A Graphical Representation of Rings via Automorphism Groups

Abstract

Let $R$ be a commutative ring with identity. We define a graph $Γ_{\aut}(R)$ on $R$, with vertices elements of $R$, such that any two distinct vertices $x,y$ are adjacent if and only if there exists $σ\in \aut$ 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 $n\ge 0$ and characterize all rings of type zero. We also characterize local rings $(R,M)$ in which either the subset of units ($\ne 1$) is connected or the subset $M-\{0\}$ is connected in $Γ_{\aut}(R)$.

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