Journal of Algebra and Related Topics
https://jart.guilan.ac.ir/
Journal of Algebra and Related Topicsendaily1Thu, 01 Dec 2022 00:00:00 +0330Thu, 01 Dec 2022 00:00:00 +0330Co-intersection graph of subacts of an act
https://jart.guilan.ac.ir/article_5647.html
In this paper, we define the co-intersection graph $G(A)$ of an \(S\)-act \(A\) which is a graph whose vertices are non-trivial subacts of \(A\) and two &nbsp;distinct &nbsp;vertices \(B_1\) and \( B_2\) are adjacent if \(B_1 \cup B_2\neq A\). &nbsp;We investigate the relationship between the algebraic properties of an &nbsp;\(S\)-act \(A\) and the properties of the graph $G(A)$.&nbsp;Centers of centralizer nearrings determined by All endomorphisms of symmetric groups
https://jart.guilan.ac.ir/article_5644.html
For $n = 5, 6$ and $E = \End S_n$, the functions in the centralizer nearring $M_E(S_n) = \{f : S_n \to S_n \ |\ f(1) = (1) \ \hbox{and} \ f \circ s = s \circ f \ \hbox{for all}\ s \in E\}$ are determined. &nbsp;The centers of these two nearrings are also described. &nbsp;Results that can be used to determine the functions in $M_E(S_n)$ and their centers for $n \geq 7$ are also presented.&nbsp;d-n-ideals of commutative rings
https://jart.guilan.ac.ir/article_5645.html
Let $R$ be a commutative ring with non-zero identity, and $\delta :\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ is the set of all ideals of $R$. In this paper, we introduce the concept of $\delta$-$n$-ideals which is an extension of $n$-ideals in commutative rings. We call a proper ideal $I$ of $R$ a $\delta$-$n$-ideal ifwhenever $a,b\in R$ with$\ ab\in I$ and $a\notin\sqrt{0}$, then $b\in \delta(I)$. For example, an ideal expansion $\delta_{1}$ is defined by $\delta_{1}(I)=\sqrt{I}.$ In this case, a $\delta_{1}$-$n$-ideal $I$ is said to be a quasi $n$-ideal or equivalently, $I$ is quasi $n$-ideal if $\sqrt{I}$ is an $n$-ideal. A number of characterizations and results with manysupporting examples concerning this new class of ideals are given. In particular, we present some results regarding quasi $n$-ideals. Furthermore, we use $\delta$-$n$-ideals to characterize fields and UN-rings.On the laskerian properties of extension functors of local cohomology modules
https://jart.guilan.ac.ir/article_5987.html
Let $R$ be a commutative Noetherian ring with identity and $M$ be an $R$-module. In this paper, we are going to generalize some results on finiteness of extension functors of local cohomology modules to the category of generalized Laskerian modules. In particular,&nbsp; we study the Laskerianness of $\Ext^1_R(R/\fa, H^t_R(M)))$ and $\Ext^2_R(R/\fa, H^t_\fa(M))$ where $t$ is a non-negative integer.ε-orthogonality preserving pairs of mappings on Hilbert C*-modules
https://jart.guilan.ac.ir/article_5988.html
Let $ \mathcal{A} $ be a standard $ C^{*} $-algebra. In this paper, we will study the continuity of $ \varepsilon $-orthogonality preserving mappings between Hilbert $ \mathcal{A} $-modules. Moreover, we will show that a local mapping between Hilbert $ \mathcal A $-modules is &nbsp;$ \mathcal A $-linear. Furthermore, we will prove that for a pair of nonzero $ \mathcal A $-linear mappings $ T,S:E\longrightarrow F $, between Hilbert &nbsp;$ \mathcal A $-modules, satisfying &nbsp;$ \varepsilon $-orthogonality preserving property, there exists $ \gamma \in \mathbb{C} $,\begin{align*}&nbsp; &nbsp; \Vert \langle T(x), S(y) \rangle-\gamma \langle x, y \rangle\Vert \leq \varepsilon \Vert T\Vert \Vert S\Vert \Vert x\Vert \Vert y \Vert, &nbsp;\quad x, y \in E.\end{align*}Our results generalize the known ones in the context of Hilbert spaces.Three Bounds For Identifying Code Number
https://jart.guilan.ac.ir/article_6001.html
Let $G=(V, E)$ be a simple graph. A set $C$ of vertices $G$ is an identifying set of $G$ if for every two vertices $x$ and $y$ belong to $V$ the sets $N_{G}[x] \cap C$ and $N_{G}[y] \cap C$ are non-empty and different. Given a graph $G,$ the smallest size of an identifying set of $G$ is called the identifying code number of $G$ and is denoted by $\gamma^{ID}(G).$ Two vertices $x$ and $y$ are twins when $N_{G}[x]=N_{G}[y].$ Graphs with at least two twin vertices are not identifiable graphs. In this paper,&nbsp; we present three bounds for identifying code number.Classical and strongly classical $n$-absorbing second submodules
https://jart.guilan.ac.ir/article_6002.html
Let $R$ be a commutative ring with identity and $M$ be an $R$-module. The main purpose of this paper is to introduce and investigate the notion of classical and strongly classical $n$-absorbing second submodules as a dual notion of classical $n$-absorbing submodules. We obtain some basic properties of these classes of modules.Construction of symmetric pentadiagonal matrix from three interlacing spectrum
https://jart.guilan.ac.ir/article_6003.html
&lrm;In this paper&lrm;, &lrm;we introduce a new algorithm for constructing a&lrm; &lrm;symmetric pentadiagonal matrix by using three interlacing spectrum&lrm;, &lrm;say $(\lambda_i)_{i=1}^n$&lrm;, &lrm;$(\mu_i)_{i=1}^n$ and $(\nu_i)_{i=1}^n$&lrm; &lrm;such that&lrm;&lrm;\begin{eqnarray*}&lrm;&lrm;0&lt;\lambda_1&lt;\mu_1&lt;\lambda_2&lt;\mu_2&lt;...&lt;\lambda_n&lt;\mu_n,\\&lrm;&lrm;\mu_1&lt;\nu_1&lt;\mu_2&lt;\nu_2&lt;...&lt;\mu_n&lt;\nu_n&lrm;,&lrm;\end{eqnarray*}&lrm;&lrm;where $(\lambda_i)_{i=1}^n$ are the eigenvalues of pentadiagonal&lrm; &lrm;matrix $A$&lrm;, &lrm;$(\mu_i)_{i=1}^n$ are the eigenvalues of $A^*$ (the&lrm;&nbsp; &nbsp;&lrm;matrix $A^*$ differs from $A$ only in the $(1,1)$ entry) and&lrm; &lrm;$(\nu_i)_{i=1}^n$ are the eigenvalues of $A^{**}$ (the matrix&lrm; &lrm;$A^{**}$ differs from $A^*$ only in the $(2,2)$ entry)&lrm;. &lrm;From the&lrm;&lrm;interlacing spectrum&lrm;, &lrm;we find the first and second columns of&lrm; &lrm;eigenvectors&lrm;. &lrm;Sufficient conditions for the solvability of the problem&lrm; &lrm;are given&lrm;. &lrm;Then we construct the pentadiagonal matrix $A$ from these&lrm; &lrm;eigenvectors and given eigenvalues by using the block Lanczos algorithm&lrm;. &lrm;We&lrm; &lrm;also give an example to demonstrate the efficiency of the algorithm&lrm;.n-Super finitely copresented and n-weak projective modules
https://jart.guilan.ac.ir/article_6004.html
Let $R$ be a &nbsp;ring and $n$ a non-negative integer. In this paper, &nbsp;we first introduce the concept of $n$-super finitely copresented &nbsp;$R$-modules and via these modules, we give a concept of $n$-weak projective modules and investigate some characterizations of these modules over any arbitrary ring. For example, we obtain that ($\mathcal{WP}^{n}(R), \mathcal{WP}^{n}(R)^{\bot}$) is a perfect hereditary cotorsion theory and for any $N\in \mathcal{WP}^n(R)^{\bot}$, there exists an $n$-weak projective cover with the unique mapping property if and only if every $R$-module is &nbsp;$n$-weak projective.Remarks on the sum of element orders of non-group semigroups
https://jart.guilan.ac.ir/article_6005.html
TThe invariant $\psi (G)$, the {\it sum of element orders} of a finite group $G$ will be generalized and defined for the finite non-group semigroups in this paper. We give an appropriate definition for the order of elements of a semigroup. As well as in the groups we denote the sum of element orders of a non-group semigroup $S$, which may possess the zero element and$/$ or the identity element, by $\psi (S)$. The non-group monogenic semigroup will be denoted by $C_{n,r}$ where $2\leq r\leq n$. In characterizing the semigroups $C_{n,r}$ we give a suitable upper bound and a lower bound for $\psi (C_{n,r})$, and then investigate the sum of element orders of the semi-direct product and the wreath product of two semigroups of this type. A natural question concerning this invariant may be posed as "For a finite non-group semigroup $S$ and the group $G$ with the same presentation as the semigroup, is $\psi (S)$ equal to $\psi (G)$ approximately?" We answer this question in part by giving classes of non-group semigroups, involving an odd prime $p$ and satisfying $\lim_{p\rightarrow \infty} \frac{\psi (S)}{\psi (G)}=1$. As a result of this study, we attain the sum of element orders of a wide class of cyclic groups, as well.Semisimple lattices with respect to filter theory
https://jart.guilan.ac.ir/article_6006.html
Since the theory of filters plays an important role in the theory of lattices, in this paper, we will make an intensive study of the notions of semisimple lattices and the socle of lattices based on their filters. The bulk of this paper is devoted to stating and proving analogues to several well-known theorems in the theory of the rings. It is shown that, if $L$ is a semisimple distributive lattice, then $L$ is finite. Also, an application of the results of this paper is given. It is shown that if $R$ is a right distributive ring, then the lattice of right ideals of $R$ is semisimple iff &nbsp;$R$ is a semisimple ring.Hypergraph associated with Lie algebra of upper triangular matrices
https://jart.guilan.ac.ir/article_5643.html
For an associated combinatorial structure with Lie algebra $\mathbf{g}_n$ of upper triangular matrices, an allowable, forbidden, and the graphs that are not associated with $\mathbf{g}_n$ of any three vertices are determined. This work also introduces a neoteric association of hypergraph with Lie algebra of upper triangular matrix $\mathcal{G}_n$ for an element of Lie algebra $\mathbf{g}_n$. The properties of this structure are analyzed, characterized and have been presented as an algorithm for finite order.On the m-extension dual complex Fibonacci p-numbers
https://jart.guilan.ac.ir/article_5646.html
In this paper, we introduced $m$-extension dual complex Fibonacci $p$-numbers. We established the properties of $m$-extension dual complex Fibonacci $p$-numbers. They are connected to complex Fibonacci numbers, complex Fibonacci $p$-numbers, and dual complex Fibonacci $p$-numbers.Some properties of pair of n-isoclinism induction
https://jart.guilan.ac.ir/article_6246.html
Let $(G,M)$ be a pair of groups, in which $M$ is a normal subgroup of a group $G.$ We study some properties of $n$-isoclinism of pair of groups. In fact, we show that the subgroups and quotient groups of two $n$-isoclinism pair of groups are $m$-isoclinic for all $m\le n.$ Moreover, the properties of $\pi$-pair and supersolvable pair of groups which are invariant under $n$-isoclinism has be studied.Perfect 2-Colorings of Cn × Cm
https://jart.guilan.ac.ir/article_6306.html
In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the generalized prism graph Cn &times; C3, where n &ge; 3,. We also present some generalized results for Cn &times; Cm, where m, n &ge; 3.\L{}ukasiewicz fuzzy ideals in BCK-algebras and BCI-algebras
https://jart.guilan.ac.ir/article_6307.html
The notion of (closed) \L{}ukasiewicz fuzzy ideal is introduced, and several properties ae investigated.The relationship between \L{}ukasiewicz fuzzy subalgebra and \L{}ukasiewiczfuzzy ideal is discussed, and characterization of a \L{}ukasiewicz fuzzy ideal is considered. Conditions for a \L{}ukasiewicz fuzzy subalgebra to be a \L{}ukasiewicz fuzzy ideal are provided, andconditions for the $\in$-set, $q$-set and $O$-set to be ideals are explored.On a question concerning the Cohen's theorem
https://jart.guilan.ac.ir/article_6308.html
Let $R$ be a commutative ring with identity, and let $M$ be an $R$-module.&nbsp; The Cohen's theorem is the classic result that a ring is Noetherian if&nbsp;and only if its prime ideals are finitely generated.&nbsp;Parkash and Kour obtained a new version of Cohen's theorem for modules, which states that a finitely generated $R$-module $M$ is&nbsp;Noetherian if and only if for every prime ideal $p$ of $R$ with $Ann(M) \subseteq p$, there exists a finitely generated submodule $N$ of $M$ such that $pM \subseteq N \subseteq M(p)$,&nbsp;where $M(p) = \{x \in M | sx \in pM \,\,\textit{for some} \,\, s \in R \backslash p\}$. In this paper, we prove this result for some classes of modules.&nbsp;