A Graphical Representation of Rings via Automorphism Groups

Details A Graphical Representation of Rings via Automorphism Groups A Graphical Representation of Rings via Automorphism Groups

2010-02-28

arxiv.org/abs/1003.0174v1

Mathematics

Commutative Algebra

13 PAGES

Scientific paper

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 $\Gamma_{\aut}(R)$.

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