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(ΓN(M)), have been introduced and studied in [2]. In this paper, A generalization of the total graph T(ΓN(M)), denoted by T(ΓN,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∈M, the vertices m and n are adjacent if and only if m∈M(N,I), where M(N,I)={m∈M:rm∈N for some r∈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.