The Submodule-Based Zero-Divisor Graph with Respect to Some Homomorphism

Let M be an R-module and 0 not equal f is an element of M+ = Hom(M, R). The graph Gamma(f) (M) is a graph with vertices Z(f) (M) = {x is an element of M \ {0} | xf(y) = 0 or yf (x) = 0 for some non- zero y 2 M}, in which non- zero elements x and y are adjacent provided that xf (y) = 0 or yf(x) = 0, which introduced and studied in left perpendicular 3 right perpendicular. In this paper we associate an undirected submodule based graph Z(N)(f) (M) for each submodule N of M with vertices Z(N)(f) (M) = {x is an element of M \ N | xf(y) is an element of N or yf (x) is an element of N for some y is an element of M \ N}, in which non-zero elements x and y are adjacent provided that xf(y) is an element of N or yf(x) is an element of N. We observe that over a commutative ring R, Gamma(f)(N) (M) is connected and diam(Gamma(f)(N) (M)) <= 3. Also we get some results about clique number and connectivity number of Gamma(f)(N) (M)