RINGS HAVING ZERO-DIVISOR GRAPHS OF SMALL DIAMETER OR LARGE GIRTH

Let R be a commutative ring possessing (non-zero) zero-divisors. There is a natural graph associated to the set of zero-divisors of R. In this article we present a charac-terisation of two types of R. Those for which the associated zero-divisor graph has diameter different from 3 and those R for which the associated zero-divisor graph has girth other than 3. Thus, in a sense, for a generic non-domain R the associated zero-divisor graph has diameter 3 as well as girth 3. Let R be a commutative ring with 1 6 = 0 and let Z(R) denote the set of non-zero zero-divisors of R. By the zero-divisor-graph of R we mean the graph with vertices Z(R) such that there is an (undirected) edge between vertices x, y if and only if x 6 = y and xy = 0 (see [1, 3, 4]). Since there is hardly any possibility of confusion, we allow Z(R) to denote the zero-divisor graph of R. Following their introduction in [3], zero-divisor graphs have received a good deal of attention. For a more comprehensive list of references the reader is requested to refer to the bibliographies of [1, 2, 4]. Zero-divisor graphs are highly symmetric and structurally very special; for example, in [4] this author has investigated the structure of cycles, the graph-automorphism-group Γ(R) and it