Journal of Algebraic Structures and Their Applications
Volume:6 Issue: 2, Summer and Autumn 2019

Let $A$ be a commutative ring with nonzero identity, and $1leq n<infty$ be an integer, and $R=Atimes Atimescdotstimes A$ ($n$ times). The total dot product graph of $R$ is the (undirected) graph $TD(R)$ with vertices $R^*=Rsetminus {(0,0,dots,0)}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xcdot y=0in A$ (where $xcdot y$ denote the normal dot product of $x$ and $y$).  Let $Z(R)$ denote the set of all zero-divisors of $R$.  Then the zero-divisor dot product graph of $R$ is the induced subgraph $ZD(R)$ of $TD(R)$ with vertices  $Z(R)^*=Z(R)setminus {(0,0,dots,0)}$. It follows that if  $Gamma(A)$ is not  perfect, then  $ZD(R)$ (and hence $TD(R)$) is not  perfect.In this paper we investigate perfectness of the graphs $TD(R)$ and $ZD(R)$.

‎Let $G$ be a finite group‎. ‎We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $chi$ of $G$ such that $chi(g)=0$‎. ‎In this paper‎, ‎we classify groups whose set of vanishing elements is exactly a conjugacy class‎.

In this work, we introduce $H^*$-condition on the set of submodules of a module. Let $M$ be a module. We say $M$ satisfies $H^*$ provided that for every submodule $N$ of $M$, there is a direct summand$D$ of $M$ such that $(N+D)/N$ and $(N+D)/D$ are cosingular. We show that over a right perfect right $GV$-ring,a homomorphic image of a $H^*$ duo module satisfies $H^*$.

Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set ofcomplete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets:begin{eqnarray*}V(Gamma) &=& {(x,y) | x in V(X) & y inV(X_x) },\ E(Gamma) &=& {(x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}.end{eqnarray*}The $X-$join graph $Gamma$ is said to be reduced if  $x, y in V(X)$, $x ne y$ and $N_X(x) setminus { y} = N_X(y) setminus { x}$ imply that $(i)$ if $xy notin E(X)$ then the graphs $X_x$ or $X_y$ are non-empty; $(ii)$ if $xy in E(X)$ then $X_x$ or $X_y$ are not complete graphs. The aim of this paper is to explore how the graph theoretical properties of  $X-$join of graphs effect on its automorphism group. Among other results we compute the automorphism group of reduced complete-empty $X-$join of graphs.

‎Let $Gamma$ be a graph with adjacency eigenvalues $lambda_1leqlambda_2leqldotsleqlambda_n$‎. ‎Then the energy of‎ ‎$Gamma$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$‎. ‎Also‎ ‎the Estrada index of $Gamma$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as $EE(Gamma)=sum_{i=1}^ne^{lambda_i}$‎. ‎In this paper‎, ‎we compute the eigenvalues‎, ‎energy and Estrada index of Cayley graphs on generalized dihedral groups‎. ‎As an application‎, ‎we‎ ‎compute these items for honeycomb toroidal graphs and Cayley graphs on dihedral groups‎.

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we will introduce the notions of 2-absorbing $I$-prime and 2-absorbing $I$-second submodules of an $R$-module $M$ as a generalization of 2-absorbing and strongly 2-absorbing second submodules of $M$ and explore some basic properties of these classes of modules.

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-prime power hypothesis, a group of type($A$),($B$) or ($C$) and certain metacyclic $p-$groups and a minimal non-metacyclic $p-$group where $p$ is a prime number. We will show that the divisibility graph $D(G)$ for all of them has no triangles.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1}ldots r_{i-1}r_{i+1}ldots r_{n-1}min N$, for some $iinlbrace 1,ldots, n-1rbrace$ $(ngeqslant 3)$.In this work, $(n-1, n)$-$phi$-classical prime submodules are studied and some results are established.

Let $R$ be a commutative Noetherian ring. We prove that  over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover$varphi:C rightarrow M$ such that $C$ is finitely generated and the projective dimension of $Kervarphi$ is finite and $varphi$ is surjective.

Let $R$ be a commutative Noetherian ring with non-zero identity, $mathfrak{a}$ an ideal of $R$, $X$ an arbitrary $R$--module, $mathcal{F}$ a filtration of $operatorname{Spec}(R)$ which admits $X$, and $s, s', t, t'$ non-negative integers such that $s+ t= s'+ t'$. In this paper, we study the membership of $R$--modules $operatorname{H}^{s}_mathfrak{a}(operatorname{H}^{t- 1}(operatorname{C}_R(mathcal{F}, X)))$ and $operatorname{H}^{s'- 1}(operatorname{H}^{t'}_mathfrak{a}(operatorname{C}_R(mathcal{F}, X)))$ in Serre subcategories of the category of $R$--modules and find some sufficient conditions which ensure the existence of an isomorphism between them, where $operatorname{C}_R(mathcal{F},X)$ is the Cousin complex for $X$ with respect to $mathcal{F}$. As applications, we give some new facts and represent some older facts about the local cohomology modules and the Cousin complexes.
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The concept of fuzzy soft $Gamma$-hyperrings introduced by  J. Zhan et al.  as a generalization of the soft rings. In this paper, we prove the equivalence relation $mu^{ast}$ defined by J. Zhan et al. is a strongly soft $Gamma$-regular relation and hyperoperations defined on quotient fuzzy soft $Gamma$-hyperrings are just operations. Also, we define the equivalence relation $mu^{ast}_I$ as a generalization the relation $mu^{ast}$ and consider quotient fuzzy soft $Gamma$-hyperrings and isomorphism theorems   by thisregular relation.
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The solvable graph associated with a finite group $G$, denoted by ${Gamma}_{rm s}(G)$, is a simple graph whose vertices are the prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge if and only if there exists a solvable subgroup of $G$ whose order is divisible by $pq$. In this paper, we give acharacterization for projective special unitary groups $U_3(q)$ with some certain conditions by the solvable graph.
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