Algebraic Structures and Their Applications
http://as.yazd.ac.ir/
Algebraic Structures and Their Applicationsendaily1Wed, 01 Feb 2023 00:00:00 +0330Wed, 01 Feb 2023 00:00:00 +0330The Noetherian dimension of modules versus their $\alpha$-small shortness
http://as.yazd.ac.ir/article_2699.html
In this article, we first consider concept of small Noetherian dimension of a module, which is dual to the small krull dimension, denoted by $sn{\rm -dim}\, A$, and defined to be the codeviation of the poset of the small submodules of $A$. We prove that if an $R$-module $A$ with finite hollow dimension, has small Noetherian dimension, then $A$ has Noetherian dimension and $ sn{\rm -dim}\, A\leq n{\rm -dim}\, A\leq sn{\rm -dim}\, A+1$. Last we introduce the concept of $\alpha$-small short modules, i.e., for each small submodule $S$ of $A$, either $n{\rm -dim}\, S \leqslant \alpha$ or $sn{\rm -dim}\,\frac{A}{S}\leqslant\alpha$ and $\alpha$ is the least ordinal number with this property and by using this concept, we extend some of the basic results of short modules to $\alpha$-small short modules. In particular, we prove that if $A$ is an $\alpha$-small short module, then it has small Noetherian dimension and $sn{\rm -dim}\, A=\alpha$ or $sn{\rm -dim}\, A=\alpha+1$. Consequently, we show that if $A$ is an $\alpha$-small short module with finite hollow dimension, then $\alpha\leq n{\rm -dim}\, A\leq\alpha+2$.On GE-ideals of bordered GE-algebras
http://as.yazd.ac.ir/article_2700.html
In this paper, the properties of GE-ideals of transitive bordered GE-algebra are studied and characterizations of GE-ideals are given. We have observed that the set of all GE-ideals of a transitive bordered GE-algebra forms a complete lattice. The notion of bordered GE-morphism is introduced and established fundamental bordered GE-morphism theorem. A congruence relation on a bordered GE-algebra with respect to GE-ideal is introduced and some bordered GE-morphism theorems are derived.Laplacian spectral characterization of setosa graphs
http://as.yazd.ac.ir/article_2703.html
A setosa graph $SG(e,f,g,h,d;b_1,b_2,\ldots,b_s)$ is a graph consisting of five cycles and $s(\geq 1)$ paths $P_{b_1+1}, P_{b_2+1},\ldots,P_{b_s+1}$ intersecting in a single vertex that all meet in one vertex, where $b_i\geq1$ (for $i=1,\ldots,s$) and $e,f,g,h,d\geq 3$ denote the length of the cycles $C_e$, $C_f$, $C_g$, $C_h$ and $C_d$, respectively. Two graphs $G$ and $H$ are $L$-cospectral if they have the same Laplacian spectrum. A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS, for short) if every graph with the same Laplacian spectrum is isomorphic to $G$. In this paper we prove that if $H$ is a $L$-cospectral graph with a setosa graph $G$, then $H$ is also a setosa graph and the degree sequence of $G$ and $H$ are the same. We conjecture that all setosa graphs are DLS.Sums of units in Baer and exchange rings
http://as.yazd.ac.ir/article_2721.html
In this paper, we prove that every element in an exchange ring $R$ with artinian primitive factors is $n$-tuplet-good iff $1_R$ is $n$-tuplet-good. Also, we show that for such rings the full matrix ring $M_n(R)$ (for $n\geq 2$) is $n$-tuplet-good. In [7], Fisher and Snider proved that every element of a strongly $\pi$-regular ring $R$ with $\frac{1}{2}\in R$ is 2-good. We prove the same result under new condition and show that these rings are twin-good. We also consider the conditions under which endomorphism ring of a finitely generated projective module $M$ over unit regular ring $L$ is 2-tuplet-good. The main result of [14] states that regular self-injective rings are $n$-tuplet-good if such rings has no factor ring isomorphic to a field $D$ with $|D|&lt;n+2$. We generalized this result to regular Baer rings proving that every regular Baer ring $R$ that has no factor ring isomorphic to a field of order less than $n+2$, is $n$-tuplet-good.On $\sim_{n}$ notion of conjugacy in semigroups
http://as.yazd.ac.ir/article_2724.html
In this paper, we study the $\sim_{n}$ notion of conjugacy in semigroups. After proving some basic results, we characterize this notion in subsemigroups of $\mathcal{P}(T)$ (partial transformation semigroup) and $\mathcal{T}(T)$ (transformation semigroup) through digraphs and their restrictive partial homomorphisms.On the power graphs of finite groups and Hamilton cycle
http://as.yazd.ac.ir/article_2748.html
The power graph $\mathcal{P}(G)$ of a finite group $G$ is a graph whose vertex set is the group $G$ and distinct elements $x,y\in G$ are adjacent if one is a power of the other, that is, $x$ and $y$ are adjacent if $x\in \langle y\rangle$ or $y \in \langle x\rangle$. In this paper, we study existence of the Hamilton cycle in the power graph of some finite nilpotent groups $G$ with a cyclic subgroup as direct factor when $G$ is written as direct product Sylow $p$-subgroups. For this purpose we use of cartesian product a spanning tree and a cycle. Finally, we determined values of $n$ such that $\mathcal{P}(U_n)$ is Hamiltonian, where $U_n$ is a group consist of all positive integers less than $n$ and relatively prime to $n$ under multiplication modulo $n$.&nbsp;Some results of $\alpha$-coset groups
http://as.yazd.ac.ir/article_2806.html
We call $G$ to be an $\alpha$-coset group, if it contains a proper $\alpha$-invariant normal subgroup $N$ such that $Nx^\alpha =\{x^g~|~ g\in G\}$, for some automorphism $\alpha$ of $G$ and any $x \in G\setminus N$. Clearly, if $\alpha$ is identity automorphism of $G$, one obtains the notion of con-cos groups, which was first introduced by Muktibodh in 2006.In the present article, we discuss some properties of the new notion. Also, we introduce the concept of $\alpha$-Camina groups and give its connection with the groups of property $\mathcal P$, where $\mathcal P$ is the class of all finite groups such that their $\alpha$-centres are the same as $\alpha$-commutator subgroups of order $p$.Some results on uniform MV-algebras
http://as.yazd.ac.ir/article_2818.html
In this paper, we study Tychonoff spaces and uniformities on MV-algebras. In particular, we find some conditions under which an MV-algebra can be made into a Tychonoff space. Also, we find uniformities that make an MV-algebra into a uniform MV-algebra. Next, we discuss some algebraic and topological properties of uniform MV-algebras. More precisely, we study the existence of closed ideals and closed filters, and examine the way these are related to uniform MV-algebras. Furthermore, we obtain some results on the uniform continuity of the operations and its consequences.&nbsp;Minimal prime filters of commutative $BE$-algebras
http://as.yazd.ac.ir/article_2840.html
In this paper we introduced the concept of minimal prime filters in commutative $BE$-algebras. A characterization theorem for minimal prime filters of $BE$-algebras is derived. Some properties of minimal prime filters of a commutative $BE$-algebras are derived with the help of congruences. A necessary and sufficient is derived for a pair of minimal prime filters to become co-maximal.On the basic properties of the compressed annihilator graph of $\mathbb{Z}_n$
http://as.yazd.ac.ir/article_2843.html
For a commutative ring $R$, the compressed annihilator graph $AG_E(R)$ is defined by, taking the equivalence classes of zero divisors of $R$ as the vertex set and two distinct vertices $[a]$ and $[b]$ are adjacent if and only if $ ann(a) \cup ann (b) \subset ann(ab)$. In this paper, we discuss some of the basic properties such as degree of the vertices, Eulerian, regularity, domination number and planarity of $AG_E(\mathbb{Z}_n)$, where $\mathbb{Z}_n$ is the ring of integer modulo $n$.Planar, outerplanar and ring graph of the intersection graph
http://as.yazd.ac.ir/article_2914.html
Let $R$ be a commutative ring and $M$ be an $R$-module. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$ is a graph with the vertex set $I(R)^*$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IM\cap JM\neq 0$. In this paper, we study $G_{R/J}(R/I)$, where $I$ and $J$ are ideals of $R$ and $I\subseteq J$. We characterize all ideals $I$ and $J$ for which $G_{R/J}(R/I)$ is planar, outerplanar or ring graph.A length for Artinian modules
http://as.yazd.ac.ir/article_2917.html
In this paper we shall introduce a theory of length for Artinian modules over an arbitrary ring (with identity), assigning to each Artinian module $A$ an ordinal number $len(A)$ which will briefly be called the length of $A$. We also demonstrate for some familiar properties of left Artinian ring be proved efficiently using length and arithmetic properties of ordinal numbers.On closedness of right(left) normal bands and left(right) quasinormal bands
http://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 $axy=ax^nay$~$[axy=ayx^ny]$~ $(n&gt;1)$, $axy=a^nxa^ry$ $[axy=ay^rxy^n]$ $(n,r\in {\bf N})$, respectively.The minimum edge dominating energy of the Cayley graphs on some symmetric groups
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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
http://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;A class of almost uniserial rings
http://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
http://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.The effect of singularity on a type of supplemented modules
http://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.On the genus of annihilator intersection graph of commutative rings
http://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
http://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
http://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 ${\rm ker}(\phi)\subseteq{\rm rad}(R)$.