On the diameter of the compressed zero-divisor graph

ABSTRACT

Let R be a commutative ring with nonzero identity. The relation on R given by a∼b if and only if $an{n}_R\left(a\right)=an{n}_R\left(b\right)$ is an equivalence relation. The compressed zero-divisor graph Γ_E(R) of R is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0]_R and [1]_R, and distinct vertices [a]_R and [b]_R are adjacent if and only if ab = 0. The distance between vertices [a]_R and [b]_R (not necessarily distinct from a) is the length of the shortest path connecting them, and the diameter of the graph, diam(Γ_E(R)), is the sup of these distances. In this paper, we continue study of the diameter of the compressed zero-divisor graph Γ_E(R). A complete characterization for the possible diameters of Γ_E(R) is given exclusively in terms of the ideals of R. Also we give a complete characterization for the possible diameters of Γ_E(R[x]) in terms of the diameters of Γ_E(R). For a reduced ring R with nonzero zero-divisors, it is shown that $1\le diam\left({\Gamma }_E\right(R\left)\right)\le diam\left({\Gamma }_E\right(R\left[x\right]\left)\right)\le diam\left({\Gamma }_E\right(R\left[\right[x\left]\right]\left)\right)\le 3$.

KEYWORDS:

• Compressed zero-divisor graph
• Noetherian rings
• Polynomial rings
• power series rings
• zero-divisor graph

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13A99
• 05C12

Acknowledgments

The authors would like to thank the referee for a very careful reading of the paper and for many constructive comments. This research was in part supported by a grant from Shahrood University of Technology.