Characterization of Automorphisms of Zero Divisor Graph of Nilradicals of Some Classes of Local Rings

Abstract

The study of zero divisor graphs G(R) of rings R was first introduced by Beck in 1988. Beck isolated all Z (R) of the ring R and considered them as the vertex set of G(R). Beck’s focus was on the chromatic number χ(G(R)) of the graph G(R). Beck illustrated that G(R) is simple, connected, and diam(G(R)) ≤ 2. Besides, Beck further established that χ(G(R)) < 4. In 1999, Anderson and Livingstone standard- ized Beck’s graph G(R) by Γ(R). They considered Z (R)∗ and showed that Γ(R) is simple, connected, and that diam(Γ(R)) ≤ 3. The study on automorphisms of Γ(R) have helped in creating a link between the commutative algebra and graph theory. Various studies have been carried out by researchers to visualize and better under- stand algebraic structures of rings in graph theory. This approach is most effective when applied to commutative local rings. The objective of this study is to char- acterize automorphisms of Γ (R) of nilradicals of some classes of local commutative finite rings utilizing either Livingstone’s and Anderson’s zero divisor graph or Mu- lay’s condensed zero divisor graph. The methodology involves using combinatorial algorithmic technique to divide the rings into disjoint subsets of nilpotent and non- nilpotent elements, then constructing Z (R)∗ graphs and using case-by-case analysis to investigate maps that preserve structure within the graph. Structure-preserving maps are then identified using mathematical induction. Results and structure of Z (N (R)) will depend primarily on the order of nilpotent elements and their connec- tions, which will give a clear picture of how these nilradicals interact. Even though automorphisms can reflect many properties of the ring, it is still unknown whether the automorphism group can distinguish between such rings, especially in higher- dimensional contexts. This study will help determine how such automorphisms are useful in categorizing different local rings.