Journal of Algebra and Its Applications

As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type $B$ which is a $\mathbb{Q}(q)$-algebra associated with the Weyl group $\mathcal{𝒲}(B)$ of type $B$, and symmetric groups ${𝔖}_l$ for $l=0,1,\dots ,m$, satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type $A$ arising from the related $q$-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type $B$ admits natural representations on certain tensor spaces. We then establish a Levi-type $q$-Schur–Weyl duality of type $B$, which reveals the double centralizer property between such duplex Hecke algebras and $ı$quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque 402 (2018)].

In this paper, we give a complete description of the maximal subgroups of the semigroup of all $n\times n$ tropical matrices under multiplication up to isomorphism. Using the description, we prove that every subgroup of the semigroup of all $n\times n$ Boolean matrices under multiplication is embedded in the symmetric group $S_n$.

In this paper, we describe some different variations of prime ideals in the context of rings of continuous functions such as strongly prime ideals, almost prime ideals, $n$-absorbing ideals, and 2-prime ideals. We characterize the strongly prime ideals of $C(X)$. We prove that an ideal $I$ of $C(X)$ is almost prime if and only if it is semiprime or equivalently if and only if it is an $(m,n)$-closed ideal, where $m$ and $n$ are positive integers. In [On $n$-absorbing ideals of commutative rings, Comm. Algebra 39 (2011) 1646–1672], Anderson and Badawi conjectured that every $n$-absorbing ideal of a commutative ring is strongly $n$-absorbing. We determine the $n$-absorbing ideals of $C(X)$ and show that the conjecture holds in the case of rings of continuous functions. We also characterize the rings of continuous functions $C(X)$ in which every pseudoprime ideal is 2-prime.

This paper deals with the graded commutative rings in which every graded prime ideal is contained in a unique graded-maximal ideal called graded-Gelfand ring. The purpose is to establish some topological and algebraic characterizations of these rings, one of which is the algebraic analogue of Urysohn’s lemma. Finally, we look at a special class of those graded rings called graded-ordered rings which can be viewed as graded rings with a Gelfand strong property.

Let $n$ be a positive integer and let ${𝔽}_q$ be a finite field with $q$ elements, where $q$ is a prime power and $gcd(n,q)=1$. In this paper, we give the explicit factorization of $x^n-1$ over ${𝔽}_q$ and count the number of its irreducible factors for the following conditions: $n,q$ are odd and $\text{rad}(n)|(q^2+1)$. First, we present a method to obtain the set of all representatives of $q$-cyclotomic cosets modulo $m$, where $m=gcd(n,q^2+1)$. This set of representatives is then used to find the irreducible factors of $x^n-1$ and the cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ over ${𝔽}_q$. The form of irreducible factors of $x^n-1$ is characterized such that the coefficients of these irreducible factors are followed by second-order linear recurring sequences.

Let $N⊴G$ be a pair of finite subgroups of ${\mathrm{\text{SL}}}_2(ℂ)$ and $V$ a finite-dimensional fundamental $G$-module. We study Kostant’s generating functions for the decomposition of the ${\mathrm{\text{SL}}}_2(ℂ)$-module $S^k(V)$ restricted to $N◃G$ in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.

In this paper, we study some sufficient and necessary conditions for the amalgamated duplication Banach algebra $X⋊B$ to be zero product determined. Precisely, we show that in the case where $X$ has a bounded approximate identity, $X⋊B$ is zero product determined if and only if $B$ is zero product determined and $X⋊B$ has the property $𝔹$. Moreover, we give a better characterization when $X$ is unital. As an application, we also construct some examples on semidirect product of Banach algebras and the $𝜃$–Lau product of Banach algebras. Finally, we raise some open problems related to the zero product determined of classical module extension Banach algebras.

In this paper, we study the modules whose projectivity domains are contained in the class of all pure-split modules and call them $p$-impecunious modules. Every quotient of a p-impecunious $R$-module is $p$-impecunious if and only if $R$ is right pure semisimple. An abelian group $A$ is p-impecunious if all p-components of $A$ are nonzero and $B_p(A)\ne 0$ for some prime $p$ where $B_p(A)$ is the basic subgroup of the $p$-component $T_p(A)$. The converse is true if $A$ is torsion. We characterize Dedekind domains over which there are injective p-impecunious modules.

The Morita equivalences of classical Brauer algebras and classical Birman–Murakami–Wenzl (BMW) algebras have been well studied. Here, we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and BMW algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of Coxeter groups, respectively.

In this note, we investigate Jacobian conjecture through investigation of automorphisms of polynomial rings in characteristic $p$. Making use of the technique of inverse limits, we show that under Jacobian condition for a given homomorphism $\phi$ of the polynomial ring k[$x_1,\dots ,x_n$], if $\phi$ preserves the maximal ideals, then $\phi$ is an automorphism.

In the spirit of some earlier studies of the authors, we discuss the alienation problem for ∨-additive and multiplicative mappings. This study enables us to answer two questions that were left open in [B. Al Subaiei and N. Jarboui, On the monoid of unital endomorphisms of a Boolean ring, Axioms 10(4) (2021) 305].

Linear algebra tools are used to give a new approach to the open problem of the classification of Bernstein algebras. We prove that any Bernstein algebra $(A,\omega )$ is isomorphic to a semidirect product $N{⋉}_{(\cdot ,\mathrm{\Omega })}k$ associated to a commutative algebra $(N,\cdot )$ such that ${(x^2)}^2=0$, for all $x\in N$ and an idempotent endomorphism $\mathrm{\Omega }={\mathrm{\Omega }}^2\in {\mathrm{\text{End}}}_k(N)$ of $N$ satisfying two compatibility conditions. The set of types of $(1+|I|)$-dimensional Bernstein algebras is parametrized by an explicitly constructed classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups $(N,+)⋉{\mathrm{\text{GL}}}_k(N)$.

Let $R$ be a commutative ring with identity and Nil$(R)$ be the set of all nilpotent elements of $R$. The nil clean graph of $R$, denoted by $N.G(R)$, is a graph whose vertices are all nonzero nil clean elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy\in \mathrm{\text{Nil}}(R)$ or $x-y\in \mathrm{\text{Nil}}(R)$. In this paper, we focus on $N.G_2(R)$, the subgraph of $N.G(R)$ induced by the set $N.G_2(R)=\{(e,n):0,1\ne e\in \mathrm{\text{Id}}(R)$ and $n\in \mathrm{\text{Nil}}(R)\}$. It is observed that $N.G_2(R)$ has a crucial role in $N.G(R)$. We prove that $N.G_2(R)$ is connected with diam($N.G_2(R))\le 3$ and gr($N.G_2(R))\in \{3,\infty \}$. We investigate also the interplay between the ring-theoretic properties of $R$ and the graph-theoretic properties of $N.G_2(R)$. Moreover, the clique number and the chromatic number of $N.G_2(R)$ for some classes of rings are determined.

Let $V_L$ be the vertex algebra associated with a non-degenerate and non-positive definite even lattice $L$ and $V_L^{+}$ the fixed point subalgebra of $V_L$ under the action of the automorphism induced from the $-1$-isometry of $L$. We determine the fusion rules for weak $V_L^{+}$-modules. As an application of this result, we classify the irreducible weak modules for the fixed point subalgebra of $V_L$ under the action of the automorphism induced from an isometry of $L$ of order $2$. We also show that every weak module for the same fixed point subalgebra is completely reducible.

Let $V$ be an $\mathbb{N}$-graded vertex algebra over a field $𝔽$, and let $g$ be an automorphism of $V$ with a finite order $T$. Assume that the field $𝔽$ contains a $T$th primitive root of unity. We construct two associative algebras, $A_g(V)$ and $R_g(V)$, and investigate their properties and interconnections. As applications, we establish the following results: (1) If $R_g(V)$ is finite-dimensional, then $A_g(V)$ is also finite-dimensional; (2) If $V$ is strongly finitely generated, then $A_g(V)$ is stably noetherian; (3) If $V$ is strongly finitely generated and $M$ is an irreducible admissible $g$-twisted $V$-module, then the endomorphism algebra ${\mathrm{\text{End}}}_V(M)$ is finite-dimensional.

Let $G$ be a group. If the set ${𝒜}(G)=\{\alpha \in \mathrm{\text{Aut}}(G):x\alpha (x)=\alpha (x)x\text{ for all }x\in G\}$ forms a subgroup of $\mathrm{\text{Aut}}(G)$, then G is called ${𝒜}$-group. Recently, it has been proven that the minimum coclass of a non-${𝒜}$ $p$-group is equal to $3$. In this paper, we deal with finite $p$-groups of coclass $3$. We prove that a finite $p$-group $G$ of order $p^n$, $n\ge 6$, of coclass $3$ for an odd prime $p$ is an ${𝒜}$-group except when the nilpotency class of the second term of upper central series $G$, $Z_2(G)$, is equal to $2$. In particular, we show that in this case, center of $G$, $Z(G)$, and commutator subgroup $G$, $G^{\prime }$, are of order $p$ and $p^{n-4}$, respectively, and $Z_2(G)$ is a group of order $p^4$. Also, we give some sufficient conditions on a finite 2-group $G$ of coclass $3$ implying that $G$ is an ${𝒜}$-group.

The enhanced power graph of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x,y$ are adjacent if $x,y\in 〈z〉$ for some $z\in G$. An $L(2,1)$-labeling of graph $\mathrm{\Gamma }$ is an integer labeling of $V(\mathrm{\Gamma })$ such that adjacent vertices have labels that differ by at least $2$ and vertices distance $2$ apart have labels that differ by at least $1$. The $\lambda$-number of $\mathrm{\Gamma }$, denoted by $\lambda (\mathrm{\Gamma })$, is the minimum range over all $L(2,1)$-labelings. In this paper, we study the lambda number of the enhanced power graph ${{𝒫}}_E(G)$ of the group $G$. This paper extends the corresponding results, obtained in [X. Ma, M. Feng and K. Wang, Lambda number of the power graph of a finite group, J. Algebraic Combin. 53(3) (2021) 743–754], of the lambda number of power graphs to enhanced power graphs. Moreover, for a nontrivial simple group $G$ of order $n$, we prove that $\lambda ({{𝒫}}_E(G))=n$ if and only if $G$ is not a cyclic group of order $n\ge 3$. Finally, we determine the lambda number of the enhanced power graphs of nilpotent groups.

In this paper, we provide a necessary and sufficient condition for a class of trivial extensions so that each Jordan derivation on such algebras can be written as the sum of a derivation and an antiderivation. Among other things, we obtain some of the known results concerning the Jordan derivations on trivial extensions with fewer assumptions. Also, as a generalization of a well-known result on triangular matrix algebras, we show that every Jordan derivation from arbitrary triangular algebra into its certain bimodule is the sum of a derivation and an antiderivation. We also apply our results for upper triangular matrix algebra ${{𝒯}}_n({ℛ})$, matrix algebra ${{ℳ}}_n({ℛ})$ and the nest algebra ${𝒯}({𝒩})$ to describe the Jordan derivations into their bimodules.

The purpose of this paper is to study cohomologies of differential $3$-Lie algebras with any weight and applications. First, we introduce representations of differential $3$-Lie algebras. Moreover, we develop cohomology theory of differential $3$-Lie algebras. We also depict the relationship between the cohomologies of a differential $3$-Lie algebra and its associated differential Leibniz algebra with weight zero. Finally, formal deformations, abelian extensions and skeletal differential $3$-Lie 2-algebras are characterized in terms of cohomology groups.

Let $G$ be a simply connected and simple algebraic group defined and split over a finite prime field ${𝔽}_p$ of $p$ elements. In this paper, using an ${𝔽}_p$-linear map splitting Frobenius endomorphism on a hyperalgebra relative to $G$, we obtain some ${𝔽}_p$-linear isomorphisms induced by multiplication in the hyperalgebra.

In this paper, we study almost Cohen–Macaulay bipartite graphs. In particular, we prove that if $G$ is an almost Cohen–Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of $deg(v)\le 2$. In particular, if $G$ is an almost Cohen–Macaulay bipartite graph and $u$ is a vertex of degree one of $G$ and $v$ its adjacent vertex, then $G\setminus \{v\}$ is almost Cohen–Macaulay. Also, we show that an unmixed Ferrers graph is almost Cohen–Macaulay if and only if it is connected in codimension two. Moreover, we give some examples.