On the extended zero divisor graph of commutative rings

In this paper we present a new graph that is closely related to the classical zero-divisor graph. In our case two nonzero distinct zero divisors x and y of a commutative ring R are adjacent whenever there exist two nonnegative integers n and m such that x n y m = 0 with x n 0 and y m 0. This yields an extension of the classical zero divisor graph ( R) of R, which will be denoted by ( R). First we distinguish when ( R) and ( R) coincide. Various examples in this context are given. We show that if ( R) ( R), then ( R) must contain a cycle. We also show that if ( R) ( R) and ( R) is complemented, then the total quotient ring of R is zero-dimensional. Among other things, the diameter and girth of ( R) are also studied.