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THE ZERO-DIVISOR GRAPH OF VON NEUMANN REGULAR RINGS

Let R be a commutative ring, and let ZðRÞ denote its set of zerodivisors. We associate a simple graph GðRÞ to R with vertices ZðRÞ 1⁄4 ZðRÞ f0g, the set of nonzero zero-divisors of R. Two distinct vertices x and y are adjacent if xy 1⁄4 0. Thus GðRÞ is the empty graph if and only if R is an integral domain. The notion of a zero-divisor graph was first introduced by I. Beck in [4] and further investigated in [1], where the authors were interested in colorings of GðRÞ, though their vertex set included the zero element. In [2, 3, 5] and [7] the authors, using the same vertex set as here, were interested in examining the interplay between the ring-theoretic properties of R and the graph-theoretic properties of GðRÞ. We examine the zero-divisor graph GðRÞ, of a Von Neumann regular ring R. Recall that a ring R is called Von Neumann regular if for each a 2 R there exists b 2 R such that ab 1⁄4 a, or equivalently if every principal ideal of R is generated by an idempotent. We show that if R and S are two such rings with GðRÞ ffi GðSÞ, then they share certain ring-theoretic properties. If the socle is essential in one, then it is essential in the other (Proposition 2.5). In Theorem 2.3 we show that if the socle of one is the direct sum of finite simple ideals, then so is the other and the socles are isomorphic as rings (without identities). In Corollary 2.4 we show that if two rings are the