University of Guilan Journal of Algebra and Related Topics 2345-3931 2 1 2014 06 01 The generalized total graph of modules respect to proper submodules over commutative rings. 27 42 58 EN N. K. Tohidi Islamic Azad University F. Esmaeili Khalil Saraei University of Tehran S. A. Jalili Islamic Azad University Journal Article 2013 08 30 Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(\Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(\Gamma_{N}(M))$, denoted by $T(\Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $M$ as vertices, and for distinct $m,n\in M$, the vertices $m$ and $n$ are adjacent if and only if $m+n\in M(N,I)$, where $M(N,I)=\{m\in M : rm\in N+IM \ for \ some \ \ r\in R-I\}$. The main purpose of this paper is to extend the definitions and properties given in [2] and [12] to a more general case. https://jart.guilan.ac.ir/article_58_08720c0a97470138baf6f3d0ccee4594.pdf