Journal of Algebra and Related Topics
Volume:10 Issue: 2, Autumn 2022

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  distinct  vertices \(B_1\) and \( B_2\) are adjacent if \(B_1 \cup B_2\neq A\).  We investigate the relationship between the algebraic properties of an  \(S\)-act \(A\) and the properties of the graph $G(A)$.

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.  The centers of these two nearrings are also described.  Results that can be used to determine the functions in $M_E(S_n)$ and their centers for $n \geq 7$ are also presented.

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.

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,  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.

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  $ \mathcal A $-linear. Furthermore, we will prove that for a pair of nonzero $ \mathcal A $-linear mappings $ T,S:E\longrightarrow F $, between Hilbert  $ \mathcal A $-modules, satisfying  $ \varepsilon $-orthogonality preserving property, there exists $ \gamma \in \mathbb{C} $,\begin{align*}    \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,  \quad x, y \in E.\end{align*}Our results generalize the known ones in the context of Hilbert spaces.

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,  we present three bounds for identifying code number.

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.

‎In this paper‎, ‎we introduce a new algorithm for constructing a‎ ‎symmetric pentadiagonal matrix by using three interlacing spectrum‎, ‎say $(\lambda_i)_{i=1}^n$‎, ‎$(\mu_i)_{i=1}^n$ and $(\nu_i)_{i=1}^n$‎ ‎such that‎‎\begin{eqnarray*}‎‎0<\lambda_1<\mu_1<\lambda_2<\mu_2<...<\lambda_n<\mu_n,\\‎‎\mu_1<\nu_1<\mu_2<\nu_2<...<\mu_n<\nu_n‎,‎\end{eqnarray*}‎‎where $(\lambda_i)_{i=1}^n$ are the eigenvalues of pentadiagonal‎ ‎matrix $A$‎, ‎$(\mu_i)_{i=1}^n$ are the eigenvalues of $A^*$ (the‎   ‎matrix $A^*$ differs from $A$ only in the $(1,1)$ entry) and‎ ‎$(\nu_i)_{i=1}^n$ are the eigenvalues of $A^{**}$ (the matrix‎ ‎$A^{**}$ differs from $A^*$ only in the $(2,2)$ entry)‎. ‎From the‎‎interlacing spectrum‎, ‎we find the first and second columns of‎ ‎eigenvectors‎. ‎Sufficient conditions for the solvability of the problem‎ ‎are given‎. ‎Then we construct the pentadiagonal matrix $A$ from these‎ ‎eigenvectors and given eigenvalues by using the block Lanczos algorithm‎. ‎We‎ ‎also give an example to demonstrate the efficiency of the algorithm‎.

Let $R$ be a  ring and $n$ a non-negative integer. In this paper,  we first introduce the concept of $n$-super finitely copresented  $R$-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  $n$-weak projective.

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  $R$ is a semisimple ring.

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.