https://jart.guilan.ac.ir/ Journal of Algebra and Related Topics en daily 1 Wed, 01 Apr 2026 00:00:00 +0330 Wed, 01 Apr 2026 00:00:00 +0330 https://jart.guilan.ac.ir/article_8183.html Let $\pounds$ be a bounded distributive lattice. Following the concept of $1$-absorbing prime ideal, we define $1$-absorbing prime filters of $\pounds$. A proper filter $F$ of $\pounds$ is called $1$-absorbing prime filter of $\pounds$ if whenever non-zero elements $a, b, c \in \pounds$ and $a \vee b \vee c \in F$, then either $a \vee b \in F$ or $c \in F$. We will make an intensive investigate the basic properties and possible structures of these filters. https://jart.guilan.ac.ir/article_8846.html The purpose of this document is to present the concept of bipolar fuzzy soft hypervector spaces and explore their fundamental characteristics‎. ‎To begin with‎, ‎a new operation and an external hyperoperation are introduced for bipolar fuzzy soft sets on the hypervector space $\mathcal{V}$‎, ‎which are connected to the operation and external hyperoperation of $\mathcal{V}$‎. ‎Then the notion of bipolar fuzzy soft hypervector space is defined‎, ‎supported by non-trivial examples‎, ‎and it is investigated whether the new bipolar fuzzy soft sets‎, ‎constructed by the mentioned operation and hyperoperation‎, ‎are bipolar fuzzy soft hypervector spaces‎. ‎Finally‎, ‎the behavior of bipolar fuzzy soft hypervector spaces under linear transformations is investigated. https://jart.guilan.ac.ir/article_8847.html In this paper, we introduce dual k-Leonardo quaternions which we call generalized dual Leonardo quaternion numbers. Some algebraic properties of these quaternions such as recurrence relation, generating function, Binet’s formula, generating function, Cassini identity, sum formulas will also be obtained. https://jart.guilan.ac.ir/article_8193.html ‎Let $\CA$ be an exact category and $\Cb_N(\CA)$ be the category of all $N$-complexes in $\CA$‎. ‎If $\mathbb{X}$ is a sufficiently nice class of objects in $\Cb_N(\CA)$‎, ‎then‎, ‎we give a characterization of elements in the right orthogonal $\mathbb{X}^\perp$ of $\mathbb{X}$ in $\Cb_N(\CA)$ with respect to the induced exact structure‎.‎ https://jart.guilan.ac.ir/article_8848.html This paper focuses on the classification of $7$-dimensional nilpotent $3$-Lie algebras. We employ a systematic approach by considering the structure of these algebras through the central ideals. Specifically, we divide the $7$-dimensional nilpotent $3$-Lie algebra by a $1$-dimensional central ideal, resulting in a $6$- dimensional nilpotent $3$-Lie algebra. Our findings reveal the relationships between $7$-dimensional structures and their $6$-dimensional counterparts, contributing to a deeper understanding of the properties and classifications of nilpotent $3$-Lie algebras. https://jart.guilan.ac.ir/article_8823.html ‎Let $R=\prod_{i\in I}R_{i}$ be the product of an infinite family of rings $\{R_{i}\}_{i\in I}$‎. ‎In this study‎, ‎we investigate the direct sum $\bigoplus_{i\in I}R_{i}$‎. ‎Special attention is paid to the relationship between the ideal $\bigoplus_{i\in I}R_{i}$ and the $\mathcal{F}_{r}$ Frechet filter in $I$‎, ‎also we show a new characterization of $\bigoplus_{i\in I}R_{i}$ by the $\mathcal{F}_{r}-\lim$‎. https://jart.guilan.ac.ir/article_9317.html ‎The current study elucidates the nature of right and left inverses of an element in the path algebra of a quiver‎. ‎A general characterisation‎ ‎of such elements has been established‎. ‎An explicit formula to calculate the‎ ‎inverse element has been formulated‎. ‎It is observed that the left and right‎ ‎inverses of an element in the non-commutative path algebraic structure coincides‎. ‎Furthermore‎, ‎it is noted that the Jacobson radical of any finite dimensional path algebra can be easily found using this characterisation‎. https://jart.guilan.ac.ir/article_8850.html The rings considered in this paper are commutative with identity and are nonzero. Let R be a ring. An ideal I of R is said to be nonnil if I is not contained in the nilradical of R. We say that R is a nonnil-zero-divisor ring if for any proper nonnil ideal I of R, the set of zero-divisors of the R-module R/I is a finite union of prime ideals of R. This paper aims to discuss some basic properties of nonnil-zero-divisor rings and to compare the ring-theoretic properties of zero-divisor rings with that of the ring-theoretic properties of nonnil-zero-divisor rings. https://jart.guilan.ac.ir/article_9431.html Let $\pounds$ be a $C$-lattice and $M$ be a lattice module over $\pounds$‎. ‎The join maximal element graph $\mathbb{G}(M)$ is a simple‎, ‎undirected graph with all proper non-zero elements of $M$ as vertices‎, ‎and two distinct vertices‎, ‎$N$ and $K$‎, ‎are adjacent if and only if $N\vee K\in Max(M)$‎, ‎where $Max(M)$ is the collection of all maximal elements of $M$‎. ‎In this paper‎, ‎some properties of the graph $\mathbb{G}(M)$ like diameter‎, ‎girth and clique number are investigated‎. ‎Also‎, ‎the interplay between the algebraic properties of $M$ and the properties of those graphs is studied‎. https://jart.guilan.ac.ir/article_8824.html ‎Sierpi\'{n}ski gasket graphs have many applications and are studied in diverse areas including fractal theory‎, ‎topology‎, ‎dynamic systems and chemistry‎. ‎In this paper we study and determine the girth of generalized Sierpi\'{n}ski gasket $S[G‎, ‎t]$ for an arbitrary simple graph $G$‎, ‎in terms of the girth of the base graph ‎$‎G‎$‎‎.‎Moreover‎, ‎we determine the planarity of $S[G‎, ‎t]$ for some famous families of graphs‎. https://jart.guilan.ac.ir/article_9287.html A Lie-Yamaguti algebra bundle is a type of algebra bundles with fibres being Lie-Yamaguti algebras, and appears naturally from geometric considerations in the work of M. Kikkawa. The aim of the present paper is to introduce the notion of generalized Reynolds operators, O-operators and Nijenhuis operators in the context of Lie-Yamaguti algebra bundle and find their applications. We also study abelian extensions of Lie-Yamaguti algebra bundles and investigate its relationship with its cohomology. https://jart.guilan.ac.ir/article_9531.html In this paper, we will study some properties of the integral closure of a filtration relative to a module in the Rees ring. https://jart.guilan.ac.ir/article_8328.html Consider the polynomial ring $S=\mathbb{K}[x_1,\ldots, x_n]$ over a field $\mathbb{K}$. For any equigenerated monomial ideal $I \subset S$ with the defining ideal $J$ of the fiber cone $\F(I)$ generated by quadratic binomials, we introduce a matrix. The key observation is that the set of binomial $2$-minors of this matrix serves as a generating set for $J$. This framework in particular provides a characterization of the fiber cone for Freiman ideals, as well as offering a specific characterization for the fiber cone of sortable ideals. https://jart.guilan.ac.ir/article_8822.html In this paper, we introduce a new class of ring called regular hereditary ring, which is a weak version of hereditaryring property. Any hereditary ring is naturally a regular hereditary ring, and in the domain context, these two forms coincide to become a Dedekind domain. We study the transfer of this notion to various context of commutative ring extensions such as localization, direct product, trivial ring extensions and pullbacks. Our results generate new families of examples of non-hereditary regular hereditary rings. https://jart.guilan.ac.ir/article_8852.html The Power graph of a group $G$ is a graph $\mathcal{P}(G)$ with vertex set $G$ and two vertices $x$ and $y$, $x \neq y$ are adjacent if there exists some integer $k$ such that $x=y^k$ or $y=x^k$. We denote the vertex connectivity of power graph $\mathcal{P}(G)$ by $\mathcal{K}(\mathcal{P}(G))$ and the algebraic connectivity of power graph $\mathcal{P}(G)$ by $\lambda_{n-1}(\mathcal{P}(G))$. This paper investigates the upper bound for the vertex connectivity and the algebraic connectivity of $\mathcal{P}(\mathbb{Z}_{n})$. Moreover, we discuss the equivalent conditions for $\mathcal{P}(\mathbb{Z}_{n})$ to be Laplacian integral. Further the conjecture for an upper bound of the algebraic connectivity of $\mathcal{P}(\mathbb{Z}_{n})$ is posed in this article. https://jart.guilan.ac.ir/article_8190.html ‎Let $\mathcal{S}$ be a commutative ring and $Z(\mathcal{S})$ be its zero-divisors set‎.‎The total graph of $\mathcal{S}$ with respect to multiplication‎, ‎denoted by $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}))$‎, ‎is an undirected graph with vertex set as the ring elements $\mathcal{S}$ and two distinct vertices $\alpha$ and $\beta$ are adjacent if and only if $\alpha\beta \in Z(\mathcal{S})$‎.‎The graph $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ is a subgraph of $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}))$ with vertex set $\mathcal{S}^*$ (set of nonzero elements of $\mathcal{S}$)‎.‎In this paper‎, ‎we characterize finite rings $\mathcal{S}$ for which $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ belongs to some well-known families of graphs‎. ‎Further‎, ‎we classify the finite rings $\mathcal{S}$ for which $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ is planar‎, ‎toroidal or double toroidal‎. ‎Finally‎, ‎we analyze the finite rings $\mathcal{S}$ for which the graph $T_{Z(\mathcal{S})}(\Gamma(\mathcal{S}^*))$ has crosscap at most two‎. https://jart.guilan.ac.ir/article_9524.html The paper introduces a novel bicomplex Fibonacci p quaternions, establishing algebraic properties, Honsberger identity, D’Ocagne’s identity, Cassini’s identity, Catalan’s identity for these quaternions. The study extends previous work on bicomplex Fibonacci quaternions [10] by incorporating bicomplex Fibonacci p quaternions. These new quaternions may have implications in applied mathematics, quantum mechanics, quantum physics, Lie groups, Kinematics and differential equations. https://jart.guilan.ac.ir/article_9526.html In this paper‎, ‎first we proved that all subvarieties of the variety of left (right) regular bands are closed in the variety of $n$-nilpotent extension of bands‎. ‎Secondly‎, ‎we proved the closedness of rectangular bands in the variety $\mathcal{V}=[ac=ab^nc]$ $(n\in \bf N)$‎, ‎of semigroups‎. ‎Further‎, ‎we have shown that all subvarieties of the variety of left (right) normal bands are closed in the variety $\mathcal{V}=[axy=a^py^qx^r]$ $(p,q,r\in \bf N)$‎, ‎of semigroups and lastly‎, ‎we proved that all subvarieties of the variety of left (right) quasinormal bands are closed in the variety $\mathcal{V}=[axy=a^px^qa^ry]$ $(p,q,r\in \bf N)$‎,‎ of semigroups. https://jart.guilan.ac.ir/article_9525.html Let $R$ be a ring. The unitary Cayley graph of a ring $R$, denoted by $\Gamma(R)$, is a graph with vertex set $R$ where two vertices $u,v\in R$ are adjacent if and only if $u-v$ is a unit of $R$. In this paper, we investigate the unitary Cayley graph of a finite ring, called a group ring, and examine its fundamental properties. We present the conditions for adjacency, the connectivity of the graph and its basic structure. Additionally, we provide the exact value of the degree of a vertex and the distance between any two vertices within the graph. https://jart.guilan.ac.ir/article_8851.html ‎Let ‎$‎(R, {\frak m})‎$ ‎be a‎ ‎Noetherian ‎local ‎ring ‎and ‎‎$‎M‎$ ‎be a ‎finitely generated ‎$‎R‎$‎-module such that ‎$‎‎{\rm Hom}_R(M,R) \cong \underset{i=1}{\overset{n}{\oplus}} C$ ‎for ‎some ‎positive ‎integer ‎‎$‎n‎$‎. We try to present new characterizations of Gorenstein rings via ‎$‎M‎$ ‎and ‎‎$‎C‎$‎. It is proved that if ‎$‎‎{\rm depth}\, R=0$ ‎and ‎‎$‎‎{\rm id}_R (M) < ‎\infty‎$ ‎then ‎‎$‎R‎$ ‎is ‎Gorenstein. Also, it is shown that‏ if ‎‎$‎M‎$ ‎is a‎ ‎Cohen-Macaulay ‎‎$‎R‎$‎-module with finite injective dimension, then ‎$‎R‎$ ‎is ‎Gorenstein. https://jart.guilan.ac.ir/article_8204.html In this paper we prove the following results. Every element of a ring R is a sum of twocommuting 6-potent elements if and only if R is isomorphic to R_1×R_2×R_3, where R_1is isomorphic to a subdirect product of Z_2's, R_2 is isomorphic to a subdirect productof Z_3's and R_3 is isomorphic to a subdirect product of Z_11's. Also if every element ofa ring R is sum of two 6-potent and one nilpotent all commute each other then R isisomorphic to R_1×R_2×R_3, where J(R_1) is nil and R_1/J(R_1) is a subdirect product ofrings isomorphic to either of the rings Z_2, F_4, M_2(F_2) and M_2(F_4), a^{81}-a is nilpotentfor every a in R2 , J(R_3) is nil and R_3/J(R_3) is a subdirect product of Z_11's. https://jart.guilan.ac.ir/article_8326.html Let $R$ be a commutative ring with non-zero identity. The comaximal graph is a graph with vertices allelements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $Rx +Ry = R$. Let $\Gamma_2(R)$ be the subgraph of the comaximal graph with vertex-set $W^{*}(R)$, where $W^{*}(R)$ is the set of all non-zero and non-unit elements of $R$. In this paper, we investigate when the graph $\Gamma_2(R)$ is a line graph. We completely present all commutative rings which their comaximal graphs are line graphs. Also, we study when the comaximal graph is the complement of a line graph. https://jart.guilan.ac.ir/article_8327.html ‎In this paper we describe the continued fraction expansions of certain infinite series over $\mathbb{F}_{q}(T)$‎, ‎where $\mathbb{F}_{q}$ is a finite field with $q$ elements‎. ‎As the first application‎, ‎we determine the continued fraction expansion of the sum of rational functions with exponential elements‎. ‎As the second application‎, ‎we exhibit the continued fraction expansions of many classes of transcendental series that have bounded partial quotients‎. https://jart.guilan.ac.ir/article_8849.html Let $G$ be a group with identity $e$. In this paper, we define and study the non-identity order divisor graph of $G$, where the vertex set is $G-\{e\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if either $O(x)|O(y)$ or $O(y)|O(x)$. We denote the order divisor graph of group $G$ by $\o(G)$. We study some basic properties of $\o(G)$ such as connectedness, completeness, bipartiteness and Eulerian property. The lower bound as well as the number of edges of $\o(G)$ are also calculated for some group $G$ and some characterizations for fundamental properties of $\o(G)$ have been obtained. Finally, we explore the relation between the order prime graph and the non-identity order divisor graph of some group $G$. https://jart.guilan.ac.ir/article_8853.html ‎A ring $R$ is called $\pi$-regular if‎, ‎for every $x\in R$‎, ‎there exists $y\in R$ such that $x^n=x^nyx^n$ for some positive integer $n$‎. ‎Here‎, ‎we shall give some characterizations of $\pi$-regular rings in which nilpotent elements lie in the center‎. ‎It is shown that these rings can be formulated in a way motivated by recent works of P‎. ‎Danchev‎, ‎leading to new insights into $\pi$-regular rings and providing partial answers to a question posed by him‎. ‎In the end‎, ‎we aim to classify this class of rings‎, ‎up to an isomorphism‎. https://jart.guilan.ac.ir/article_9284.html Let $R$ be a commutative semiring (ring) with identity $1 \neq 0$‎. ‎A vertex $a$ in a simple graph $G$ is said to be a Smarandache vertex (or S-vertex for short) provided that there exist three distinct vertices $x$‎, ‎$y$‎, ‎and $b$ (all different from $a$) in $G$ such‎ ‎that $x$---$a$‎, ‎$a$---$b$‎, ‎and $b$---$y$ are edges in $G$‎, ‎but there is no edge between $x$ and $y$‎. ‎In this interdisciplinary subject‎, ‎we investigate‎ ‎the interplay between the algebraic properties of the commutative semirings and their associated zero-divisor graphs‎, ‎denoted by $\Gamma(R)$‎, ‎using the notion of the S-vertices in connection with the nonexistence of S-vertices in $\Gamma(R)$‎. ‎We discuss when $\Gamma(R)$ is a complete bipartite graph‎ ‎together with some of its other graph-theoretic properties and their relation to the nonexistence of S-vertices of $\Gamma(R)$. https://jart.guilan.ac.ir/article_9285.html Let $\mathcal{G}_{\epsilon}$ (resp. $\mathcal{W}_{\epsilon}$) with $\epsilon=0$ or $\frac{1}{2}$ be the complete spectrum-generating superalgebra (resp. the centerless super Virasoro algebra). In this paper, the super-skewsymmetric super-biderivations on $\mathcal{G}_{\epsilon}$ and $\mathcal{W}_{\epsilon}$ are completely determined. In particular, we show that every super-skewsymmetric super-biderivation $\varphi$ (resp. $\phi$) of $\mathcal{G}_{\epsilon}$ (resp. $\mathcal{W}_{\epsilon}$) is inner. Based on the results of super-biderivations, we shall give the certain forms of all linear super-commuting maps on $\mathcal{G}_{\epsilon}$ and $\mathcal{W}_{\epsilon}$ https://jart.guilan.ac.ir/article_9286.html In this paper‎, ‎we introduce and investigate new notions of injectivity and essentiality for right $S$-acts‎, ‎defined relative to a multiplicatively closed subset $T$ of a monoid $S$‎. ‎We study the concepts of $T$-injective and $T_\cap$-injective $S$-acts‎, ‎along with $T$-essential and $T_\cap$-essential subacts‎. ‎We first establish foundational definitions and illustrate the differences between $T$-essential and $T_\cap$-essential subacts with examples‎. ‎Our study shows that $T$-injectivity does not necessarily imply the existence of a zero element in the $S$-act‎, ‎which contrasts with classical results on injective $S$-acts‎. ‎We proved that every $S$-act admits a $T_\cap$-injective hull‎, ‎satisfying a universal property analogous to classical injective envelopes‎. ‎We study closure properties of the classes of $T$-injective and $T_\cap$-injective $S$-acts under categorical constructions such as products‎, ‎retracts‎, ‎and direct limits‎. ‎Moreover‎, ‎we demonstrate that pushouts preserve $T_\cap$-essential extensions‎, ‎while pullbacks may not‎, ‎highlighting an asymmetry in categorical behavior‎. https://jart.guilan.ac.ir/article_9318.html We define hom-Jacobi algebras as an extension of hom-Poisson algebras and we give some examples. We describe the universal property of first-order hom-differential operators as well as the universal property of first-order hom-differential multi-operators. By using these universal properties, we prove the existence and uniqueness of a canonical purely hom-Jacobi form associated to purely hom-Jacobi algebra. https://jart.guilan.ac.ir/article_9319.html ‎Let $M$ be a nilpotent quotient of a free monoid‎. ‎satisfying the‎ ‎monoid ring $R[M]$ in the McCoy's theorem for any semiprime or right‎ ‎APP ring $R$ is proven‎. ‎Also‎, ‎it is shown that $R[M]$ is right McCoy‎ ‎for any reduced ring $R$‎.‎ https://jart.guilan.ac.ir/article_9320.html This paper investigates the additive and multiplicative properties of complemented and completely regular $\Gamma-$ semirings‎. ‎We prove many results on different structures of $\Gamma-$ semirings like anti-inverse‎, ‎quasi-separative‎, ‎distributive‎, ‎and partial order‎. ‎Boolean $\Gamma-$ semiring is demonstrated by applying the concept of a completely regular $\Gamma-$ semiring‎. ‎Finally‎, ‎by using the idea of simple and completely regular $\Gamma-$ semiring‎, ‎we define a relation $\leq$ on $R$ such that $x \leq y$ if and only if $x+y+1 = x\alpha y$ for all $x,y \in R‎, ‎\alpha \in \Gamma $ and prove that $R$ is a partially ordered $\Gamma-$ semiring. https://jart.guilan.ac.ir/article_9321.html Let $\ell$ be a non-zero integer, a set of $m$ distinct positive integers $\left\{a_{1},a_{2},\ldots,a_{m}\right\}$ is called a $D(\ell)$-Diophantine $m$-tuple, if $a_{i}a_{j}+\ell$ is a perfect square for any distinct $i,j\in\left\{1,2,...,m\right\}$. Let $P_n$ denote the $n^{th}$ Pell number, defined by the recurrence relation $P_{n+1}=2P_{n}+P_{n-1}$, with initial conditions $P_{0}=0$ and $P_{1}=1$. This paper investigates the extendibility of the pair $P_{2n+4}$, $4P_{2n+2}$ to a D(4)-Diophantine triple by another Pell number. https://jart.guilan.ac.ir/article_9322.html The major goal of the present paper is to introduce and study skew derivationson partially ordered sets (posets), but with new restrictions. Some related propertiesof such derivations are introduced and investigated. Moreover, several examples are givento illustrate that various restrictions forced within the assumptions over these results can’tbe ignored. https://jart.guilan.ac.ir/article_9323.html One approach to construct a model structure on $C_N(\mathcal{A})$, the category of $N$-complexes over an abelian category $\mathcal{A}$, is to start with a complete hereditary cotorsion pair $(\mathcal{F},\mathcal{C})$ in $\mathcal{A}$ and then introduce Hovey pairs on $C_N(\mathcal{A})$. There are three important pairs of cotorsion pairs in the literature. In this paper, we employ a different technique by considering $\mathcal{A}$ as a Grothendieck category to introduce these Hovey pairs. For these pairs of cotorsion pairs, we omit the hereditary conditions, the conditions of having enough $\mathcal{F}$-objects as well as the condition of being closed under direct limits for the class $\mathcal{F}$. So we can construct Hovey pairs on categories that do not necessarily have enough $\mathcal{F}$-objects or where the class of objects is not closed under direct limits such as the category of Cartesian modules over small categories and the category of quasi-coherent sheaves on a scheme $\mathbb{X}$. https://jart.guilan.ac.ir/article_9383.html In this paper, we have $R$ a commutative Noetherian ring, with nonzero identity, and $\mathfrak{a}$ an ideal of $R$. Here, we give some results of the theory of modules for local cohomology involving the edge ideal. We introduce the concept of $\mathfrak{a}$-edge minimax $R$-modules and also the concept of $\mathfrak{a}$-edge cominimax $R$-modules, together with the edge ideal of a simple and finite graph, with no isolated vertices. We put results involving these new concepts and present relationships that exist between them. https://jart.guilan.ac.ir/article_9432.html In a previous paper‎, ‎the author investigated the structure of a right (left) quasi-adequate semigroup with a particular normal medial idempotent‎. ‎The subject of the current paper is an order analogue‎. ‎Namely‎, ‎we provide a structure theorem for a naturally ordered quasi-adequate semigroup $(S,\leq)$ with a maximum idempotent $u$ in which $uSu$ is an adequate subsemigroup of $S$ with the property that the relations $\L$ and $\R$ are abundant‎. ‎We achieve this result as a combination of those in the one-sided case‎. https://jart.guilan.ac.ir/article_9434.html ‎We present some new bounds for signed domination numbers‎. ‎Let $G=(V‎, ‎E)$ be a simple and undirected graph‎. ‎For a function $ f‎ : ‎V \longrightarrow \lbrace‎ -‎1‎ , ‎1\rbrace‎, ‎$ the weight of is $ f $ defined by $ w(f) = \sum_{ v\in V} f(v)‎. ‎$ For a vertex $ v $ in $ V‎, ‎$ we define $ f [v] = \sum_{u\in N[v]} f(u)‎. ‎$ A signed domination function of $ G $ is a function $ f‎ : ‎V \longrightarrow \lbrace‎ -‎1‎ ,‎1\rbrace $ such that $ f[v] \geq 1 $ for all $ v \in V‎. ‎$ The signed domination number $ \gamma_{s}(G) $ of $ G $ is the minimum weight among all signed domination functions of $ G‎. ‎$ In this paper‎, ‎we study the signed domination problem of the general graph‎, ‎and obtain some bounds of the signed domination number of $ G‎. ‎$ We also establish upper and lower bounds of the signed domination number of subdivision construction $ S(G)‎. ‎$ https://jart.guilan.ac.ir/article_9435.html ‎Given a module $\mathcal{M}$ over a commutative ring $\mathcal{R}$‎, ‎we can construct a simple graph‎ ‎$EG\mathcal{(M)}$ with the vertex set $\mathcal{Z_R(M)} \setminus \mathcal{{\rm Ann}_R(M)}$‎. ‎Two distinct vertices $x‎, ‎y$ are connected whenever ${\rm Ann}_{\mathcal{M}}(xy)$ is an essential submodule‎ ‎of $\mathcal{M}$‎. ‎The present study provides a detailed analysis of planar zero-divisor and planar essential graphs‎, ‎especially when they possess a universal vertex‎. ‎It demonstrates that such graphs can be represented‎ ‎as join of some known graphs‎. ‎Additionally‎, ‎it examines that whether‎ ‎the zero-divisor and the essential graphs of $\mathbb{Z}_n$ are planar or not‎. ‎ https://jart.guilan.ac.ir/article_9457.html Let $\mathcal{L}$ be a complete lattice. The identity meet graph of elements of $\mathcal{L}$, denoted by $\mathbb{IMG} (\mathcal{L})$, is an undirected simple graph whose vertices are all nontrivial elements of $\mathcal{L}$ and two distinct elements $x$ and $y$ are adjacent if and only if $x \vee y = 1$ and $x \wedge y \neq 0$. The basic properties and possible structures of the graph $\mathbb{IMG}(\mathcal{L})$ are investigated. The connectedness, clique number, domination number, independence number, chromatic number of $\mathbb{IMG}(\mathcal{L})$ and their relations to algebraic properties of $\mathcal{L}$ are explored. https://jart.guilan.ac.ir/article_9458.html We show that commutative inverse posemigroups and finite monogenic posemigroups are saturated in the category of all posemigroups. Further, we show that the variety of pobands satisfying the identity $axya=ayxa$ is closed as well as saturated. Finally, we show that the convex finite monogenic posemigroups and inverse posemigroups are absolutely closed in the category of all commutative posemigroups. https://jart.guilan.ac.ir/article_9459.html ‎Let an $S$-module $J$ be the ideal of $S$‎. ‎An $S$-module ‎$‎X‎$‎ is called $J_{\delta_{ss}}$-supplemented provided that there is a direct summand W of X with $X=Y+W$‎, ‎$Y\cap W\leq Soc_{\delta}(W)$ and $Y\cap W \subseteq WJ$ for each submodule $Y$ of a right module‎. ‎In this article‎, ‎the important features of this notion is presented‎, ‎its comparison with $\oplus_{ss}$-supplemented and $\delta{ss}$-supplemented modules is given‎. https://jart.guilan.ac.ir/article_9522.html The main objective of this paper is to study the concept of 1-absorbing prime ideals in commutative semirings as a generalization of 1-absorbing prime ideals in commutative rings‎. ‎We examine several characterizations and examples of 1-absorbing prime ideals and also investigate the 1-absorbing Prime Avoidance Theorem for such ideals‎. ‎Moreover‎, ‎we study the concept of 1-absorbing primary ideals in commutative semirings and studying a method for constructing 1-absorbing primary ideals that are not necessarily primary ideals‎. ‎Finally‎, ‎we examine various characterizations and examples of 1-absorbing primary ideals in commutative semirings‎. https://jart.guilan.ac.ir/article_9523.html Bachmuth defined an IA-automorphism of a group G as an automorphism of G which induces the identity automorphism on G/G'. Ghumde and Ghate introduced S(G) and Ivar(G) subgroups. In this paper, we first introduce a new series on the IA-central subgroup and verify the relationships of the members of this series. Also, we give a new definition for S(G)-autonilpotency on this series. Then, we discuss some properties of these concepts with some theorem and their corollaries. We investigate the members of Ivar(G) fixing the center element-wise. At the end of this paper, we study the conditions in which Ivar(G) relate to Inner automorphisms in S(G)-autonilpotent groups. Among the paper's innovations and contributions, we can mention the works [7] and [5] and the generalization of S(G)-Autonilpotent groups [6]. https://jart.guilan.ac.ir/article_9532.html A subgroup $H$ of a finite group $G$ is said to be $S$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $HT$ is $S$-permutable in $G$ and ${H \cap T \leq H_{sG }}$‎. ‎$H$ is said to be $S$-semiembedded in $G$ if there is an $S$-permutable subgroup $T$ of $G$ such that $HT$ is $S$-permutable in $G$ and $H \cap T \leq H_{\overline{s}G}$; where $H_{\overline{s}G}$ is an $S$-semipermutable subgroups of $G$ contained in $H$‎. ‎In this paper‎, ‎we present some new characterization of supersolubility and nilpotency of finite groups under $S$-semiembedding property. https://jart.guilan.ac.ir/article_9533.html ‎Let $ S= \mathbb{F}_{q}[u]‎/ ‎\langle u^2\rangle$ = $ \mathbb{F}_{q}+u\mathbb{F}_{q}$ and $R=\mathbb{F}_{q}[u]‎/ ‎\langle u^3\rangle$ = $\mathbb{F}_{q}+u\mathbb{F}_{q}‎ + ‎u^{2}\mathbb{F}_{q}$ are two finite chain rings‎, ‎where $ u^{2}=0=u^{3} $ and $ q $ is a power of a prime number‎. ‎We construct a class of $ SR $-additive cyclic codes generated by pairs of polynomials‎, ‎where $ S $ is a $ R $-algebra and $ SR $-additive cyclic code is a $ R $-submodul of $ S^{\alpha} \times R^{\beta} $‎ . ‎Based on probabilistic arguments‎, ‎we study the asymptotic behaviour of the rates and relative minimum distances of a certain class of the codes‎. ‎We show that there exists an asymptotically good infinite seqence of $ SR $-additive cyclic codes with the relative minimum distance of the code is convergent to $ \delta $‎, ‎and the rat is convergent to $ \frac{2}{q+q^{2}} $ for $ 0 < \delta < \frac{1}{1+q} $‎. https://jart.guilan.ac.ir/article_9534.html ‎In this paper‎, ‎we extend the concept of primary ideals in commutative rings by introducing a new class‎, ‎namely $2$-nil primary ideals‎, ‎in a commutative semiring $S$ with $1 \neq 0$‎. ‎We further investigate the structure of ideals by considering related classes such as $2$-nil ideals‎, ‎$2$-absorbing ideals‎, ‎and quasi-primary ideals‎. ‎A comprehensive framework has been developed to emphasize the significance of $2$-nil primary ideals in semiring theory‎, ‎focusing on their properties and interrelations‎. ‎We provide arguments and illustrative examples that demonstrate how $2$-nil primary ideals are connected to several well-established classes of ideals‎, ‎including prime ideals‎, ‎primary ideals‎, ‎$n$-ideals‎, ‎$2$-absorbing ideals‎, ‎and quasi-primary ideals‎. ‎Finally‎, ‎Theorem \ref{6} together with Corollaries \ref{q} and \ref{r} illustrates the applications of these ideals and highlights their significance within the theory of semirings‎. https://jart.guilan.ac.ir/article_9535.html In this paper, we introduce affine-type pairs on locally compact hypergroups and study their ergodicity and weakly ergodicity. Among other results, we give some applicable sufficient conditions for that an affine-type pair to be strongly totally dissipative. Inspiring the main results and concepts of this paper, the ergodicity on hypergroups can be studied extensively. https://jart.guilan.ac.ir/article_9557.html In this paper‎, ‎for any subsets $D,C$ of a pseudo $BCI$-algebra $A$ the notion of generalized annihilator of $D$ with respect to $C$ and $(\ast,\diamond)$ (resp‎: ‎to $C$ and ($\diamond‎, ‎\ast))$‎, ‎denoted by $(C‎: ‎D)^{(\ast‎, ‎\diamond)}$ (resp‎: ‎$(C‎: ‎D)^{(\diamond‎, ‎\ast)}$) is introduced and its related properties are investigated‎. ‎Also‎, ‎a necessary and sufficient condition for $BCI$-algebra to be p-semisimple or pseudo $BCK$-algebra are given‎. ‎Moreover‎, ‎it is shown that the equation $(C‎: ‎D)^{(\ast,\diamond)} =(C‎: ‎D)^{(\diamond‎, ‎\ast)}$ hold for every p-semisimple $BCI$-algebra‎. ‎Finally‎, ‎it is proved that the set of all involutory ideal‎, ‎denoted by $S_{C}^{(\ast‎ , ‎\diamond)}(A)$ forms a distributive lattice‎. https://jart.guilan.ac.ir/article_9596.html Let $G$ be a finite simple group‎, ‎and $X$ be a non-trivial conjugacy class of $G.$ The $rank$ of $X$ in $G$‎, ‎denoted by $rank(G~{:}~X)$‎, ‎is defined to be the minimum number of elements of $X$ generating $G.$ In this paper‎, ‎we investigate the ranks of the non-trivial classes of the symplectic simple group $Sp(6,2).$ We use the structure constants method to determine these ranks‎. ‎The Groups‎, ‎Algorithms and Programming (GAP) \cite{GAP} and the Atlas of finite group representations \cite{Wil} were used in our computations‎. https://jart.guilan.ac.ir/article_9597.html A Cheban loop $(G‎, ‎\circ)$ is characterized by the identities $(z\circ yx)x= zx\circ xy$ and $x(xy\circ z)= yx\circ xz$ for all $x,y,z\in G$‎. ‎It was‎ ‎established that the left‎, ‎right‎, ‎and middle nuclei of a Cheban loop coincide‎, ‎and the nucleus of a Cheban loop is the set of elements $a$ whose middle inner mappings $T_a$ are automorphisms‎. ‎The generators of the inner mapping group of a Cheban were refined in terms of one of the generators of the total inner mapping group of a Cheban loop‎. ‎Necessary and sufficient conditions regarding the inner mapping group (associators) for a loop to be a Cheban loop were established‎. ‎It was shown that‎, ‎in a Cheban loop‎, ‎the mapping $a\mapsto T_a$ is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping‎. ‎Additionally‎, ‎a Cheban loop was proved to be a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group‎. ‎Furthermore‎, ‎a Cheban loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop)‎. ‎Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Cheban loop were derived‎, ‎and based on these‎, ‎the generators of the total inner mapping group of a Cheban loop were fine-tuned‎. ‎Finally‎, ‎it was shown that a Cheban loop is a totally automorphic loop (TA-loop) if and only if it is a commutative and flexible loop‎. ‎These results above were used to give a partial answer to a 2013 question and an apparent solution to the 2015 problem in the case of a Cheban loop‎.