A MODULE THEORETIC APPROACH TO ZERO-DIVISOR GRAPH WITH RESPECT TO (FIRST) DUAL

Let M be an R-module and 0 not equal f is an element of M* = Hom(M, R). We associate an undirected graph Gamma(f) (M) to M in which non-zero elements x and y of M are adjacent provided that xf(y) = 0 or y f (x) = 0. We observe that over a commutative ring R, Gamma(f) (M) is connected and diam(Gamma(f) (M)) <= 3. Moreover, if Gamma(f) (M) contains a cycle, then gr(Gamma(f) (M)) <= 4. Furthermore, if vertical bar Gamma(f)(M)vertical bar >= 1, then Gamma(f) (M) is finite if and only if M is finite. Also if Gamma(f) (M) = empty set, then f is monomorphism (the converse is true if R is a domain). If M is either a free module with rank(M) >= 2 or a non-finitely generated projective module, there exists f is an element of M* with rad(Gamma(f) (M)) = 1 and diam(Gamma(f) (M)) <= 2. We prove that for a domain R, the chromatic number and the clique number of Gamma(f) (M) are equal. Finally, we give answer to a question posed in [M. Baziar, E. Momtahan and S. Safaeeyan, A zero-divisor graph for modules with respect to their (first) dual, T. Algebra Appl. 12 (2013), no. 2, 11 pages].