Journal of Algebra and Its Applications

We say that an $R$-module $M$ satisfies epi-ACC on submodules if in every ascending chain of submodules of $M$, except probably a finite number, each module in chain is a homomorphic image of the next one. Noetherian modules, semisimple modules and Prüfer $p$-groups have this property. Direct sums of modules with epi-ACC on submodules need not have this property. If $E{(R_R)}^{(\mathbb{N})}$ satisfies epi-ACC on submodules, then $R$ is quasi-Frobenius. As a consequence, a ring $R$ in which all modules satisfy epi-ACC on submodules is an artinian principal ideal ring. Dually, we say that an $R$-module $M$ satisfies epi-DCC on submodules if in every descending chain of submodules of $M$, except probably a finite number, each module in chain is a homomorphic image of the preceding. Artinian modules, semisimple modules and free modules over commutative principal ideal domains are examples of such modules. A semiprime right Goldie ring satisfies epi-DCC on right ideals if and only if it is a finite product of full matrix rings over principal right ideal domains. A ring $R$ for which all modules satisfy epi-DCC on submodules must be an artinian principal ideal ring.

Let $G$ be a finite nonabelian group, let $f(G)$ be the minimal degree of a nonlinear irreducible character of $G$ and suppose that $\left|G^{n+1}\right|\le f(G)$ for some positive integer $n$. Then $G$ is nilpotent.

Let $R$ be a ring, ${{𝒞}}_R$ and ${}^{\prime }{{𝒞}}_R$ be the set of regular and left regular elements of $R$ (${{𝒞}}_R{\subseteq }^{\prime }{{𝒞}}_R$). Goldie’s Theorem is a semisimplicity criterion for the classical left quotient ring $Q_{l,cl}(R):={{𝒞}}_R^{-1}R$. Semisimplicity criteria are given for the classical left regular left quotient ring ${}^{\prime }{Q}_{l,cl}(R):{=}^{\prime }{{𝒞}}_R^{-1}R$. As a corollary, two new semisimplicity criteria for $Q_{l,cl}(R)$ are obtained (in the spirit of Goldie).

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension $p^{n}m$, where $p$ is a prime number, $m$ is a square-free natural number and $\mathrm{\text{gcd}}(p,m)=1$. We prove that, if $n\le 5$ or $m$ is prime with $m<p$, then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than $1125$ is group-theoretical.

Let $K$ be a field and $R=K[x_1,\dots ,x_n]$ be a polynomial ring in the variables $x_1,\dots ,x_n$. In this paper, we introduce two classes of monomial ideals of $R$, which have the following properties:

• ⁽ⁱ⁾The (strong) persistence property of associated prime ideals.
• ⁽ⁱⁱ⁾There exists a strongly superficial element.
• ⁽ⁱⁱⁱ⁾Ratliff–Rush closed.

Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.

We propose a new, widely generalized context for the study of the zero-divisor type (annihilating-ideal) graphs, where the vertices of graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set [lattice] (imitating the lattice of ideals of a ring), equipped with a commutative (not necessarily associative) binary operation (imitating the product of ideals of a ring). We discuss, when ${𝔸𝔾}_z(L)$ (the annihilation graph of the commutator poset [lattice] $L$ with respect to an element $z\in L$) is a complete bipartite graph together with some of its other graph-theoretic properties. In contrast to the case of rings, we construct a commutator poset whose ${𝔸𝔾}_z(L)$ contains a cut-point. We provide some examples to show that some conditions are not superfluous assumptions. We also give some examples of a large class of lattices, such as the lattice of ideals of a commutative ring, the lattice of normal subgroups of a group, and the lattice of all congruences on an algebra in a variety (congruence modular variety) by using the commutators as the multiplicative binary operation on these lattices. This shows that how the commutator theory can define and unify many zero-divisor type graphs of different algebraic structures as a special case of this paper.

Adjoint functors between the categories of crossed modules in dialgebras and Leibniz algebras are constructed. The well known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.

Let $D$ be the field of fractions of the group algebra $k\mathrm{\Gamma }$ of the Heisenberg group $\mathrm{\Gamma }$, over the field $k$ of characteristic $\ne 2$. We show that for some involutions of $D$ that are not induced from involutions of $\mathrm{\Gamma }$, $D$ contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is a maximal Cohen–Macaulay approximation of the trivial module and is the tensor identity of the stable category of Gorenstein-projective modules over the category algebra. We give conditions on when the trivial module is Gorenstein-projective.

The work studies the line-transitive point-imprimitive automorphism groups of finite linear spaces, and is underway on the situation when the numbers of points are products of two primes. Let ${𝒮}$ be a non-trivial finite linear space with $cd$ points, where $c$ and $d$ are two primes. We prove that if $G\le \text{Aut}({𝒮})$ is line-transitive point-imprimitive, then $G$ is solvable.

In this paper we study finite $p$-groups $G$ for which the $p$-part of $|\mathrm{\text{Aut}}(G):\mathrm{\text{Inn}}(G)|$ has the least possible value $p$. We characterize the groups in some special cases, including $p$-groups of nilpotency class $2$, of maximal class, of order at most $p^5$, with cyclic Frattini subgroup and $p$-groups $G$ in which $|G:Z(G)|\le p^4$.

An LPI domain is a domain in which every locally principal ideal is invertible. In this paper, we give a negative answer to a question of Anderson–Zafrullah. We show that if $R$ is an LPI domain and $S$ is a multiplicatively closed set, then $R_S$ is not necessarily an LPI domain. Concerning a question of Bazzoni, we give a characterization of finite character for a finitely stable LPI-domain. We show that if $R$ is a finitely stable LPI-domain, then every nonzero nonunit element of $R$ is contained in only finitely many maximal ideals which are in ${𝒯}(R)$, and $R$ is of finite character if and only if each nonzero nonunit element is contained in only finitely many nonstable maximal ideals. A maximal ideal $m$ is in ${𝒯}(R)$ provided there is a finitely generated ideal $I$ such that $m$ is the only maximal ideal containing $I$.

We discuss an algebraic problem (Separation and Irreducibility Conjecture) which arises from the study of the nonlinear Schrödinger equation (NLS for short). This problem is about separation and irreducibility (over the ring of integers) of the characteristic polynomials of the graphs, describing blocks of a normal form for the NLS. For the cubic NLS the problem has been completely solved (see [C. Procesi, M. Procesi and B. Van Nguyen, The energy graph of the nonlinear Schrödinger equation, Rend. Lincei Mat. Appl. 24(2) (2013) 229–301]), meanwhile for higher degree NLS it is still open, even in small dimensions (see [C. Procesi, The energy graph of the non-linear Schrödinger equation, open problems. Int. J. Algebra Comput. 23(4) (2013) 943–962]). In this work, the author will give a partial answer for this problem, in particular, the author will prove Separation and Irreducibility Conjecture for NLS of arbitrary degree on one-dimensional and two-dimensional tori.

In this paper, we establish a strong connection between groups and gyrogroups, which provides the machinery for studying gyrogroups via group theory. Specifically, we prove that there is a correspondence between the class of gyrogroups and a class of triples with components being groups and twisted subgroups. This in particular provides a construction of a gyrogroup from a group with an automorphism of order two that satisfies the uniquely 2-divisible property. We then present various examples of such groups, including the general linear groups over $ℝ$ and $ℂ$, the Clifford group of a Clifford algebra, the Heisenberg group on a module, and the group of units in a unital C${}^{\ast }$-algebra. As a consequence, we derive polar decompositions for the groups mentioned previously.

An element in a ring is strongly nil-clean, if it is the sum of an idempotent and a nilpotent element that commute. A ring $R$ is strongly unit nil-clean, if for any $a\in R$ there exists a unit $u\in R$, such that $ua$ is strongly nil-clean. We prove, in this paper, that a ring $R$ is strongly unit nil-clean, if and only if every element in $R$ is equivalent to a strongly nil-clean element, if and only if for any $a\in R$, there exists a unit $u\in R$, such that $ua\in R$ is strongly $\pi$-regular. Strongly unit nil-clean matrix rings are investigated as well.

This paper deals with properties of modular $C_{11}$ lattices involving hereditary preradicals on hereditary classes of modular lattices. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

The concept of internally cancellable rings has been extensively studied in the literature. This paper seeks to continue the study of these rings and find some new characterizations. It is proved that $R$ is “IC”, if and only if for each regular element $a\in R$, and idempotent element $b\in R$ with $Ra+Rb=R$, there exists $x\in R$ such that $a+xb$ is a unit (alternatively, unit-regular element) in $R$ and $aR\cap xR=0$. In case the ring $R$ has the summand sum property, we indicate that $R$ is IC, if and only if for each regular element $a\in R$, and element $b\in R$ with $Ra+Rb=R$, there exists an idempotent $e\in R$, such that $a+eb$ is a unit in $R$ and $aR\cap eR=0$.

Let $R$ be an associative ring and $e,f$ idempotent elements of $R$. In this paper we introduce the notion of $(e,f)$-invertibility for an element of $R$ and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra $H$ and an algebra $A$, we present sufficient conditions for $A$ to admit an inner action of $H$. We also prove that if $A$ is a left $H$-module algebra then $H$ acts innerly on the smash product $A\#H$ if and only if $H$ is a quantum commutative weak Hopf algebra.

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-$p$ groups of nilpotency class smaller than $p$ and finitely generated nilpotent ${\mathbb{Z}}_p$-Lie algebras of nilpotency class smaller than $p$. Denote by $H_{\mathrm{\text{Gr}}}^i$ and $H_{\mathrm{\text{Lie}}}^i$ the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for $i=0$, $1$ and $2$, and for a given category of modules the cohomology functors $H_{\mathrm{\text{Gr}}}^i\circ exp$ and $H_{\mathrm{\text{Lie}}}^i$ are naturally equivalent. A similar result is proved for $i=3$ with the relative cohomology groups.

We study linearized systems of equations in characteristic $p\ne 0$ of the form $Ax=x^{(q)}$ where $A$ is a square matrix and $q=p^m$. We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when $A$ has entries in ${𝔽}_q$, the finite field of $q$ elements, we explore the relationships between the minimal and characteristic polynomials of $A$ and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when $A$ has entries in an arbitrary field of characteristic $p$, we obtain generalizations of the characteristic polynomial of a matrix and the Cayley–Hamilton theorem to square linearized systems.