’s Lemma. Many classes of groups are investigated such as abelian groups, hypercentral groups, groups having abelian subgroup of finite index and finitely generated soluble groups.

The purpose of this paper is to study the structure of Lotka–Volterra algebras, the set of their idempotent elements and their group of automorphisms. These algebras are defined through antisymmetric matrices and they emerge in connection with biological problems and Lotka–Volterra systems for the interactions of neighboring individuals.

The aim of this paper is to present new results and generalize some results about indigent modules. The commutative rings whose simple modules are indigent or injective are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. We show that a reduced module is indigent if and only if it is a Whitehead test module for injectivity over commutative hereditary noetherian rings. Furthermore, Dedekind domains are characterized by test modules for injectivity by subinjectivity.

We consider the category of comodules over a smash coproduct coalgebra $C>⊲H$. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from $C>⊲H$-colinear, $H$-colinear and rational $C$-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

In this paper, we initiate the study of the total zero-divisor graph over a commutative ring with unity. This graph is constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring and give a joint insight of the structure of zero-divisors in a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graph of the ring ${\mathbb{Z}}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb{Z}}_m$ and ${\mathbb{Z}}_n$ are isomorphic if and only if $m=n$.

New upper and lower bounds on the Castelnuovo–Mumford regularity are given in terms of the Hilbert coefficients. Examples are provided to show that these bounds are in some sense nearly sharp.

A zero forcing set is a new concept in Graph Theory which was introduced in recent years. In this paper, we investigate the relationship between zero forcing sets and algebraic hyperstructures. To this end, we present some new definitions by considering a zero forcing process on a graph $G$. These definitions help us analyze the zero forcing process better and construct various hypergroups and join spaces on the vertex set of graph $G$. Finally, we give some examples to clarify these hyperstructures.

Let $A$ be an Artin algebra. It is well known that $A$ is selfinjective if and only if every finitely generated $A$-module is reflexive. In this paper, we pose and motivate the question whether an algebra $A$ is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.

Let $\mathrm{\Lambda }$ be a finite dimensional algebra over a field $k$ and $H$ a subgroup of a finite group $G$. In this paper, we consider the Gorenstein global dimensions and representation dimensions of the skew group algebras $\mathrm{\Lambda }G$ and $\mathrm{\Lambda }H$. Under the assumption that $\mathrm{\Lambda }G$ is a separable extension over $\mathrm{\Lambda }H$, we show that $\mathrm{\Lambda }G$ and $\mathrm{\Lambda }H$ share the same Gorenstein global dimensions and representation dimensions. As an application, we give an affirmative answer for a conjecture raised in [Adjoint functors and representation dimensions, Acta Math. Sinica 22(2) (2006) 625–640]. Several known results are then obtained as corollaries.

In this paper, we study a class of Leibniz conformal algebras called quadratic Leibniz conformal algebras. An equivalent characterization of a Leibniz conformal algebra $R=ℂ[\partial ]V$ through three algebraic operations on $V$ is given. By this characterization, several constructions of quadratic Leibniz conformal algebras are presented. Moreover, one-dimensional central extensions of quadratic Leibniz conformal algebras are considered using some bilinear forms on $V$. In particular, we also study one-dimensional Leibniz central extensions of quadratic Lie conformal algebras.

We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras.

In this paper, we consider the algebra $A({\mathrm{\Gamma }}_{♢})$ associated to Hasse graph of a star polygon. We determine the automorphism group for this algebra and the graded traces ${\mathrm{\text{Tr}}}_{\sigma }(t)$ for each $\sigma \in \mathrm{\text{Aut}}(A({\mathrm{\Gamma }}_{♢}))$, which are the graded trace generating functions of $A({\mathrm{\Gamma }}_{♢})$. Furthermore, we study the representations of $\mathrm{\text{Aut}}(A({\mathrm{\Gamma }}_{♢}))$ acting on each homogeneous component of $A({\mathrm{\Gamma }}_{♢})$ and we apply the same technique to the dual algebra $A{({\mathrm{\Gamma }}_{♢})}^{!}$ of $\mathrm{\text{gr}}(A({\mathrm{\Gamma }}_{♢}))$. More precisely, we consider the algebras associated to Hasse graph of star polygons $\{\frac{n}{p}\}$ with $n\ge 5$ odd.

For a ring $R$, an extension ring $\mathrm{\Gamma }$, and a fixed right $R$-module $C$, we prove the induced left $\mathrm{\Gamma }$-module $U=\mathrm{\Gamma }{\otimes }_{R}C$ is a Wakamatsu tilting module when ${}_{R}C$ is a Wakamatsu tilting module.

Let $G$ be a finite group and let $r\in \mathbb{N}$. An $r$-coloring of $G$ is any mapping $\chi :G\to \{1,\dots ,r\}$. A coloring $\chi$ is symmetric if there is $g\in G$ such that $\chi (g{x}^{-1}g)=\chi (x)$ for every $x\in G$. We show that if $G$ is Abelian and $f(r)$ is the polynomial representing the number of symmetric $r$-colorings of $G$, then the number of symmetric $r$-colorings of $G\times {\mathbb{Z}}_2$ is $f(r^2)$. We also extend this result to the dihedral group $D(G)$.

Let $\sigma =\{{\sigma }_i|i\in I\}$ be a partition of the set $\mathbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $\sigma$-central if the semidirect product $(H/K)⋊(G/C_G(H/K))$ is a ${\sigma }_i$-group for some $i=i(H/K)$. $G$ is called $\sigma$-nilpotent if every chief factor of $G$ is $\sigma$-central. We say that $G$ is semi-$\sigma$-nilpotent (respectively, weakly semi-$\sigma$-nilpotent) if the normalizer $N_G(A)$ of every non-normal (respectively, every non-subnormal) $\sigma$-nilpotent subgroup $A$ of $G$ is $\sigma$-nilpotent. In this paper we determine the structure of finite semi-$\sigma$-nilpotent and weakly semi-$\sigma$-nilpotent groups.