Zero-divisor graphs of polynomials and power series over commutative rings

Abstract. We recall several results of zero divisor graphs of com-mutative rings. We then examine the preservation of diameter and girth of the zero divisor graph under extension to polynomial and power series rings. The concept of the graph of the zero-divisors of a ring was first introduced by Beck in [3] when discussing the coloring of a commutative ring. In his work all elements of the ring were vertices of the graph. D.D. Anderson and Naseer used this same concept in [1]. We adopt the approach used by D.F. Anderson and Livingston in [2] and consider only nonzero zero-divisors as vertices of the graph. In the first section we provide the reader with a few known results concerning the girth and diameter of the graph of zero-divisors of a commutative ring. In the second section we examine the preservation and lack thereof of the diameter of the zero-divisor graphs under extensions to polynomial and power series rings, while in the third section we consider the effects o