Journal of Algebra and Its Applications

Let ${𝒞}$ be small category and ${𝒜}$ an arbitrary category. Consider the category ${𝒞}({𝒜})$ whose objects are functors from ${𝒞}$ to ${𝒜}$ and whose morphisms are natural transformations. Let ${ℬ}$ be another category, and again, consider the category ${𝒞}({ℬ})$. Now, given a functor $F:{𝒜}\to {ℬ}$ we construct the induced functor $F_{{𝒞}}:{𝒞}({𝒜})\to {𝒞}({ℬ})$. Assuming ${𝒜}$ and ${ℬ}$ to be abelian categories, it follows that the categories ${𝒞}({𝒜})$ and ${𝒞}({ℬ})$ are also abelian. We have two main goals: first, to find a relationship between the derived category $D({𝒞}({𝒜}))$ and the category ${𝒞}(D({𝒜}))$; second relate the functors $R(F_{{𝒞}})$ and ${(RF)}_{{𝒞}}:{𝒞}(D({𝒜}))\to {𝒞}(D({ℬ}))$. We apply the general results obtained to the special case of quiver sheaves.

Kayal and Nezhmetdinov introduced in [Factoring groups efficiently, Int. Colloquium on Automata$,$ Languages$,$ and Programming (Springer, Berlin, Heidelberg, 2009), pp. 585–596] the concept of an irreducible conjugacy class and defined, for any finite non-abelian group, a simple graph on the set of all such classes. This graph is a key ingredient in their proposed algorithm for solving the group decomposition problem. This paper considers the class of finite groups for which the KN-graph is complete (KN-complete groups). The main result is a necessary and sufficient condition for a given group to be KN-complete, formulated in terms of its minimal non-central normal subgroups. This condition is used to obtain a more detailed description of the rather rich class of KN-complete groups.

In this paper, we mainly consider three classes of quantum B-algebras such as locally unital quantum B-algebras, two-sided quantum B-algebras and implication-commutative quantum B-algebras. We first prove that a quantum B-algebra can be embedded into a unital quantum B-algebra if and only if it is a locally unital quantum B-algebra. Next, we introduce two-sided (implication-commutative) quantum B-algebras and give their characterization. Based on which we finally investigate the filters of quantum B-algebras.

The aim of this paper is to give a new characterization of cyclic modules over local rings, namely, we show that if $N$ is a nonzero finitely generated module over a local (Noetherian) ring $(R,𝔪)$ such that for some integer $n\ge 1$, the submodule ${𝔪}^{n}N$ of $N$ is cyclic, then every submodule of $N$ is cyclic and the ring $R/{\mathrm{\text{Ann}}}_R(N)$ is a $\mathrm{\text{D}}VR$, whenever $\mathrm{\text{depth}}N>0$. As a consequence, it follows that $R/{\mathrm{\text{Ann}}}_R(N/{\mathrm{\Gamma }}_{𝔪}(N))$ is a $\mathrm{\text{D}}VR$ and every submodule of $N/{\mathrm{\Gamma }}_{𝔪}(N)$ is cyclic.

In this paper we introduce dual $F$-Baer modules and give a characterization of them. Let $M$ be a module and $F$ a fully invariant submodule of $M$. $M$ is called dual $F$-Baer if for every family of endomorphisms ${\{g_{\alpha }\}}_{\alpha \in I}$ of $M$, ${\sum }_{\alpha \in I}{g}_{\alpha }(F)$ is a direct summand of $M$. We prove that $M$ is dual $F$-Baer if and only if $M=F\oplus N$ for some submodule $N$ of $M$ with $F$ dual Baer. We obtain a positive solution for the Schröder–Bernstein problem for certain dual $F$-Baer modules.

We consider HNN extensions $〈B,t:t^{-1}Ht=K〉$, where $H,K$ are normal subgroups of $B$ and $H\cap K=1$. We show that class-preserving automorphisms of those HNN extensions are all inner if $B$ satisfies certain conditions. From Grossman’s result, it follows that outer automorphism groups of such conjugacy separable HNN extensions are residually finite.

In this paper, we investigate the properties of well-rounded twists of a given ideal lattice of an imaginary quadratic field $K$. We show that every ideal lattice $I$ of $K$ has at least one well-rounded twist lattice. Moreover, we provide an explicit algorithm to compute all well-rounded twists of $I$.

Rota–Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota–Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota–Baxter algebras and braided dendriform algebras, under the commutativity condition. We further generalize the notion of dendriform Hopf algebra to the braided context and show that quantum shuffle algebra gives a braided dendriform Hopf algebra. Enveloping braided commutative Rota–Baxter algebras of braided commutative dendriform algebras are obtained.

In this paper, we introduced the concept of crossed modules for Hom–Lie antialgebras. It is proved that the category of crossed modules for Hom–Lie antialgebras and the category of ${\mathrm{\text{Cat}}}^1$-Hom–Lie antialgebras are equivalent to each other. The relationship between the crossed modules extension of Hom–Lie antialgebras and the third cohomology group is investigated.

In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize them by ∗-almost super-SH domains, where a domain $D$ is called a ∗-almost super-SH domain if every nonzero proper principal ideal of $D$ is a ∗-product of ∗-almost super-homogeneous ideals. We prove that (1) a domain $D$ is a ∗-almost IRKT if and only if $D$ is a ∗-almost super-SH domain, (2) a domain is a ∗-almost GKD if and only if $D$ is a type 1 ∗-almost super-SH domain and (3) a domain $D$ is a ∗-almost IRKT and an AGCD-domain if and only if $D$ is a ∗-afg-SH domain. Further, we characterize them by their integral closures. For example, we prove that a domain $D$ is an almost IRKT if and only if $D\subseteq \overline{D}$ is a root extension with $D$ $t$-linked under $\overline{D}$ and $\overline{D}$ is an IRKT. Examples are given to illustrate the new concepts.

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, the notion of $S$-pseudo-irreducible ideals of $R$ as a generalization of pseudo-irreducible ideals is introduced and some related properties are investigated. Then, we consider the decomposition of proper ideals of a ring into a product of pairwise comaximal $S$-pseudo-irreducible ideals. Some other results are obtained too.

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a Gröbner–Shirshov basis theory framework for modules over associative conformal algebras and apply this technique to Poisson algebras considered as conformal modules over appropriate associative conformal envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gröbner–Shirshov basis in a Poisson algebra.

A few years ago the so-called oriented subgroup $\overrightarrow{F}$ of the Thompson group $F$ was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of $F$ it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, $\overrightarrow{F}$ provided the first example of a maximal subgroup of infinite index in $F$ different from the parabolic subgroups that fix a point in $(0,1)$. In this paper we investigate possible analogues of $\overrightarrow{F}$ in higher Thompson groups $F_k,k\ge 2$, with $F=F_2$, introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup ${\overrightarrow{F}}_3$ of $F_3$, as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in $F_3$ and that the corresponding quasi-regular representation is irreducible.

In this paper we solve the problem of finding the length of group algebras of arbitrary finite abelian groups in the case when the characteristic of the ground field does not divide the order of the group. We show that these group algebras have maximal possible lengths for infinite fields and sufficiently large finite fields since they are one-generated. In case of small fields we prove that the length is bounded from above by a logarithmic function of the order of the group.

Let $R$ be a unital ring, let $E$ be a directed graph and recall that the Leavitt path algebra $L_R(E)$ carries a natural $\mathbb{Z}$-gradation. We show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.

There is a special local ring $E$ of order $4,$ without identity for the multiplication, defined by $E=〈a,b|2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉.$ We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over $E,$ and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as

In this paper, we introduce a class of commutative rings which is a generalization of $\mathrm{\text{ZD}}$-rings and rings with Noetherian spectrum. A ring $R$ is called strongly $J$-Noetherian whenever the ring $R_r$ is $J$-Noetherian for every non-nilpotent $r\in R$. We give some characterizations for strongly $J$-Noetherian rings and, among the other results, we show that if $R[[x]]$ is strongly $J$-Noetherian, then $R$ has Noetherian spectrum, which is a generalization of Theorem 2 in Gilmer and Heinzer [The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980) 13–16].

We give an example to show that, for nonunital rings $R$, the direct sum $aR\oplus bR=(a+b)R$ with $a+b$ regular has no in general right-left symmetry. It is then proved that the right-left symmetry actually holds in a left and right faithful ring.

Let $T=\left(\right.\begin{array}{cc}\hfill A\hfill & \hfill 0\hfill \\ \hfill U\hfill & \hfill B\hfill \end{array}\left)\right.$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B,A)$-bimodule. We give some computing formulas of homological dimensions of special $T$-modules. As an application, we describe the structures of $n$-tilting left $T$-modules.

Given a finite-dimensional algebra $\mathrm{\Lambda }$ and $A\ge 1$, we construct a new algebra ${\tilde{\mathrm{\Lambda }}}_A$, called the stretched algebra, and relate the homological properties of $\mathrm{\Lambda }$ and ${\tilde{\mathrm{\Lambda }}}_A$. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that $\mathrm{\Lambda }$ has (Fg) if and only if ${\tilde{\mathrm{\Lambda }}}_A$ has (Fg). We also consider projective resolutions and apply our results in the case where $\mathrm{\Lambda }$ is a $d$-Koszul algebra for some $d\ge 2$.

An extension $R\subseteq S$ of commutative rings is strongly inert (respectively, inert) if for nonzero $a,b\in S$, $ab\in R$ implies $a,b\in R$ (respectively, there exists a unit $u\in S$ with $ua,u^{-1}b\in R$). We investigate strongly inert and inert extensions and various related extensions, where we either require the product $ab$ to be nonzero or that $a$ and $b$ be nonzerodivisors. Special emphasis is given to the relationship between factorization properties of $R$ and $S$.