• DOI:10.4134/JKMS.2014.51.1.087
• Corpus ID: 119913919

A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

@article{Safaeeyan2014AGO,
  title={A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES},
  author={Saeed Safaeeyan and Mohammad Reza Baziar and Ehsan Momtahan},
  journal={Journal of Korean Medical Science},
  year={2014},
  volume={51},
  pages={87-98}
}

• S. Safaeeyan, M. Baziar, E. Momtahan
• Published 2014
• Mathematics
• Journal of Korean Medical Science

Abstract. Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say Γ(M), such thatwhen M = R, Γ(M) is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F.Anderson and S. B. Mulay, in [6], have been generalized for Γ(M) in thepresent article. We show that Γ(M) is connected with diam(Γ(M)) ≤ 3.We also show that for a reduced module M with Z(M) ∗ 6= M \ {0},gr(Γ(M)) = ∞ if and only if Γ(M) is a star graph. Furthermore, weshow that for a finitely generated semisimple R-module M such that itshomogeneous components are simple, x,y∈ M\ {0} are adjacent if andonly if xRTyR = (0). Among other things, it is also observed thatΓ(M) = ∅ if and only if M is uniform, ann(M) is a radical ideal, andZ(M) ∗ 6= M\{0}, if and only if ann(M) is prime and Z(M) ∗ 6= M\{0}. 1. IntroductionAll rings in this paper are commutative with identity and all modules areunitary right modules. Let G be an undirected graph. We say that G isconnected if there is a path between any two distinct vertices. For distinctvertices x and y in G, the distance between x and y, denoted by d(x,y), is thelength of a shortest path connecting x and y (d(x,x) = 0 and d(x,y) = ∞ ifno such path exists). The diameter of G isdiam(G) = sup{d(x,y) | x and y are vertices of G}.A cycle of length n in G is a path of the form x