The Zero-Divisor Graph of a Commutative Ring

Ž . Ž . Let R be a commutative ring with 1 and let Z R be its set of Ž . Ž . zero-divisors. We associate a simple graph G R to R with vertices Ž . Ž . 4 Z R * s Z R y 0 , the set of nonzero zero-divisors of R, and for disŽ . tinct x, y g Z R *, the vertices x and y are adjacent if and only if xy s 0. Ž . Thus G R is the empty graph if and only if R is an integral domain. The main object of this paper is to study the interplay of ring-theoretic Ž . properties of R with graph-theoretic properties of G R . This study helps Ž . Ž . illuminate the structure of Z R . For x, y g Z R , define x ; y if xy s 0 or x s y. The relation ; is always reflexive and symmetric, but is usually Ž . not transitive. The zero-divisor graph G R measures this lack of transitivŽ . ity in the sense that ; is transitive if and only if G R is complete. The idea of a zero-divisor graph of a commutative ring was introduced w x by I. Beck in 2 , where he was mainly interested in colorings. This investigation of colorings of a commutative ring was then continued by D. w x D. Anderson and M. Naseer in 1 . Their definition was slightly different than ours; they let all elements of R be vertices and had distinct x and y adjacent if and only if xy s 0. We will denote their zero-divisor graph of R Ž . Ž . by G R . In G R , the vertex 0 is adjacent to every other vertex, but 0 0 Ž . Ž . non-zero-divisors are adjacent only to 0. Note that G R is a induced Ž . Ž . Ž . subgraph of G R . Our results for G R have natural analogs to G R ; 0 0