Comaximal graph of commutative rings

Abstract

Let $R$ be a commutative ring with identity. Let $\Gamma (R)$ be a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. In this paper we consider a subgraph ${\Gamma }_2(R)$ of $\Gamma (R)$ which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph $Γ_2(R)\setminus\J(R)$. In addition, it is shown that for two finite semi-local rings $R$ and $S$, if $R$ is reduced, then $\Gamma (R)\cong \Gamma (S)$ if and only if $R\cong S$.

8 Pages