@article { author = {Naghipour, A.}, title = {THE ZERO-DIVISOR GRAPH OF A MODULE}, journal = {Journal of Algebraic Systems}, volume = {4}, number = {2}, pages = {155-171}, year = {2017}, publisher = {Shahrood University of Technology}, issn = {2345-5128}, eissn = {2345-511X}, doi = {10.22044/jas.2017.858}, abstract = {Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, sayΓ(RM), such that when M=R, Γ(RM) coincide with the zero-divisor graph of R. Many well-known results by D.F. Anderson and P.S. Livingston have been generalized for Γ(RM). We Will show that Γ(RM) is connected withdiam Γ(RM)≤ 3 and if Γ(RM) contains a cycle, then Γ(RM)≤4. We will also show that Γ(RM)=Ø if and only if M is aprime module. Among other results, it is shown that for a reduced module M satisfying DCC on cyclic submodules,gr (Γ(RM))=∞ if and only if Γ(RM) is a star graph. Finally, we study the zero-divisor graph of freeR-modules. }, keywords = {Annilhilator,diameter,girth,reduced module,zero-divisor graph}, url = {https://jas.shahroodut.ac.ir/article_858.html}, eprint = {https://jas.shahroodut.ac.ir/article_858_dc9a03e1918e0e0bd28530d1103281ff.pdf} }