Journal of Algebra and Its Applications

In this paper, we introduce (Rees) artinian $S$-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that (Rees) artinian $S$-acts are those acts whose factor acts are all finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian $S$-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain conditions on monoids. Finally, we prove that every right Rees artinian, commutative monoid is right Rees noetherian.

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number $\rho (n,m,r)$ associated with the maximal order in the quaternion algebra ${(-1,-1)}_{\mathbb{Q}}$. As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun’s “1-3-5 conjecture”. Furthermore, we show that “a perfect square” in the integral condition version of Sun’s conjecture can be replaced by “a power of 4”.

Let $G$ be a finite group. For a given pair of non-negative integers $n$ and $k$, we aim to give an explicit formula to count the number of elements of ${\mathrm{\Lambda }}_k(G)$, the set of all irreducible representations of $G$ whose degree is not divisible by $2^k$. In this paper, we give formulas for $|{\mathrm{\Lambda }}_k(G)|$, when $G$ is symmetric group, generalized symmetric group, and alternating group. Further, we also have a complete characterization of the elements in ${\mathrm{\Lambda }}_2(S_n)$, ${\mathrm{\Lambda }}_1(G(n,r))$, and the self-conjugate partitions of $2^l$ and $2^l+1$ with degree not divisible by $4$.

This paper is devoted to defining and studying Whittaker modules and high order Whittaker modules for the $N=1$ super-BMS₃ algebra. We also classify the simple Whittaker modules and obtain the necessary and sufficient conditions for the irreducibility of these modules.

The aim of this paper is to investigate the interplay between the algebraic properties of a skew Poincaré–Birkhoff–Witt extesion ring $A=\sigma (R)〈x_1,\dots ,x_n〉$ and the graph-theoretic properties of its zero-divisor graph. We are interested in studying the diameter of the zero-divisor graph of skew PBW extension rings. Among other results, we give a complete characterization of the possible diameters of $\mathrm{\Gamma }(A)$ in terms of the diameter of $\mathrm{\Gamma }(R)$.

Let $R$ be a commutative ring with nonzero identity. A proper ideal $I$ of $R$ is called a 1-absorbing prime ideal (respectively, 1-absorbing primary ideal) if whenever nonunit elements $a,b,c\in R$ with $abc\in I,$ then $ab\in I$ or $c\in I$ (respectively, $ab\in I$ or $c\in \sqrt{I}$). The purpose of this paper is to study the transfer of certain 1-absorbing-like properties to amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by $A{\bowtie }^{f}J$), introduced and studied by D’Anna, Finocchiaro and Fontana. Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties.

Let $G$ be a finite group. We denote by ${\psi }^{\prime }(G)$ the integer ${\prod }_{g\in G}o(g),$ where $o(g)$ denotes the order of $g\in G$. It is verified that some finite groups are uniquely determined by their orders and their product of element orders. In this paper, we prove that, if $p$, $q$, $r$, $t$ are different prime numbers, then ${\mathbb{Z}}_{p^3}$, ${\mathbb{Z}}_{pq}$, ${\mathbb{Z}}_{p^{2}q}$ and ${\mathbb{Z}}_{pqrt}$ are characterized by ${\psi }^{\prime }(G)$ and if ${\psi }^{\prime }(G)={\psi }^{\prime }({\mathbb{Z}}_{pqr})$, then $G\cong {\mathrm{\text{A}}}_5$ or ${\mathbb{Z}}_{pqr}$.

Let $M_q(n)$ be the standard quantized matrix algebra introduced by Faddeev, Reshetikhin, and Takhtajan. It is shown that some structural properties of $M_q(n)$ and their modules may be established and realized in a constructive-computational way.

Let $G$ be a non-compact classical semisimple Lie group and let $G/V$ be the adjoint orbit with respect to a fixed element in $G$. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on $G/V$ purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type $A_{\ell }$, $B_{\ell }$, $C_{\ell }$, $D_{\ell }$, where $\ell$ denotes the rank of the Lie algebra.

In this paper, we study the relation between the images of polynomial derivations and their simplicity. We prove that the images of simple Shamsuddin derivations are not Mathieu–Zhao spaces. In addition, we show that the images of some simple derivations in dimension three are not Mathieu-Zhao spaces. Thus, we conjecture that the images of simple derivations in dimension greater than one are not Mathieu–Zhao spaces. We also prove that locally nilpotent derivations are not simple in dimension greater than one.

Let $q$ be a prime number and $K=Q(𝜃)$ be an algebraic number field with $𝜃$ a root of an irreducible polynomial $x^{q^s}-ax-b$ having integer coefficients. In this paper, we provide some explicit conditions involving only $s,q,a,b$ for which $K$ is non-monogenic. As an application, in the special case of $a=0$ and $q=2$, we show that if $s\ge 2$ and $32$ divides $b-1$, then $K$ is not monogenic. We illustrate our results through examples.

Let ${\mathrm{\Lambda }}_{(0,0)}=\left(\right.\begin{array}{cc}\hfill A\hfill & \hfill {}_A{V}_B\hfill \\ \hfill {}_B{U}_A\hfill & \hfill B\hfill \end{array}{\left)\right.}_{(0,0)}$ be a Morita context ring with zero bimodule homomorphisms. We establish sufficient and necessary conditions for modules over ${\mathrm{\Lambda }}_{(0,0)}$ to be Gorenstein flat.

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a $\lambda$-TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative $\lambda$-TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the $\lambda$-TD shuffle product. We then show that the free commutative $\lambda$-TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative $\lambda$-TD algebra is connected and filtered, and thus is a left counital Hopf algebra.

This paper explores different notions of extreme states on operator systems and on operator spaces. We prove that boundary points of the generalized state space of an operator system in finite dimensions are precisely the restrictions of boundary representations of the generated $C^{\ast }$-algebra. We study weak TRO-extreme states on operator spaces in ternary rings of operators as a non self-adjoint version of $C^{\ast }$-extreme states on operator systems in $C^{\ast }$-algebras. Rectangular boundary point for a rectangular matrix convex set is introduced and its connection with boundary representations of the associated operator space is established in finite dimensional setting. Also, we show that the set of all rectangular boundary points is the minimal spanning set for a compact rectangular matrix convex set in finite dimensions.

The cyclic graph $\mathrm{\Gamma }(S)$ of a semigroup $S$ is the simple undirected graph whose vertex set is $S$ and two vertices $x,y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we determine the clique number of $\mathrm{\Gamma }(S)$ for an arbitrary semigroup $S$. Further, we obtain the independence number of $\mathrm{\Gamma }(S)$ if $S$ is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of $\mathrm{\Gamma }(S)$ if $S$ is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.

Let G be a finite group, we define the average codegree of the irreducible characters of G as $\mathrm{\text{acod}}(G)=\frac{1}{|\mathrm{\text{Irr}}(G)|}{\sum }_{\chi \in \mathrm{\text{Irr}}(G)}\mathrm{\text{cod}}(\chi )$, where $\mathrm{\text{cod}}(\chi )=\frac{|G:ker\chi |}{\chi (1)}$. We prove that if $G$ is non-solvable, then $\mathrm{\text{acod}}(G)\ge 68/5$, and the equality holds if and only if $G\cong A_5$. Also, we show that if $G$ is non-supersolvable, then $\mathrm{\text{acod}}(G)\ge 11/4$, and the equality holds if and only if $G\cong A_4$. In addition, we obtain that if $p$ is the smallest prime divisor of $\left|G\right|$, then $\mathrm{\text{acod}}(G)<p$ if and only if $G$ is an elementary abelian $p$-group.

In this paper, we introduce and study a generalization of coretractable modules. An $R$-module $M$ is called coretractable relative to an R-module N if ${\mathrm{\text{Hom}}}_R(M/X,N)\ne 0$, for any proper submodule $X$ of $M$. It is shown that a right $R$-module $M$ is coretractable relative to ${\oplus }_{i\in I}{N}_i$ if and only if $M$ is coretractable relative to $N_i$, for some $i\in I$. Also we show that if $R$ is a right quasi-duo ring, then there exists a right $R$-module that it is coretractable relative to $N$, for any nonzero right $R$-module $N$ if and only if $R$ is a left perfect local ring.

In this paper, we continue the study of Hurwitz series over a commutative unital ring that was begun in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. In particular, we introduce the notion of multivariate Hurwitz series. The underlying idea is that multivariate Hurwitz series are to Hurwitz series as studied in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] as formal power series in several indeterminates are to formal power series in only one indeterminate. The elementary aspects of the theory follow along the lines of [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. The treatment of substitution and divided powers introduces special problems not encountered in [W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] and requires special attention to subtle details. However, we are able to establish analogous results. With substitution and divided powers in place, we construct and study the analog to the so called inner transformations of [S. Bochner and W. T. Martin, Several Complex Variables (Princeton University Press, 1948)]. Finally, we are able to establish analogs to many of the fundamental results of single and multivariate calculus.

Let ${𝒢}$ be a 2-torsion free unital generalized matrix algebra with center $Z({𝒢})$, and $\mathrm{\Phi }$ be a linear mapping on ${𝒢}$ satisfying the condition

Action of a pomonoid on partially ordered sets ($S$-posets) has beautiful aspects in practical subjects such as automata theory, projection algebra and theoretical computer science which makes it always capture the interest of mathematicians. On the other hand, the study of different kinds of weakly injectivity (which category theory inherited from homological and commutative algebra) is an interesting subject for mathematicians. One of the important kinds of weakly injectivity is quasi-injectivity. In this paper, we study quasi-injectivity in the category of $S$-posets with respect to special kind of order embeddings, namely, down-closed embeddings (dc-quasi-injectivity).

Let $G$ be the direct product of a generalized extraspecial $\mathbb{Q}$-group and finitely many copies of ${\mathbb{Q}}_{\pi }$, where $\pi$ is a set of primes. It is proved that every polynomial automorphism of $G$ is an inner automorphism. As an application of this result, the structure of the group generated by all polynomial automorphisms of an extension of $\mathbb{Q}$ by a direct sum of finitely many copies of $\mathbb{Q}$ is determined.