Journal of Algebra and Its Applications

In this paper, we deal with integral domains with only a few semistar operations. Mainly, we give complete characterizations of integral domains with exactly three (respectively, four) semistar operations (respectively, semistar operation of finite character). Examples to illustrate our main theorems are given.

In this paper, we study the class of ${\mathbb{Z}}_p$-double cyclic codes of length $n=r+s.$ We give a closed formula for the number of ${\mathbb{Z}}_p$ -double cyclic codes of length $n=r+s,$ for any integers $r$ and $s$ that are relatively prime to $p.$ Moreover, we give a closed formula for the number of quasi-cyclic (QC) codes of length $n=2s$ and index $2.$ We also provide formulas for the number of separable and non-separable ${\mathbb{Z}}_p$-double cyclic codes of length $n.$ In order to illustrate the results, we calculate the number of some codes with different $r$ and $s$. Moreover, we list optimal parameter ${\mathbb{Z}}_2$-double cyclic codes for specific values of $r$ and $s$.

The so-called Frobenius number in the famous linear Diophantine problem of Frobenius is the largest integer such that the linear equation $a_1{x}_1+\cdots +a_k{x}_k=n$ ($a_1,\dots ,a_k$ are given positive integers with $gcd(a_1,\dots ,a_k)=1$) does not have a non-negative integer solution $(x_1,\dots ,x_k)$. The generalized Frobenius number (called the $p$-Frobenius number) is the largest integer such that this linear equation has at most $p$ solutions. That is, when $p=0$, the $0$-Frobenius number is the original Frobenius number.

In this paper, we introduce and discuss $p$-numerical semigroups by developing a generalization of the theory of numerical semigroups based on this flow of the number of representations. That is, for a certain non-negative integer $p$, $p$-gaps, $p$-symmetric semigroups, $p$-pseudo-symmetric semigroups, and the like are defined, and their properties are obtained. When $p=0$, they correspond to the original gaps, symmetric semigroups and pseudo-symmetric semigroups, respectively.

We are dealing with extensions of commutative rings $R\subseteq S$ whose chains of the poset $[R,S]$ of their subextensions are finite (i.e. $R\subseteq S$ has the FCP property) and such that $[R,S]$ is a distributive lattice, that we call distributive FCP extensions. Note that the lattice $[R,S]$ of a distributive FCP extension is finite. This paper is the continuation of our earlier papers where we studied catenarian and Boolean extensions. Actually, for an FCP extension, the following implications hold: Boolean $\Rightarrow$ distributive $\Rightarrow$ catenarian. A comprehensive characterization of distributive FCP extensions actually remains a challenge, essentially because the same problem for field extensions is not completely solved. Nevertheless, we are able to exhibit a lot of positive results for some classes of extensions. A main result is that an FCP extension $R\subseteq S$ is distributive if and only if $R\subseteq \overline{R}$ is distributive, where $\overline{R}$ is the integral closure of $R$ in $S$. A special attention is paid to distributive field extensions.

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA $A$ a derivative series of IGWAs, $A\to A^{\prime }\to A^{″}\to \cdots \to A^{(\alpha )}\to \cdots$, is attached where $\alpha$ is an arbitrary ordinal. In general, all rings $A^{(\alpha )}$ are distinct. A new construction of rings, the inner $(\sigma ,\tau ,a)$-extension of a ring, is introduced (where $\sigma$ and $\tau$ are endomorphisms of a ring $D$ and $a\in D$).

The classes ${{𝒟}}_{{𝒬}}$ of flat relative Mittag-Leffler modules are sandwiched between the class ${ℱ}{ℳ}$ of all flat (absolute) Mittag-Leffler modules, and the class ${ℱ}$ of all flat modules. Building on the works of Angeleri Hügel, Herbera, and Šaroch, we give a characterization of flat relative Mittag-Leffler modules in terms of their local structure, and show that Enochs’ Conjecture holds for all the classes ${{𝒟}}_{{𝒬}}$. In the final section, we apply these results to the particular setting of $f$-projective modules.

Let $G$ be a finite group and $K$ a splitting field of $G$ of characteristic $p>0$. Denote by $KG$ the group algebra of $G$ over $K$ and $Z(KG)$ the center of $KG$. Let $V_G$ be the $K$-subspace of trace-zero elements of $KG$. We give some numerical sufficient and necessary conditions for $V_G$ and $V_G\cap Z(KG)$, respectively, to be Mathieu subspaces of $KG$ in terms of the degrees of irreducible representations of $G$ over $K$. The same numerical conditions also characterize the finite groups $G$ that $KG$ has no nonzero trace-zero idempotents and the finite groups $G$ that $KG$ has no nonzero central trace-zero idempotents, respectively.

The concept of standard basis for a module is generalized to that for a mixed module that can be written as sums of more than one modules over different rings. An algorithm to compute a standard basis for a mixed module is presented, along with a proof of its correctness and termination in finite steps. The algorithm is demonstrated in several examples appearing in computations of codimensions of divergent diagrams. A parametric extension of the standard basis (comprehensive standard system, CSS) for a mixed-module is also presented, along with a proof of its correctness and termination in finite steps and its computational algorithms. The algorithm is demonstrated in several examples in the classification of divergent diagrams including applications of Mather’s lemma and the complete transversal theorem.

In the present paper, non-weight modules over the super-BMS₃ algebras ${𝔖}^{𝜖}$ ($𝜖=0$ or $\frac{1}{2}$) are constructed. Assume that $𝔱=ℂL_0\oplus ℂW_0\oplus ℂG_0$ and $𝔗=ℂL_0\oplus ℂW_0$ are the Cartan subalgebra (modulo center) of ${𝔖}^0$ and ${𝔖}^{\frac{1}{2}}$, respectively. These modules over ${𝔖}^0$ when restricted to the $𝔱$ are free of rank $1$, while these modules over ${𝔖}^{\frac{1}{2}}$ when restricted to the $𝔗$ are free of rank $2$. Then we determine the necessary and sufficient conditions for these modules being simple, as well as determining the necessary and sufficient conditions for two ${𝔖}^{𝜖}$-modules being isomorphic.

Staic defined symmetric cohomology of groups and studied that the secondary symmetric cohomology group is corresponding to group extensions and the injectivity of the canonical map from symmetric cohomology to classical cohomology. In this paper, we define symmetric cohomology and symmetric Hochschild cohomology for cocommutative Hopf algebras. The first one is a generalization of symmetric cohomology of groups. We give an isomorphism between symmetric cohomology and symmetric Hochschild cohomology, which is a symmetric version of the classical result about cohomology of groups by Eilenberg and MacLane and cohomology of Hopf algebras by Ginzburg and Kumar. Moreover, to consider the condition that symmetric cohomology coincides with classical cohomology, we investigate the projectivity of a resolution which gives symmetric cohomology.

In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain.

It is well known that a ring $R$ is right Kasch if each simple right $R$-module embeds in a projective right $R$-module. In this paper we study the dual notion and call a ring $R$ right dual Kasch if each simple right $R$-module is a homomorphic image of an injective right $R$-module. We prove that $R$ is right dual Kasch if and only if every finitely generated projective right $R$-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field $k$, a finite dimensional $k$-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.

Let $(R,\cdot ,\circ )$ be a braided Hopf brace, i.e. a Hopf brace in the category of Yetter–Drinfeld modules over a Hopf brace $(H,\cdot ,\circ )$. In this note, we give the sufficient and necessary conditions which make Radford’s biproduct $(R♯H,\cdot )$ and $(R♯H_{\circ },\circ )$ become a Hopf brace.

Abelian track categories can be classified via the third Baues–Wirsching cohomology of small categories. This approach is used in this paper to derive the different generalizations of the obstruction theory of non-abelian group extensions, due to Cegarra, Garzón and Grandjean, Cegarra, Garzón and Ortega, and Chen, Du and Wang as particular cases of a general theory of abelian track categories and to give unified proofs of the main properties of these obstruction classes.

The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees, by itself an ambitious objective. As understood, sizing up means looking closely at the two available parameters: the shifts and the Betti numbers. Since, in general, bounds for the shifts can behave quite steeply, we filter the difficulty by the subadditivity of the syzygies. The method we applied is hopefully new and sheds additional light on the structure of the minimal free resolution. For the Betti numbers, we apply the Boij–Söderberg techniques in order to get polynomial upper bounds for them. It is expected that the landscape of hypersurface singularities already contains most of the difficult corners of the arbitrary case. In this regard, we treat some facets of this case, including an improvement of one of the basic results of a recent work by Busé–Dimca–Schenck–Sticlaru.

In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let ${𝒜}$ be a $3$-dimensional DG Sklyanin algebra with ${{𝒜}}^{\#}=S_{a,b,c}$, where $(a,b,c)\in {\mathbb{P}}_k^2-𝔇$ and

We define and study the so-called CI-extending property for Abelian groups, thus somewhat extending results due to Birkenmeier et al. in Commun. Algebra (2001) concerning the FI-extending property.

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of $m$-systems. The notion of $m$-system is also studied from the topological point of view.

We introduce and investigate the algebras of steadily growing length, that is the class of algebras, where the length is bounded by a linear function of the dimension. In particular we show that Malcev algebras belong to this class and establish the exact upper bound for its length.

Let $𝔤$ be a finite-dimensional simple Lie algebra over $ℂ$. A $Y(𝔤)$-module is said to be weight if it is a weight $𝔤$-module. We give a complete classification of simple weight modules for $Y({𝔰𝔩}_2)$ which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical ${𝔰𝔩}_2$-representation theory, we show that there exists a class of $Y({𝔰𝔩}_2)$ irreducible modules which have uniformly two-dimensional weight spaces.

In a previous paper, the second and third named authors introduced the concept of the complete cycle index and discussed a relation with the complete weight enumerator in coding theory. In this paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator, respectively, in coding theory. Moreover, the notion of the average intersection numbers is given, and we discuss a relation with the average intersection numbers in coding theory.