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