Algebraic Structures and Their Applications
https://as.yazd.ac.ir/
Algebraic Structures and Their Applicationsendaily1Tue, 01 Aug 2023 00:00:00 +0430Tue, 01 Aug 2023 00:00:00 +0430On closedness of right(left) normal bands and left(right) quasinormal bands
https://as.yazd.ac.ir/article_2918.html
It is well known that all subvarieties of the variety of all semigroups are not absolutely closed. So, it is worth to find subvarieties of the variety of all semigroups that are closed in itself or closed in the containing varieties of semigroups. We have gone through this open problem and able to determine that the varieties of right~[left] normal bands and left~[right] quasinormal bands are closed in the varieties of semigroups defined by the identities $axy = xa^ny~[axy = ay^nx],~axy = x^nay~[axy = ayx^n]$ $(n&gt;1)$; and $axy=ax^nay$~$[axy=ayx^ny]$~ $(n&gt;1)$, $axy=a^nxa^ry$ $[axy=ay^rxy^n]$ $(n,r\in&nbsp;\mathbb{N})$, respectively.The minimum edge dominating energy of the Cayley graphs on some symmetric groups
https://as.yazd.ac.ir/article_3001.html
The minimum edge dominating energy of a graph $G$ is defined as the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of $G$. In this paper, for some finite symmetric groups $\Gamma$ and subset $S$ of $\Gamma$, the minimum edge dominating energy of the Cayley graph of the group $\Gamma$, denoted by $Cay(\Gamma, S)$, is investigated.Hopfcity and Jacobson small submodules
https://as.yazd.ac.ir/article_3002.html
The study of modules by properties of their endomorphisms has long been of interest. In this paper, we introduce the notion of jacobson weakly Hopfian modules. It is shown that over a ring $R$, every projective (free) $R$-module is jacobson weakly Hopfian if and only if $R$ has no nonzero semisimple projective $R$-module. Let $L$ be a module such that $L$ satisfies ascending chain conditions on jacobson-small submodules. Then it is shown that $L$ is jacobson weakly Hopfian. Some basic characterizations of projective jacobson weakly Hopfian modules are proved.Genus $g$ Groups of Diagonal Type
https://as.yazd.ac.ir/article_3003.html
A transitive subgroup $G\leq S_n$ is called a genus $g$ group if there exist non identity elements $x_1,...,x_r\in G$ satisfying $G=\langle x_1,x_2,...,x_r\rangle$, $\prod_{i=1}^r {x_i}=1$ and $\sum_{i=1}^r ind\, x_i=2(n+g-1)$. The Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ is the space of genus $g$ covers of the Riemann sphere $\mathbb{P}^1\mathbb{C}$ with $r$ branch points and the monodromy group $G$. Isomorphisms of such covers are in one to one correspondence with genus $g$ groups.In this article, we show that $G$ possesses genus one and two group if it is diagonal type and acts primitively on $\Omega$. Furthermore, we study the connectedness of the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ for genus 1 and 2.Regular divisors of a submodule
https://as.yazd.ac.ir/article_3004.html
In this article, we extend the concept of divisors to ideals of Noetherian rings, more generally, to submodules of finitely generated modules over Noetherian rings. For a submodule $N$ of a finitely generated module $M$ over a Noetherian ring, we say a submodule $K$ of $M$ is a regular divisor of $N$ in $M$ if $K$ occurs in a regular prime extension filtration of $M$ over $N$. We show that a submodule $N$ of $M$ has only a finite number of regular divisors in $M$. We also show that an ideal $\mathfrak b$ is a regular divisor of a non-zero ideal $\mathfrak a$ in a Dedekind domain $R$ if and only if $\mathfrak b$ contains $\mathfrak a$. We characterize regular divisors using some ordered sequences of prime ideals and study their various properties. Lastly, we formulate a method to compute the number of regular divisors of a submodule by solving a combinatorics problem.Sheffer stroke R$_{0}-$algebras
https://as.yazd.ac.ir/article_3006.html
The main objective of this study is to introduce Sheffer stroke R$_{0}-$algebra (for short, SR$_{0}-$ algebra). Then it is stated that the axiom system of a Sheffer stroke R$_{0}-$algebra is independent. It is indicated that every Sheffer stroke R$_{0}-$algebra is R$_{0}-$algebra but specific conditions are necessarily for the inverse. Afterward, various ideals of a Sheffer stroke R$_{0}-$algebra are defined, a congruence relation on a Sheffer stroke R$_{0}-$algebra is determined by the ideal and quotient Sheffer stroke R$_{0}-$algebra is built via this congruence relation. It is proved that quotient Sheffer stroke R$_{0}-$algebra constructed by a prime ideal of this algebra is totally ordered and the cardinality is less than or equals to 2. After all, important conclusions are obtained for totally ordered Sheffer stroke R$_{0}-$algebras by applying various properties of prime ideals.Results on generalized derivations in prime rings
https://as.yazd.ac.ir/article_3009.html
A prime ring ${S}$ with the centre ${Z}$ and generalised derivations that meet certain algebraic identities is considered. Let's assume that $\Psi$ and $\Phi$ are two generalised derivations associated with $\psi$ and $\phi$ on ${S},$ respectively. In this article, we examine the following identities: (i) $\Psi(a)b-a\Phi(b)\in {Z},$ (ii) $\Psi(a)b-b\Phi(a)\in {Z},$ (iii) $\Psi(a)a-b\Phi(b)\in {Z},$ (iv) $\Psi(a)a-a\Phi(b)\in {Z},$ (v) $\Psi(a)a-b\Phi(a)\in {Z},$ for every $a, b\in {J},$ where ${J}$ is a non-zero two sided ideal of ${S}.$ We also provide an example to show that the condition of primeness imposed in the hypotheses of our results is essential.Characterization of ${\rm Alt}(5) \times \mathbb{Z}_p$, where $p \in \{ 17, 23\}$, by their product element orders
https://as.yazd.ac.ir/article_3010.html
We denote the integer $ \prod_{g \in G} o(g) $ by $\psi^{\prime}(G)$ where $o(g)$ denotes the order of $g \in G$ and $G$ is a finite group. In [14], it was proved that some finite simple group can be uniquely determined by its product of element orders. In this paper, we characterize ${\rm Alt}(5) \times \mathbb{Z}_p$, where $p \in \{ 17, 23\}$, by their product of element orders.Modal operators on $L$-algebras
https://as.yazd.ac.ir/article_3016.html
The main goal of this paper is to introduce analogously modal operators on $L$-algebras and study their properties. To begin with, we introduce the notion of modal operators on $L$-algebras and investigate some important properties of this operator. In order for the kernel of modal operator to be ideal, we investigate what conditions are required. Relations between modal operator and endomorphism of $L$-algebras are investigated. Also, we define the concept of positive $L$-algebra and some characterizations of positive $L$-algebra are established. Finally, we introduce a map $k_{a}$ and show that $k_{a}$ is a modal operator and we prove that the set of all $k_{a}$ on a positive $L$-algebra makes a dual BCK-algebra.Characterization of zero-dimensional rings such that the clique number of their annihilating-ideal graphs is at most four
https://as.yazd.ac.ir/article_3020.html
The rings considered in this article are commutative with identity which are not integral domains. Let $R$ be a ring. An ideal $I$ of $R$ is said to be an annihilating ideal of $R$ if there exists $r\in R\backslash \{0\}$ such that $Ir = (0)$. Let $\mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $\mathbb{A}(R)^{*} = \mathbb{A}(R)\backslash \{(0)\}$. Recall that the annihilating-ideal graph of $R$, denoted by $\mathbb{AG}(R)$, is an undirected graph whose vertex set is $\mathbb{A}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent in this graph if and only if $IJ = (0)$. The aim of this article is to characterize zero-dimensional rings such that the clique number of their annihilating-ideal graphs is at most four.&nbsp;Commutative True-False ideals in BCI/BCK-algebras
https://as.yazd.ac.ir/article_3027.html
The notion of a (limited) commutative $T\&amp;F$-ideal in BCK-algebras and BCI-algebras is introduced, and their properties are investigated. A relationship between a $T\&amp;F$-ideal and a commutative $T\&amp;F$-ideal in BCK-algebras and BCI-algebras is established, and examples to show that any $T\&amp;F$-ideal may not be commutative are given. Proper conditions for a $T\&amp;F$-ideal to be commutative are provided. Using a commutative ideal of a BCK-algebra and a BCI-algebra, a commutative $T\&amp;F$-ideal is established. The closed $T\&amp;F$-ideal in a BCI-algebra is introduced, and a condition for a closed $T\&amp;F$-ideal to be commutative is discussed. Characterization of a commutative $T\&amp;F$-ideal in a BCI-algebra is considered.&nbsp;The effect of singularity on a type of supplemented modules
https://as.yazd.ac.ir/article_3044.html
Let $R$ be a ring, $M$ a right $R$-module, and $S = End_R(M)$ the ring of all $R$-Endomorphisms of $M.$ We say that $M$ is Endomorphism $\delta$-$H$-supplemented (briefly, $E$-$\delta$-$H$-supplemented) provided that for every $\phi\in S,$ there exists a direct summand $D$ of $M$ such that $M = Im\phi + X$ if and only if $M = D + X$ for every submodule $X$ of $M$ with $M/X$ singular. In this paper, we prove that a non-$\delta$-cosingular module $M$ is $E$-$\delta$-$H$-supplemented if and only if $M$ is dual Rickart. We also show that every direct summand of a weak duo $E$-$\delta$-$H$-supplemented module inherits the property.A class of almost uniserial rings
https://as.yazd.ac.ir/article_3039.html
An $R-$module $M$ is called almost uniserial if any two non-isomorphic submodules of $M$ are comparable. A ring $R$ is an almost left uniserial ring if $_R R$ is almost uniserial. In this paper, we introduce a class of artinian almost uniserial rings. Also we give a classification of almost uniserial modules over principal ideal domains.Local automorphisms of $n$-dimensional naturally graded quasi-filiform Leibniz algebra of type I
https://as.yazd.ac.ir/article_3040.html
The notions of a local automorphism for Lie algebras are defined as similar to the associative case. Every automorphism of a Lie algebra $\mathcal{L}$ is a local automorphism. For a given Lie algebra $\mathcal{L}$, the main problem concerning these notions is to prove that they automatically become an automorphism or to give examples of local automorphisms of $\mathcal{L}$, which are not automorphisms. In this paper, we study local automorphisms on quasi-filiform Leibniz algebras. It is proved that quasi-filiform Leibniz algebras of type I, as a rule, admit local automorphisms which are not automorphisms.On the genus of annihilator intersection graph of commutative rings
https://as.yazd.ac.ir/article_3067.html
Let $R$ be a commutative ring with unity and $A(R)$ be the set of annihilating-ideals of $R$. The annihilator intersection graph of $R$, represented by $AIG(R)$, is an undirected graph with $A(R)^*$ as the vertex set and $\mathfrak{M} \sim \mathfrak{N}$ is an edge of $AIG(R)$ if and only if $Ann(\mathfrak{M}\mathfrak{N}) \neq Ann(\mathfrak{M}) \cap Ann(\mathfrak{N})$, for distinct vertices $\mathfrak{M}$ and $\mathfrak{N}$ of $AIG(R)$. In this paper, we first defined finite commutative rings whose annihilator intersection graph is isomorphic to various well-known graphs, and then all finite commutative rings with a planar or toroidal annihilator intersection graph were characterized.Hybrid ideals on a lattice
https://as.yazd.ac.ir/article_3074.html
The fuzzy set is a fantastic tool for expressing hesitancy and dealing with uncertainty in real-world circumstances. Soft set theory has recently been developed to deal with practical problems. The soft and fuzzy sets were combined by Jun et al. to generate hybrid structures. The idea of hybrid ideals on a distributive lattice is discussed in this work. The relation between hybrid congruences and hybrid ideals on a distributive lattice is also examined. In addition, the product of hybrid ideals and its numerous results are discussed.On higher order $z$-ideals and $z^\circ$-ideals in commutative rings
https://as.yazd.ac.ir/article_3083.html
A ring $R$ is called radically $z$-covered (resp. radically $z^\circ$-covered) if every $\sqrt z$-ideal (resp. $\sqrt {z^\circ}$-ideal) in $R$ is a higher order $z$-ideal (resp. $z^\circ$-ideal). In this article we show with a counter-example that a ring may not be radically $z$-covered (resp. radically $z^\circ$-covered). Also a ring $R$ is called $z^\circ$-terminating if there is a positive integer $n$ such that for every $m\geq n$, each $z^{\circ m}$-ideal is a $z^{\circ n}$-ideal. We show with a counter-example that a ring may not be $z^\circ$-terminating. It is well known that whenever a ring homomorphism $\phi:R\to S$ is strong (meaning that it is surjective and for every minimal prime ideal $P$ of $R$, there is a minimal prime ideal $Q$ of $S$ such that $\phi^{-1}[Q] = P$), and if $R$ is a $z^\circ$-terminating ring or radically $z^\circ$-covered ring then so is $S$. We prove that a surjective ring homomorphism $\phi:R\to S$ is strong if and only if ${\rm ker}(\phi)\subseteq{\rm rad}(R)$.Some results on the strongly annihilating submodule graph of a module
https://as.yazd.ac.ir/article_3117.html
Let M be a module over a commutative ring R. We continue our study of strongly annihilating submodule graph SAG(M) introduced in [9]. In addition to providing the more properties of this graph, we introduce the subgraph SAG&lowast;(M) of SAG(M) and compare the properties of SAG&lowast;(M) with SAG(M) and AG(M) (the annihilating submodule graphof M introduced in [5])Right-left induced hyperlattices and the genetic code hyperlattices
https://as.yazd.ac.ir/article_3123.html
&lrm;&lrm;&lrm;In this paper first we introduce right(resp&lrm;. &lrm;left) induced hyperlattices and investigate some of their properties&lrm;. &lrm;Especially&lrm; &lrm;a characterization of the smallest strongly regular relation for the class of distributive right/left induced hyperlattice is investigated&lrm;. &lrm;Next we propose and study the generated hyperlattices from hyperlattices&lrm;. &lrm;Finally&lrm;, &lrm;the right induced hyperlattices of two Boolean lattices of four DNA bases and physico-chemical properties of amino acids of four DNA bases are investigated&lrm;.The duals of annihilator conditions for modulesâ€Ž
https://as.yazd.ac.ir/article_3139.html
Let $R$ be a commutative ring with identity and let $M$ be an $R$-module&lrm;. &lrm;The purpose of this paper is to introduce and investigate the submodules of an $R$-module $M$ which satisfy the dual of Property $\mathcal{A}$&lrm;, &lrm;the dual of strong Property $\mathcal{A}$&lrm;, &lrm;and the dual of proper strong Property $\mathcal{A}$&lrm;. &lrm;Moreover&lrm;, &lrm;a submodule $N$ of $M$ which satisfy Property $\mathcal{S_J(N)}$ and Property $\mathcal{I^M_J(N)}$ will be introduced and investigated&lrm;.Modular group algebra with upper Lie Nilpotency index $11p-9$
https://as.yazd.ac.ir/article_3140.html
Let $KG$ be the modular group algebra of a group $G$ over a field $K$ of characteristic $p&gt;0$. Recently, we have seen the classification of group algebras $KG$ with upper Lie nilpotency index $t^{L}(KG)$ up to $10p-8$. In this paper, our aim is to classify the modular group algebra $KG$ with upper Lie nilpotency index $11p-9$, for $G'= \gamma_{2}(G)$ as an abelian group.A study on constacyclic codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$
https://as.yazd.ac.ir/article_3145.html
This paper studies $\lambda$-constacyclic codes and skew $\lambda$-constacyclic codes over the finite commutative non-chain ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=0$ for $\lambda= (1+2u+2u^2)$ and $(3+2u+2u^2)$. We introduce distinct Gray maps and show that the Gray images of $\lambda$-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over $\mathbb{Z}_4$. It is also shown that the Gray images of skew $\lambda$-constacyclic codes are quasi-cyclic codes of length $2n$ and index 2 over $\mathbb{Z}_4$. Moreover, the structure of $\lambda$-constacyclic codes of odd length $n$ over the ring $R$ is determined and give some suitable examples.Semihypergroups that every hyperproduct only contains some of the factors
https://as.yazd.ac.ir/article_3150.html
Breakable semihypergroups, defined by a simple property: every non-empty subset of them is a subsemihypergroup. In this paper, we introduce a class of semihypergroups, in which every hyperproduct of $n$ elements is equal to a subset of the factors, called $\pi_n$-semihypergroups. Then, we prove that every semihypergroup of type $\pi_{2k}$, ($k\geq 2$) is breakable and every semihypergroup of type $\pi_{2k+1}$ is of type $\pi_3$. Furthermore, we obtain a decomposition of a semihypergroup of type $\pi_n$ into the cyclic group of order 2 and a breakable semihypergroup. Finally, we give a characterization of semi-symmetric semihypergroups of type $\pi_n$.On the Ree groups $^2{}G_2(q)$ characterized by a size of a conjugacy class
https://as.yazd.ac.ir/article_3151.html
One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group $^2{}G_2(q)$, where $q\pm\sqrt{3q}+1$ is a prime number can be uniquely determined by its order and one conjugacy class size.Non-commutative hypergroupoid obtained from simple graphs
https://as.yazd.ac.ir/article_3163.html
The purpose of this paper is the study of non-weak commutative hypergroups associated with hypergraphs. In this regards, we construct a hyperoperation on the set of vertices of hypergraph and obtain some results and characterizations of them. Moreover, according to this hyperoperation, we investigate conditions under which the hypergroupoid is a join space hypergroup. Finally, we present an application to marketing social network.
&nbsp;Edge geodetic sequence in graphs
https://as.yazd.ac.ir/article_3181.html
In this paper, we introduced the concept of edge geodetic sequences in graph and its generating function. Some general properties satisfied by this concept are studied. It is shown that for every generating function$$ G(x)=\sum_{i=1}^{\infty} {a}^{i-1}{x^{i-1}} \quad a\in N-\left\lbrace 1\right\rbrace,$$there exists a recurrence graph $G$ with edge geodetic decomposition $\pi=\{G_{1},G_{2},\ldots ,G_{n}\ldots\}$.