Journal of Algebra and Its Applications

In this paper, copure-injectively poor modules are introduced as modules whose copure-injectivity domains are minimal. Rings over which every module is copure-injectively poor are CDS rings. Examples showing the relations between poor and copi-poor modules are given. It is shown that over a commutative (co-)noetherian ring copi-poor modules coincide with pi-poor modules. Also, some properties of copure-split modules are given.

In this paper, Zariski compactness of the minimal spectrum and flat compactness of the maximal spectrum are characterized.

To determine if a poset of type $\{k_1,\dots ,k_{d-1}\}$ is a directly regular or chiral polytope, it is necessary to test whether or not its rotation group (as a quotient of the orientation-preserving subgroup of the Coxeter group $[k_1,\dots ,k_{d-1}]$) satisfies the so-called intersection condition of chiral form. However, due to the fact that many cases need to be checked, this process is often very tedious and takes much time. In this paper, under certain circumstances, we give some simplifications for checking the intersection condition, which leads to certain constructions for directly regular or chiral polytopes.

According to Mao et al. (2016), let $H$ be a subgroup of a finite group $G$; then $H$ is said to be generalized $S\mathrm{\Phi }$-supplemented in $G$ if there exists a subnormal subgroup $T$ of $G$ such that $G=HT$ and $(H\cap T)H_{sG}/H_{sG}\le \mathrm{\Phi }(H/H_{sG})$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $S$-permutable in $G$. In this paper, we extend the concept of generalized $S\mathrm{\Phi }$-supplemented subgroups and generalize the main result of Mao et al. (2016).

Let $𝔤=𝔰𝔩(3,k)$ be a simple Lie algebra of type $A_2$ over an algebraically closed field $k$ of prime characteristic $p$ and $U_{\chi }(𝔤)$ be the reduced enveloping algebra of $𝔤$. In this paper, when the $p$-character $\chi$ is regular nilpotent and has standard Levi form, we precisely determine the Lowey series of principal indecomposable $U_{\chi }(𝔤)$-modules and the dimensions for the self-extension of irreducible $U_{\chi }(𝔤)$-modules.

In this paper, we introduced the concept of a ring of a right (left) dyadic range 1. We proved that a Bezout ring of right (left) dyadic range 1 is a ring of stable range 2. And we proved that a commutative Bezout ring is an elementary divisor ring if and only if it is a ring of dyadic range 1.

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.

Let $G$ be a group. If the set ${𝒜}(G)=\{\alpha \in \mathrm{\text{Aut}}(G):x\alpha (x)=\alpha (x)x$ for all $x\in G\}$ forms a subgroup of $\mathrm{\text{Aut}}(G)$, then $G$ is called ${𝒜}$-group. Let $p$ be an odd prime. Recently it has been proven that a finite $p$-group of almost maximal class is an ${𝒜}$-group. For finite 2-groups of almost maximal class, the situation is much more complicated. This paper deals with the case $p=2$. We prove that a 2-group of almost maximal class is an ${𝒜}$-group. We show the minimum coclass of a non-${𝒜}$, $p$-group is equal to 3. We also discuss the direct product of certain ${𝒜}$-groups and subsequently we give some applications of our results.

In this paper, we study the depth and the Castelnuovo–Mumford regularity of the powers of edge ideals which are unmixed and whose underlying graphs are bipartite. In particular, we prove that the depth of the powers of the edge ideal stabilizes when the exponent is the same as half the number of vertices in the underlying connected bipartite graph. We also define the idea of “drop” in the sequence of depth of powers of ideals. Further, we show that the sequence of depth of the powers of such edge ideals may have any number of “drops”. In the process of proving these results we put forward some interesting examples and some questions for future research. As for regularity, we establish a formula for the regularity of the powers of such edge ideals in terms of the regularity of the edge ideal itself.

In this paper we classify finite non-abelian groups having a unique non-abelian proper subgroup.

In this paper, we define and study Whittaker modules for the super-Viraoro algebras, including the Neveu-Schwarz algebra and the Ramond algebra. We classify the simple Whittaker modules and obtain necessary and sufficient conditions for irreducibility of these modules.

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of $2\times 2$ block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

In this paper, we introduce the concept of the index of a generator in a cyclic hypergroup, and show that a single power cyclic hypergroup is generated by an element with index $2$. Also, a characterization of the fundamental relation on a cyclic hypergroup is given. Finally, we study corresponding quotient structures induced by regular (strongly regular) relations on cyclic hypergroups. As an application, the corresponding results on single power hypergroups are obtained.

We prove that all quasi-cotilting modules are pure-injective and cofinendo. It follows that the class $\mathrm{\text{Cogen}}M$ is always a covering class whenever $M$ is a quasi-cotilting module. Some characterizations of quasi-cotilting modules are given. As a main result, we prove that there is a bijective correspondence between the equivalent classes of quasi-cotilting modules and torsion-free covering classes.

This paper studies properties of certain hypersurfaces in prime characteristic: we give sufficient and necessary conditions for some classes of such hypersurfaces to have Finite $F$-representation Type (FFRT) and we compute the $F$-signatures of these hypersurfaces. The main method used in this paper is based on finding explicit matrix factorizations.

We prove that the inverse limits of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, “numericalsgps”: A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps].

Let $f$ be a monic polynomial over a finite field ${𝔽}_q$, and $k\ge 2$ an integer. A digraph $G(f,k)$ is one whose vertex set is $\frac{{𝔽}_q[x]}{〈f〉}$ and for which there is a directed edge from a polynomial $g_1$ to $g_2$ if $g_1^k=g_2$ in $\frac{{𝔽}_q[x]}{〈f〉}$. If $f=P_1^{e_1}{P}_2^{e_2}\dots P_r^{e_r}$, where the $P_i$’s are distinct monic irreducible polynomials over ${𝔽}_q$, then $G(f,k)$ can be factorized as $G(f,k)\cong G(P_1^{e_1},k)\times G(P_2^{e_2},k)\times \cdots \times G(P_r^{e_r},k)$. In this work, we investigate the structure of these power digraphs. The semiregularity property is examined, and its relationship with the symmetric property is established. In addition, we look into the uniqueness of factorization of trees attached to a fixed point.

Let $𝔤$ be symmetrizable Kac–Moody Lie algebra. In this paper, we describe a new class of central operators generalizing the Casimir operator. We also prove some properties of these operators and show that these operators move highest weight vectors to new highest weight vectors.

In this paper, we give a characterization of the quasi-linear left cycle sets $A$ with $\mathrm{\text{Rad}}(A)\subseteq \mathrm{\text{Soc}}(A)$ via unitary metahomomorphisms and a complete description of those with $\mathrm{\text{Rad}}(A)\subseteq \mathrm{\text{Fix}}(A)=\mathrm{\text{Soc}}(A)$ improving the results obtained in [F. Catino and M. M. Miccoli, Construction of quasi-linear left cycle sets, J. Algebra Appl. 14(1) (2015), Article ID:1550001, 1–7]. Moreover, we develop a theory of dynamical extensions of quasi-linear left cycle sets to provide new set-theoretic solutions of the Yang–Baxter equation that are non-degenerate, involutive and multipermutational.