Journal of Algebra and Its Applications

The aim of this paper is to study the projective modules of the $\chi$-reduced enveloping superalgebra $U_{\chi }(\mathrm{\text{W(2)}})$, where $W(n)$ is the Lie superalgebra of superderivations on the Grassmann superalgebra $\mathrm{\Lambda }(n)$, over an algebraically closed field k of characteristic $p>2$. Mainly, the Cartan invariants and the dimensions of indecomposable projective modules of $U_{\chi }(\mathrm{\text{W(2)}})$ are determined for any $p$-character $\chi$ up to isomorphism.

For a nontrivial finite group $G$ different from a cyclic group of prime order, the intersection graph $\mathrm{\Gamma }(G)$ of $G$ is the simple undirected graph whose vertices are the nontrivial proper subgroups of $G$ and two vertices are joined by an edge if and only if they have a nontrivial intersection. In this paper we characterize all finite groups with planar intersection graphs. It turns out that few solvable groups have planar intersection graphs. Also we classify finite groups whose intersection graphs are bipartite, triangle free and forests.

Motivated by the definition of Duflo involution for fiat $2$-categories, we define certain analogues of Duflo involution for arbitrary finitary $2$-categories and show that such Duflo involutions exist for two classes of finitary $2$-categories associated with tree path algebras. Additionally, we describe the quiver for the algebra underlying the principal $2$-representation for these two classes of finitary $2$-categories.

Let $R$ be a commutative Noetherian ring, $d\ge 0$ be an integer and let $\sum$ denote the ideals of $R$ of dimension $\le d$. For an integer $i\ge 0$ and $R$-modules $M,N$, we define the $i$th cohomology module $H_d^i(M,N)={\underrightarrow{lim}}_{𝔞\in \sum }{\mathrm{\text{Ext}}}_R^i(M/𝔞M,N)$ and study some of its properties. Some vanishing results will be proved and the connection between the vanishing of this cohomology modules with the dimension of certain subsets of $\mathrm{\text{Supp}}(M)$ will be investigated.

Given a semidualizing module $C$ over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205(2) (2006) 423–445] investigate some connections between $C$-Gorenstein dimensions of an $R$-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” $R⋉C$. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra 216(1) (2012) 115–118].

The aim of this paper is to establish a connection between the standard Koszul and the quasi-Koszul property in the class of self-injective special biserial algebras. Furthermore, we give a characterization of standard Koszul symmetric special biserial algebras in terms of quivers and relations.

In this paper we present a technical lemma about localization at countably infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.

Yassemi’s “second submodules” are dualized and properties of its spectrum are studied. This is done by moving the ring theoretical setting to a lattice theoretical one and by introducing the notion of a (strongly) topological lattice ${ℒ}=(L,\wedge ,\vee )$ with respect to a proper subset $X$ of $L.$ We investigate and characterize (strongly) topological lattices in general in order to apply it to modules over associative unital rings. Given a non-zero left $R$-module $M,$ we introduce and investigate the spectrum ${\mathrm{\text{Spec}}}^{\mathrm{\text{f}}}(M)$ of first submodules of $M$ as a dual notion of Yassemi’s second submodules. We topologize ${\mathrm{\text{Spec}}}^{\mathrm{\text{f}}}(M)$ and investigate the algebraic properties of $M$ by passing to the topological properties of the associated space.

A characterization of a class of congruence-simple semirings with zero (now containing some infinite cases) is presented.

We consider the arithmetic background of integral representations of finite groups over $p$-adic and algebraic number rings. Some infinite series of integral pairwise inequivalent absolutely irreducible representations of finite $p$-groups with the extra congruence conditions are constructed, and some applications are given. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed.

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

Let $D$ be a division ring. In this paper, we investigate properties of subgroups of an arbitrary subnormal subgroup of the multiplicative group $D^{\ast }$ of $D$. The new obtained results generalize some previous results on subgroups of $D^{\ast }$.

This paper introduces the notion of $2$-prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain $R$ is a valuation ring if and only if every ideal of $R$ is $2$-prime. On the other hand, we will prove that the normalization $\overline{R}$ of $R$ is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of $R$ is a 2-prime ideal. At the end of this paper, we will give a generalization of some results of Gilmer and Heinzer by studying the properties of domains in which every primary ideal is an integrally closed 2-prime ideal.

In this paper, we study a finite group $G$ such that $|H:H_G|$ is a prime for each non-normal subgroup $H$ of $G$. We prove that such a group must contain a big abelian subgroup. More specifically, if such a group $G$ is not supersoluble, then there is an abelian subgroup $A$ such that $|G:A||2pq$, and if $G$ is supersoluble, then there is an abelian subgroup $A$ such that $|G:A||2^5$ or $|G:A||p^2{q}^2$, where $p$ and $q$ are primes.

We introduce the notion of Drinfeld polynomials of two-parameter quantum affine algebras and establish a one-to-one correspondence between finite-dimensional irreducible representations and sets of $l$-tuples of pairs of polynomials with certain conditions.

We classify the Hilbert polynomial of a local ring $(R,𝔪)$ satisfying ${𝔪}^4=0$ which admits a doubly-infinite eventually linear resolution $C$ which is “partially” complete — that is, for which $H^i{Hom}_R(C,R)$ vanishes for all $\left|i\right|\gg 0$. As a corollary to our main result, we show that an ${𝔪}^4=0$ local ring can admit certain classes of asymmetric partially complete resolutions only if its Hilbert polynomial is symmetric. Moreover, we show that the Betti sequence associated to an eventually linear partially complete resolution over an ${𝔪}^4=0$ local ring cannot be periodic of period two or three.

Let $G$ be a finite $p$-group and let Aut($G$) denote the full automorphism group of $G$. In the recent past, there has been interest in finding necessary and sufficient conditions on $G$ such that certain subgroups of Aut($G$) are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.

For a finite group $G$, the symbol $\pi (G)$ denotes the set of the prime divisors of $\left|G\right|,{ℳ}(G)$ denotes the number of conjugacy classes of maximal subgroups of $G$. Let $\tau (G)$ denote the number of conjugacy classes of non-abelian subgroups of $G$ and $\mathrm{\Gamma }(G)$ denote the number of conjugacy classes of all non-normal non-abelian subgroups of $G$. In this paper, we consider the finite groups with $\tau (G)\le {ℳ}(G)+1$ or $\mathrm{\Gamma }(G)\le |\pi (G)|-1$. We show these groups are solvable.