MMN-1569
An extension of total graph over a module
Ahmad Abassi; A. Ramin;Abstract
Let $R$ be a commutative ring with nonzero identity and $U(R)$ its multiplicative group of units. Let $M$ be an R-module where the collection of prime submodules is non-empty and let $N_{\Lambda}$ be an arbitrary union of prime submodules. Also, suppose that $c\in U(R)$ such that $c^{-1}=c$. We define the extended total graph of $M$ as a simple graph $T\Gamma_{c}(M,N_{\Lambda})$ with vertex set $M$, and two distinct elements $x,y\in M$ are adjacent if and only if $x+cy\in N_{\Lambda}$. This graph provides an extension of the total graph of a commutative ring and total graph of a module. In this paper, we will study some graph theoretic results of $T\Gamma_{c}(M,N_{\Lambda})$. We also show that if $M\neq M(L)$ for some proper submodule $L$ of $M$, then $M(L)$ is a union of prime submodules of $M$ where $M(L)=\{m\in M\vert rm\in L$ for some $r\in R\setminus(L:_{R}M)\}$.
Vol. 18 (2017), No. 1, pp. 17-29