ZERO-DIVISOR GRAPHS WITH RESPECT TO PRIMAL AND WEAKLY PRIMAL IDEALS

Abstract

Abstract. We consider zero-divisor graphs with respect to primal, non-primal, weakly prime and weakly primal ideals of a commutative ring Rwith non-zero identity. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ I (R) forsome ideal I of R. Also we show that the zero-divisor graph with respectto primal ideals commutes by localization. 1. IntroductionThe idea of associating a graph with the zero-divisors of a commutative ringwas introduced by Beck in 1988, where the author talked about the colorings ofsuch graphs. By the definition he gave, every element of the ring R was a vertexin the graph, and two vertices x,y were adjacent if and only if xy = 0 ([4]).We adopt the approach used by D. F. Anderson and P. S. Livingston ([2]) andconsider only non-zero zero-divisors as vertices of the graph. The zero-divisorgraph of a commutative ring has been studied extensively by several authors(see, for example, [2, 4, 5, 10, 11, 12]).Redmond[13]introduced the definition ofthe zero-divisorgraphwith respectto an ideal. Let I be an ideal of a ring R. The zero-divisor graph of Rwith respect to I is an undirected graph, denoted by Γ