Journal of Algebra and Related Topics
https://jart.guilan.ac.ir/
Journal of Algebra and Related Topicsendaily1Wed, 01 Jun 2022 00:00:00 +0430Wed, 01 Jun 2022 00:00:00 +0430S-small and S-essential submodules
https://jart.guilan.ac.ir/article_5549.html
This paper is concerned with S-comultiplication modules which are a generalization of comultiplication modules.In section 2, we introduce the S-small and S-essential submodules of a unitary $R$-module $M$ over a commutative ring $R$ with $1\neq 0$ such that S is a multiplicatively closed subset of $R$. We prove that if $M$ is a faithful S-strong comultiplication $R$-module and $N\ll ^{S}M$, then there exist an ideal $I\leq ^{S}_{e}R$ and an $t\in S$ such that $t(0 :_{M}I)\leq N\leq (0 :_{M}I)$. The converse is true if $S\subseteq {\rm U}(R)$ such that ${\rm U}(R)$ is the set of all units of $R$. Also, we prove that if $M$ is a torsion-free S-strong comultiplication module, then $N\leq ^{S}_{e} M$ if and only if there exist an ideal $I\ll ^{S}R$ and an $s\in S$ such that $s(0 :_{M} I)\leq N\leq (0 :_{M} I)$. In section 3, we introduce the concept of S-quasi-copure submodule $N$ of an $R$-module $M$ and investigate some results related to this class of submodules.Generalization of $n$-ideals
https://jart.guilan.ac.ir/article_5614.html
Let f:A&rarr;B be a ring homomorphism and let Jbe an ideal of B. We proved some results concerning n-ideals and(2; n)-ideals of A⋈^f J. Then we recall a proper ideal I of A as &radic;(&delta;(0))-ideal if ab ϵ I then b &isin; I or a &isin; &radic;(&delta;(0)) for every a; b &isin; A. We investigate some properties of &radic;(&delta;(0))-ideal with similar n-idealsand J-ideals.Mappings between the lattices of varieties of submodules
https://jart.guilan.ac.ir/article_5615.html
Let $R$ be a commutative ring with identity and $M$ be an $R$-module. It is shown that the usual lattice $\mathcal{V}(_{R}M)$ of varieties of submodules of $M$ is a distributive lattice. If $M$ is a semisimple $R$-module and the unary operation $^{\prime}$ on $\mathcal{V}(_{R}M)$ is defined by $(V(N))^{\prime}=V(\tilde{N})$, where $M=N\oplus \tilde{N}$, then the lattice $\mathcal{V}(_{R}M)$ with $^{\prime}$ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$, in particular considering when these mappings are lattice homomorphisms. It is shown that if $M$ is a faithful primeful $R$-module, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic lattices, and therefore $\mathcal{V}(_{R}M)$ and the lattice $\mathcal{R}(R)$ of radical ideals of $R$ are anti-isomorphic lattices. Moreover, if $R$ is a semisimple ring, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Boolean algebras, and therefore $\mathcal{V}(_{R}M)$ and $\mathcal{L}(R)$ are anti-isomorphic Boolean algebras.Generalized orthogonal graphs of characteristic a power of 2
https://jart.guilan.ac.ir/article_5616.html
Let $R$ be a finite local ring of characteristic a power of $2$ with the residue field $k$.In this paper, we define a generalized orthogonal graph on a module of rank at least $2$ over $R$. Then we study its graph properties via the same graph over $k$. The number of vertices and the valency of each vertex in this graph over $R$ are computed. We also prove that this graph is arc transitive and find its diameter. Moreover, the first subconstituent of this orthogonal graph is considered. We show that it is a generalized strongly regular graph.*-alpha-derivation on prime *-rings
https://jart.guilan.ac.ir/article_5617.html
Let $\Re$ be an associative ring with involution $*$. An additive map $\lambda\rightarrow \lambda^{*}$ of $\Re$ into itself is called an involution if the following conditions are satisfied $(i) (\lambda\mu)^{*}=\mu^{*}\lambda^{*}$, $(ii) (\lambda^{*})^{*}=\lambda ~~ \mbox{for all}~ \lambda,\mu\in \Re$. A ring equipped with an involution is called an $*$-ring or ring with involution. The aim of the present paper is to establish some results on $*$-$\alpha$-derivations in $*$-rings and investigate the commutativity of prime $*$-rings admitting $*$-$\alpha$-derivations on $\Re$ satisfying certain identities also prove that if $\Re$ admits a reverse $*$-$\alpha$-derivation $\delta$ of $\Re$, then $\alpha\in Z(\Re)$ and some related results have also been discussed.A new radical in free modules
https://jart.guilan.ac.ir/article_5618.html
An $R$-module $M$ is called torsion-free, if $rx=0$ for $r\in R$ and $x\in M$ implies that $r=0$ or $x=0$. In this paper, we introduce the notions semi torsion-free modules and quasi torsion-free modules. We show that a submodule $N$ of an $R$-module $M$ is a $P$-primary submodule if and only if $\dfrac{R}{P}$-module $\dfrac{M}{N}$ is semi torsion-free. Also we define a new radical in free modules and find some characterizations of it. We prove that for $P$-submodule $N$ of a free $R$-module $F$ which $\sqrt N \subsetneqq F$, we have for any $r \in R$ and $m \in F$, $rm \in N$ implies $r \in \sqrt P$ or $m \in \sqrt N$ if and only if $\dfrac{R}{P}$-module $\dfrac{F}{N}$ is quasi torsion free.On Dominions and Determination of Closed Varieties of Semigroups
https://jart.guilan.ac.ir/article_5302.html
&nbsp; It is known that all subvarieties of variety of all semigroups are not absolutely closed. So, it is a natural question to find out those subvarieties of variety of all semigroups that are closed in itself or close in larger subvarieties of variety of all semigroups. We have gone through this open problem and able to determine some closed varieties of semigroups defined by the identities $axy=yxax~[axy=xyxa]$ and $axy=yxxa$ by using Isbell's zigzag theorem as an essential tool. Further, we partially generalize a result of Isbell on semigroup dominions from the class of commutative semigroups to some generalized classes of commutative semigroups by showing that dominions of such semigroups belong to the same class.Jordan algebra bundles and Jordan Rings
https://jart.guilan.ac.ir/article_5619.html
In this paper, We define Jordan algebra bundles of finite type and we give one-one correspondence between Jordan algebra bundles of finite type and Jordan rings.Some results on domination in the generalized total graph of a commutative ring
https://jart.guilan.ac.ir/article_5620.html
Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R/H is a saturated multiplicatively closed subset of R. Anderson and Badawi [4] introduced the generalized total graph of R as an undirected simple graph GTH(R) with vertex set as R and any two distinct vertices x and y are adjacent if and only if x + y ϵ H. The main objective of this paper is to study the domination properties of the graph GTH(R). We determine the domination number of GTH(R) and its induced subgraphs GTH(H) and GTH(R/H). We establish a relationship betweenthe domination number of GTH(R) and the same of GTH(R/H). We also establish a relationship between diameter and domination number of GTH(R/H). In addition,we obtain the bondage number of GTH(R). Finally, a relationship between girth and bondage number of GTH(R/H) has been established.Integral closure of a filtration relative to a Noetherian module
https://jart.guilan.ac.ir/article_5621.html
&nbsp;Let &nbsp;$M$ be a Noetherian $R-$module.In this paper we will introduce the integral closure of a filtration ${\mathcal{F}}=\{I_{n}\}_{n\geq 0}$ &nbsp;relative to the Noetherian module &nbsp;$M$ and prove some related results.\\&nbsp;The integral closure of a filtration ${\mathcal{F}}=\{I_{n}\}_{n\geq 0}$ &nbsp;relative to $M$ &nbsp;is a filtration and it has an interesting relationship with &nbsp;the integral closure of the filtration&nbsp;${\widetilde{\mathcal{F}}}=\{\widetilde{I}_{n}\}_{n\geq 0}$, where $\widetilde{I}_{n}$ is the image of $I_n$ &nbsp;under the natural ring homomorphism &nbsp;$R\rightarrow R/(Ann_R(M))$ for every $n\geq 0$.On Special Amalgams and Closed Varieties of Posemigroups
https://jart.guilan.ac.ir/article_5622.html
In this paper we extend a result of Scheiblich by showing that variety of po-normal bands is closed. We also extend the well known results to posemigroups namely, that pogroups and inverse posemigroups have special amalgamation property in the category of all posemigroups and commutative posemigroups, respectively. Finally, we find some varieties of posemigroups which are closed if they are self convex.A note on generalized derivations on prime ideals
https://jart.guilan.ac.ir/article_5623.html
This paper investigates the structure of the quotient ring $\mathscr{R}/ \mathscr{P}$, where $\mathscr{R}$ is an arbitrary ring and $\mathscr{P}$ is a prime ideal of $\mathscr{R}$ . We show that the structure of this class of rings has a relationship with the behaviour of generalized derivations satisfying algebraic identities involving prime ideals.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.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. 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.
&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&nbsp;the concept of $\delta$-$n$-ideals which is an extension of $n$-ideals in&nbsp;commutative rings. We call a proper ideal $I$ of $R$ a $\delta$-$n$-ideal if&nbsp;whenever $a,b\in R$ with$\ ab\in I$ and $a\notin\sqrt{0}$, then $b\in&nbsp;\delta(I)$. 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&nbsp;to be a quasi $n$-ideal or equivalently, $I$ is quasi $n$-ideal if $\sqrt{I}$&nbsp;is an $n$-ideal. A number of characterizations and results with many&nbsp;supporting examples concerning this new class of ideals are given. In&nbsp;particular, we present some results regarding quasi $n$-ideals. Furthermore,&nbsp;we use $\delta$-$n$-ideals to characterize fields and UN-rings.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.Co-intersection graph of subacts of an act
https://jart.guilan.ac.ir/article_5647.html
&lrm;&nbsp;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 $G(A)$.&nbsp;