The generalized total graph of modules respect to proper submodules over commutative rings.

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 [۲]. 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 [۲] and [۱۲] to a more general case.

Keywords:

Total graph , prime submodule , module