Journal of Algebra and Its Applications

For an integer $N\ge 2$ and three subcategories $\mathcal{𝒲},\mathcal{𝒳},\mathcal{𝒴}$ of modules, we study the notion of $(\mathcal{𝒲},\mathcal{𝒳},\mathcal{𝒴})$-Gorenstein $N$-complexes. We show that under certain hypotheses, an $N$-complex $C$ is $(\mathcal{𝒲},\mathcal{𝒳},\mathcal{𝒴})$-Gorenstein if and only if $C_n$ is a $(\mathcal{𝒲},\mathcal{𝒳},\mathcal{𝒴})$-Gorenstein module for each $n\in \mathbb{Z}$, ${\mathrm{\text{Hom}}}_R(V,C)$ and ${\mathrm{\text{Hom}}}_R(C,U)$ are $N$-acyclic for any $V\in \widetilde{\mathcal{𝒴}}$ and $U\in \widetilde{\mathcal{𝒳}}$, where $\widetilde{\mathcal{𝒳}}$ and $\widetilde{\mathcal{𝒴}}$ denote the subcategory of $\mathcal{𝒳}$$N$-complexes and $\mathcal{𝒴}$$N$-complexes, respectively. Also, the stability of $(\mathcal{𝒲},\mathcal{𝒳},\mathcal{𝒴})$-Gorenstein $N$-complexes is considered. As applications, not only some known results are encompassed, but also some new results are obtained, for instance, some equivalent characterizations for Ding injective $N$-complexes and Gorenstein AC-injective $N$-complexes are given.

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of $1$-absorbing prime ideals which is a generalization of prime ideals. A proper ideal $I$ of $R$ is called $1$-absorbing prime if for all nonunit elements $a,b,c\in R$ such that $abc\in I$, then either $ab\in I$ or $c\in I$. Some properties of $1$-absorbing prime are studied. For instance, it is shown that if $R$ admits a $1$-absorbing prime ideal that is not a prime ideal, then $R$ is a quasi–local ring. Among other things, it is proved that a proper ideal $I$ of $R$ is $1$-absorbing prime if and only if the inclusion $I_1{I}_2{I}_3\subseteq I$ for some proper ideals $I_1,I_2,I_3$ of $R$ implies that $I_1{I}_2\subseteq I$ or $I_3\subseteq I$. Also, $1$-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

In this paper, we study the generalized right near domains and its relations with sharply $2$-transitive groups. We find a necessary and sufficient condition when a generalized right near domain is a field. Also, we have found the isomorphism classes of right near domains in a sharply $2$-transitive group.

This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension $6$ with rational field of definition for which it is a division algebra. The approach has other applications, one of which shows noncommutative association scheme of rank $7$ must have at least three symmetric relations.

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg. 220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of $(N-2)$-planes $Z$ in ${\mathbb{P}}^N$ with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals $I(Z)$ of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of $I(Z)$, as well as the Hilbert function of $Z$.

In this paper, we investigate the resolving resolution dimension with respect to the recollements of abelian categories. Let $({𝒜},{ℬ},{𝒞})$ be a recollement of abelian categories such that ${ℬ}$ and ${𝒞}$ have enough projective objects and let ${{𝒳}}_{{𝒜}}$, ${{𝒳}}_{{ℬ}}$, ${{𝒳}}_{{𝒞}}$ be resolving subcategories of ${𝒜}$, ${ℬ}$ and ${𝒞}$, respectively. We establish some upper and lower bounds of ${{𝒳}}_{{ℬ}}$-resolution dimension of ${ℬ}$ in terms of the ${{𝒳}}_{{𝒜}}$-resolution dimension of ${𝒜}$ and ${{𝒳}}_{{𝒞}}$-resolution dimension of ${𝒞}$. Based on these upper and lower bounds, we study the Gorensteinness of abelian categories involved in $({𝒜},{ℬ},{𝒞})$. Under some suitable assumptions, we show that if ${𝒜}$ and ${𝒞}$ are Gorenstein, then ${ℬ}$ is Gorenstein. As applications, we apply our results to ring theory and the triangular matrix artin algebras, we study the quasi-Frobenius and Gorenstein hereditary properties of the ring $R$ and $eRe$, where $e$ is an idempotent element of $R$. We also investigate Gorensteinness of the triangular matrix artin algebras, some known results are obtained as corollaries. At the end of this paper, we give two examples to illustrate our results.

Let $R$ be a commutative ring with identity. An edge labeled graph is a graph with edges labeled by ideals of $R$. A generalized spline over an edge labeled graph is a vertex labeling by elements of $R$, such that the labels of any two adjacent vertices agree modulo the label associated to the edge connecting them. The set of generalized splines forms a subring and module over $R$. Such a module is called a generalized spline module. We show the existence of a flow-up basis for the generalized spline module on an edge labeled graph over a principal ideal domain by using a new method based on trails of the graph. We also give an algorithm to determine flow-up bases on arbitrary ordered cycles over any principal ideal domain.

For a composition $\lambda$ of $n$ we consider the Kazhdan–Lusztig cell in the symmetric group $S_n$ containing the longest element of the standard parabolic subgroup of $S_n$ associated to $\lambda$. In this paper, we extend some of the ideas and results in [Beiträge zur Algebra und Geometrie, 59(3) (2018) 523–547]. In particular, by introducing the notion of an ordered $k$-path, we are able to obtain alternative explicit descriptions for some additional families of cells associated to compositions. This is achieved by first determining the rim of the cell, from which reduced forms for all the elements of the cell are easily obtained.

If $A\subseteq B$ are (commutative) rings, $[A,B]$ denotes the set of intermediate rings and $(A,B)$ is called an almost valuation (AV)-ring pair if each element of $[A,B]$ is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let $R\subseteq S$ be rings, with ${\overline{R}}_S$ denoting the integral closure of $R$ in $S$. Then $(R,S)$ is an AV-ring pair if and only if both $(R,{\overline{R}}_S)$ and $({\overline{R}}_S,S)$ are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions $R\subseteq S$. If $(R,S)$ is an AV-ring pair, then $R\subseteq S$ is a P-extension. The AV-ring pairs $(R,S)$ arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of $(B,I,D)$ type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for $(R,S)$ being an AV-ring pair to entail that $S$ is an overring of $R$, but there exist domain-theoretic counter-examples to such a conclusion in general. If $(R,S)$ is an AV-ring pair and $R\subseteq S$ satisfies FCP, then each intermediate ring either contains or is contained in ${\overline{R}}_S$. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

An endomorphism $\alpha$ of a group $G$ is called a cyclic endomorphism if the subgroup $〈x,x^{\alpha }〉$ is cyclic for all elements $x$ of $G$. It can be proved that every cyclic endomorphism is normal, i.e. it commutes with every inner automorphism of $G$ (see [F. de Giovanni, M. L. Newell and A. Russo, On a class of normal endomorphisms of groups, J. Algebra its Appl. 13 (2014) 6pp.]). In this paper, some further properties of cyclic endomorphisms will be pointed out. Moreover, the structure of a group $G$ in which the group $C\mathrm{\text{Aut}}(G)$ of cyclic automorphisms has finite index in $\mathrm{\text{Aut}}(G)$ will be investigated.

A Kuribayashi quartic curve

Quantum $N$-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper, we construct a level-one vertex representation of the quantum $N$-toroidal algebra for type $C$. In particular, we also obtain a level-one module of the quantum toroidal algebra for type $C$ as a special case.

In this paper, we introduce and study injective and projective (Krasner) hypermodules. In this regards, we introduce and study various kinds of projective and injective hypermodules based on different kinds of epic morphisms in the category of hypermodules. As main results, we introduce the notions of normal Baerian injectivity and Baerian injectivity as the versions of Baer’s criterion in the category of hypermodules. Finally, we investigate some basic properties of these hypermodules.

We show that except in several cases conjugacy classes of classical Weyl groups $W(B_n)$ and $W(D_n)$ are of type $D$. We prove that except in three cases Nichols algebras of irreducible Yetter–Drinfeld (YD) modules over the classical Weyl groups are infinite dimensional.

In this paper, we introduce the annihilator graph $\mathrm{\text{AG}}(A)$ of an MV-algebra $A$. We show that $\mathrm{\text{AG}}(A)$ contains the zero-divisor graph $\mathrm{\text{ZG}}(A)$ as a spanning subgraph. We then prove that $\mathrm{\text{AG}}(A)=\mathrm{\text{ZG}}(A)$ if and only if $\left|A\right|\le 4$. Moreover, we obtain that the girth $\mathrm{\text{gr}}(\mathrm{\text{AG}}(A))\in \{3,4,\infty \}$.

A module $M$ is said to be lifting if, for any submodule $X$ of $M$, there exists a decomposition $M=A\oplus B$ such that $A\subseteq X$ and $X/A$ is a small submodule of $M/A$. A lifting module is defined as a dual concept of the extending module. A module $M$ is said to have the finite internal exchange property if, for any direct summand $X$ of $M$ and any finite direct sum decomposition $M=M_1\oplus \cdots \oplus M_n$, there exists a direct summand $N_i$ of $M_i$ $(i=1,\dots ,n)$ such that $M=X\oplus N_1\oplus \cdots \oplus N_n$.

This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any $d$-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\left(\right.\begin{array}{cc}\hfill A\hfill & \hfill 0\hfill \\ \hfill U\hfill & \hfill B\hfill \end{array}\left)\right.$ be a triangular matrix ring. We first give an explicit description for Gorenstein injective $T$-modules over the triangular matrix ring $T$, and then construct a (right) recollement of stable categories of Gorenstein injective modules.

A ring $R$ is called weakly principally quasi Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left $s$-unital by left semicentral idempotents, which implies that $R$ modulo the right annihilator of any principal right ideal is flat. We study the relationship between the weakly p.q.-Baer property of a ring $R$ and those of the skew inverse series rings $R((x^{-1};\sigma ,\delta ))$ and $R[[x^{-1};\sigma ,\delta ]]$, for any automorphism $\sigma$ and $\sigma$-derivation $\delta$ of $R$. Examples to illustrate and delimit the theory are provided.

In this paper, we classify the smooth orientation preserving cyclic $p$-group actions on the real projective space $ℝ{\mathbb{P}}^3$ up equivalence, where two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms. We view $ℝ{\mathbb{P}}^3$ as the lens space $L(2,1)$. We show that any such action on $ℝ{\mathbb{P}}^3$ is conjugate to a standard action explicitly defined, and we identify the quotient spaces of these actions. In addition, we enumerate the equivalence classes.

This paper aims to study the concept of $\varphi$-ideals of a finite-dimensional Lie algebra which is analogous to the concept of $c$-ideal and $c$-normal subgroup. We compile some basic properties of $\varphi$-ideals and consider the influence of this concept on the structure of a finite-dimensional Lie algebra, especially its solvability and supersolvability. We also show that the counterpart of some results for $c$-ideals of Lie algebras and $c$-normal subgroups are not valid here.

The aim of this paper is to introduce and study Yetter–Drinfeld category over a weak monoidal Hom–Hopf algebra $H$. We first show that the category ${}_H^H{𝒲}{ℳ}{\mathscr{H}}{𝒴}{𝒟}$ of Yetter–Drinfeld modules over $H$ with a bijective antipode is a braided monoidal category. Secondly, we discuss some properties on the symmetries of the category ${}_H^H{𝒲}{ℳ}{\mathscr{H}}{𝒴}{𝒟}$. Finally, we prove that the representation category of triangular weak monoidal Hom–Hopf algebra is a symmetric braided monoidal subcategory of ${}_H^H{𝒲}{ℳ}{\mathscr{H}}{𝒴}{𝒟}$. Furthermore, a class of weak monoidal Hom–Yetter–Drinfeld modules are constructed by a quasitriangular weak monoidal Hom–Hopf algebra.