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 ...

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 ...

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 $b|a$. ...

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 ...

‎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 ...

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<br />$varphi:C rightarrow ...

Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set of<br />complete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets:<br />begin{eqnarray*}<br />V(Gamma) &=& {(x,y) ...

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 $s, s', t, t'$ ...

Let $A$ be a commutative ring with nonzero identity, and $1leq n<infty$ be an integer, and <br />$R=Atimes Atimescdotstimes A$ ($n$ times). The total dot product graph of $R$ is the (undirected) graph $TD(R)$ with vertices ...

‎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 $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$‎. ...