Abstract. Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T (Γ I (R)). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ S(I). The total graph of a commutative ring, that denoted by T (Γ(R)), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y ∈ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, T (Γ I (R)) = T (Γ(R)); this is an important result on the definition.
Let R be a commutative ring with nonzero identity and M be an R-module. Quasi-prime submodules of M and the developed Zariski topology on qSpec(M ) are introduced. We also, investigate the relationship between the algebraic properties of M and the topological properties of qSpec(M ). Modules whose developed Zariski topology is respectively T 0 , irreducible or Noetherian are studied, and several characterizations of such modules are given.
Let R be a commutative ring and M be an R-module with a proper submodule N. The total graph of M with respect to N, denoted by T(ΓN(M)), is investigated. The vertex set of this graph is M and for all x, y belonging to M, x is adjacent to y if and only if x + y ∈ M(N), where M(N) = {m ∈ M : rm ∈ N for some r ∈ R - (N : M)}. In this paper, in addition to studying some algebraic properties of M(N), we investigate some graph theoretic properties of two important subgraphs of T(ΓN(M)) in the cases depending on whether or not M(N) is a submodule of M.
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