Journal of Algebra and Its Applications

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree.

Let $X$ be a nonempty set and $T(X)$ denote the full transformation semigroup on $X$. For a fixed nonempty subset $Y$ of $X$, let

For a rooted cluster algebra $\mathcal{𝒜}(Q)$ over a valued quiver $Q$, a symmetric cluster variable is any cluster variable belonging to a cluster associated with a quiver $\sigma (Q)$, for some permutation $\sigma$. The subalgebra of $\mathcal{𝒜}(Q)$ generated by all symmetric cluster variables, is called the symmetric mutation subalgebra and is denoted by $\mathcal{ℬ}(Q)$. In this paper, we identify the class of cluster algebras that satisfy $\mathcal{ℬ}(Q)=\mathcal{𝒜}(Q)$, which contains almost every quiver of finite mutation type. In the process of proving the main result, we provide a classification of quivers mutations classes that relates their maximum weights to the shapes of the initial quivers. Furthermore, some properties of symmetric mutation subalgebras are given.

Let $\mathrm{\Delta }=\left(\right.\begin{array}{cc}\hfill A\hfill & \hfill {}_A{N}_B\hfill \\ \hfill {}_B{M}_A\hfill & \hfill B\hfill \end{array}\left)\right.$ be a Morita ring with $M{\otimes }_{A}N=0=N{\otimes }_{B}M$. We first study how to construct (complete) duality pairs of $\mathrm{\Delta }$-modules using (complete) duality pairs of $A$-modules and $B$-modules, generalizing the result of Mao (Commun. Algebra 12 (2020) 5296–5310) about the duality pairs over a triangular matrix ring. Moreover, we construct Gorenstein projective modules relative to complete duality pairs of $\mathrm{\Delta }$-modules. Finally, we give an application to Ding projective modules.

In this paper, we provide the necessary and sufficient conditions for the edge binomials of the tree forming a $d$-sequence in terms of the degree sequence notion of a graph. We study the regularity of powers of the binomial edge ideals of trees generated by $d$-sequence edge binomials.

Let $R$ be a commutative ring with identity and $S$ be a multiplicatively closed subset of $R$. The purpose of this paper is to introduce the concept of weakly $S$-primary ideals as a new generalization of weakly primary ideals. An ideal $I$ of $R$ disjoint with $S$ is called a weakly $S$-primary ideal if there exists $s\in S$ such that whenever $0\ne ab\in I$ for $a,b\in R$, then $sa\in \sqrt{I}$ or $sb\in I$. The relationships among $S$-prime, $S$-primary, weakly $S$-primary and $S$-$n$-ideals are investigated. For an element $r$ in any general ZPI-ring, the (weakly) $S_r$-primary ideals are characterized where $S_r=\{1,r,r^2,\dots \}$. Several properties, characterizations and examples concerning weakly $S$-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly $S$-decomposable ideals and $S$-weakly Laskerian rings which are generalizations of $S$-decomposable ideals and $S$-Laskerian rings are introduced.

Proxy signature schemes can be used in all the scenarios where a signer is not able to sign an important message. In some cases which involve a group of proxy signers, the anonymity of the proxy signer is required. We propose the first multivariate anonymous proxy signature scheme over finite field, and discuss all its security properties such as anonymity, unforgeability, verifiability, distinguishability, and undeniability. Additionally, we give a toy example to demonstrate the mathematical functioning of the proposed scheme. We compare our scheme with the existing post-quantum anonymous proxy signature scheme.

Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this paper, we compute the nef and pseudoeffective cones of divisors on the Grassmann bundle Gr${}_X(k,E)$ parametrizing $k$-dimensional subspaces of the fibers of $E$, where $1\le k\le$ rk($E$), under assumptions on $X$ as well as on the vector bundle $E$. In particular, we show that nef cone and the pseudoeffective cone of Gr${}_X(k,E)$ coincide if and only if nef cone and pseudoeffective cone of $X$ coincide under the assumption that $E$ is a slope semistable bundle on $X$ with $c_2$(End($E$))=0. We also discuss about the nefness and ampleness of the universal quotient bundle $Q_k$ on Gr${}_X(k,E)$.

A subgroup $H$ of a finite group $G$ is said to be $CSS$-subgroup of $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $SS$-quasinormal in $G$. In this paper, we investigate the $p$-nilpotency of the finite groups under the assumption that there is a subgroup $D$ of $P$ such that $1<\left|D\right|<\left|P\right|$ and every subgroup of $P$ with order $\left|D\right|$ or $2\left|D\right|$ (whenever $\left|D\right|=2$) is $CSS$-subgroup, where $p$ is a prime dividing the order of $G$ and $P$ is a Sylow $p$-subgroup of $G$. As an application of the results, a related problem posed by Heliel et al. in [Finite groups with minimal CSS-subgroups, Eur. J. Pure Appl. Math. 14 (2021) 1002–1014] is solved.

A multiplication on an Abelian group $G$ is a homomorphism $\mu :G\otimes G\to G$. The set $\mathrm{\text{Mult}}G$ of all multiplications on an Abelian group $G$ itself is an Abelian group with respect to addition; this group is called the multiplication group of $G$. In the paper, the class $\mathcal{𝒬}\mathcal{𝒟}1$ of quotient divisible Abelian groups of rank 1 is considered. The group $\mathrm{\text{Mult}}G$ is described for $G\in \mathcal{𝒬}\mathcal{𝒟}1$. In the group $\mathrm{\text{Mult}}G$, multiplications defining comparable rings on $G$ are described for any group $G\in \mathcal{𝒬}\mathcal{𝒟}1$. The isomorphism problem is solved in $\mathcal{𝒬}\mathcal{𝒟}1$: multiplications defining isomorphic rings on $G$ are described for any $G\in \mathcal{𝒬}\mathcal{𝒟}1$.

Given a (quasi-)twilled pre-Lie algebra, we first construct a differential graded Lie algebra ($L_{\infty }$-algebra). Then we study the twisting theory of (quasi-)twilled pre-Lie algebras and show that the result of the twisting by a linear map on a (quasi-)twilled pre-Lie algebra is also a (quasi-)twilled pre-Lie algebra if and only if the linear map is a solution of the Maurer–Cartan equation of the associated differential graded Lie algebra ($L_{\infty }$-algebra). In particular, the relative Rota–Baxter operators (twisted relative Rota–Baxter operators) on pre-Lie algebras are solutions of the Maurer–Cartan equation of the differential graded Lie algebra ($L_{\infty }$-algebra) associated to the certain quasi-twilled pre-Lie algebra. Finally, we use the twisting theory of (quasi-)twilled pre-Lie algebras to study quasi-pre-Lie bialgebras. Moreover, we give a construction of quasi-pre-Lie bialgebras through symplectic Lie algebras, which is parallel to that a Cartan $3$-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.

In this work, we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras $n_{n,1}$ and $Q_{2n}.$ We find all one-dimensional extensions of solvable Lie algebras with nilradical $n_{n,1}$. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical $n_{n,1}$ of maximal codimension. Moreover, all one-dimensional extensions of solvable Lie algebras with nilradical $n_{n,1}$ whose codimension is equal to one are found and we compared these solvable algebras with the solvable algebras with nilradicals that are one-dimensional central extension of algebra $n_{n,1}$.

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

Given a construction $f$ on groups, we say that a group $G$ is $f$-realisable if there is a group $H$ such that $G\cong f(H)$, and completely$f$-realisable if there is a group $H$ such that $G\cong f(H)$ and every subgroup of $G$ is isomorphic to $f(H_1)$ for some subgroup $H_1$ of $H$ and vice versa. In this paper, we determine completely $\mathrm{\text{Aut}}$-realisable groups. We also study $f$-realisable groups for $f=Z,F,M,D,\mathrm{\Phi }$, where $Z(H)$, $F(H)$, $M(H)$, $D(H)$ and $\mathrm{\Phi }(H)$ denote the center, the Fitting subgroup, the Chermak–Delgado subgroup, the derived subgroup and the Frattini subgroup of the group $H$, respectively.

Roos’ direct sum decomposition of a cyclic convolutional code based upon a suitable construction of a minimal encoder is extended to a broader class of convolutional codes endowed with additional algebraic structure. The two (equivalent) usual concepts of a convolutional code are taken into account, as a vector space and as a module.

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra $A_{N2n+1}^{\mu \lambda }$ and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, Super type $A_2(q;{𝕀}_2)$ and the Nichols algebra $𝔲𝔣𝔬(8)$. And the involved finite-dimensional Nichols algebras of non-diagonal type are $12$, $4m$ and $m^2$-dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over $A_{13}^{+\lambda }$.

Let $h={\prod }_{i=1}^t{p}_i^{s_i}$ be its decomposition into a product of powers of distinct primes, and ${\mathbb{Z}}_h$ be the residue class ring modulo $h$. Let ${\mathbb{Z}}_h^n$ be the $n$-dimensional row vector space over ${\mathbb{Z}}_h$. A generalized Grassmann graph over ${\mathbb{Z}}_h$, denoted by $G_r(m,n,{\mathbb{Z}}_h)$ ($G_r$ for short), has all $m$-subspaces of ${\mathbb{Z}}_h^n$ as its vertices, and two distinct vertices are adjacent if their intersection is of dimension $>m-r$, where $2\le r\le m+1\le n$. In this paper, we determine the clique number and geometric structures of maximum cliques of $G_r$. As a result, we obtain the Erdős–Ko–Rado theorem for ${\mathbb{Z}}_h^n$.

Let $G$ be a finite group and Irr($G$) be the set of irreducible characters of $G$. The number cod($\chi$) = $|G:$ker $\chi |/\chi (1)$ is called the character codegree of $\chi \in$ Irr($G$). In this paper, we show that if $G$ has at most two composite character codegrees, then $G$ is solvable. We also obtain sufficient numerical conditions on the character codegrees and character degrees for the solvability and supersolvability of a group.

We introduce the notions of Gorenstein projective $\tau$-rigid modules, Gorenstein projective support $\tau$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi–Iyama–Reiten’s bijection theorem on support $\tau$-tilting modules. More precisely, for an algebra $\mathrm{\Lambda }$, we prove that there is a bijection between the set of Gorenstein projective support $\tau$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$\tau$-tilting finite algebras and show that $\mathrm{\Lambda }$ is CM-$\tau$-tilting finite if and only if ${\mathrm{\Lambda }}^{\mathrm{\text{op}}}$ is CM-$\tau$-tilting finite. Moreover, we show that the Bongartz completion of a Gorenstein projective $\tau$-rigid module need not be a Gorenstein projective $\tau$-tilting module.

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called $\varphi$-u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On $\varphi$-exact sequence and $\varphi$-projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of $\varphi$-flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On $\varphi$-Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on $\varphi$-Dedekind rings and $\varphi$-Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly $\varphi$-flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the $\varphi$-injective dimension, $\varphi$-projective dimension and $\varphi$-flat dimension for modules, and also the $\varphi$-(weak) global dimension of rings. Then, using these dimensions, we characterize several $\varphi$-rings ($\varphi$-Prüfer, $\varphi$-chained, $\varphi$-von Neumann, etc). Finally, we study the $\varphi$-(weak) global dimension of the trivial ring extensions defined by some conditions.

Let $R$ be a commutative ring with identity and $S$ be a multiplicative subset of $R$. We say that $R$ has $S$-Noetherian spectrum if for every ideal $I$ of $R,$ $sI\subseteq \sqrt{J}\subseteq \sqrt{I}$ for some $s\in S$ and some finitely generated ideal $J.$ In this paper, we study rings with $S$-Noetherian spectrum. Among other things, we give a necessary and sufficient condition for Nagata’s idealization $R(+)M$, where $M$ is an $R$-module, to satisfy the $S$-Noetherian spectrum. We also investigate when the amalgamated algebra along an ideal has $S$-Noetherian spectrum. Several examples are given to illustrate the concepts and results.