A Graphical Representation of Rings via Automorphism Groups

TL;DR

This paper introduces a graph-based approach to study automorphism groups of rings, characterizing rings of type zero and analyzing connectivity properties of units and maximal ideals in the automorphism graph.

Contribution

It defines the notion of a ring of type n and characterizes all rings of type zero, providing new insights into the structure of rings via automorphism graphs.

Findings

Characterized all rings of type zero.

Analyzed connectivity of units and maximal ideals in automorphism graphs.

Provided a new graph-theoretic perspective on ring automorphisms.

Abstract

Let $R$ be a commutative ring with identity. We define a graph $\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in \aut$ such that $\sigma(x)=y$. The idea is to apply graph theory to study orbit spaces of rings under automorphisms. In this article, we define the notion of a ring of type $n$ for $n\geq 0$ and characterize all rings of type zero. We also characterize local rings $(R,M) $ in which either the subset of units ($\neq 1 $) is connected or the subset $M- \{0\}$ is connected in $\Gamma_{\aut}(R)$.