Abstract view
On the Comaximal Graph of a Commutative Ring


Published:20131223
Printed: Jun 2014
Karim Samei,
Department of Mathematics, Bu Ali Sina University, Hamedan, Iran
Abstract
Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M.
Bhatwadekar, A note on graphical representation of rings, J.
Algebra 176(1995) 124127], Sharma and Bhatwadekar defined a
graph on $R$, $\Gamma(R)$, 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 nonunit elements. We investigate
the behavior of $\Gamma_2(R)$ and $\Gamma_2(R) \setminus \operatorname{J}(R)$,
where $\operatorname{J}(R)$ is the Jacobson radical of $R$. We associate the
ring properties of $R$, the graph properties of $\Gamma_2(R)$ and
the topological properties of $\operatorname{Max}(R)$. Diameter, girth, cycles
and dominating sets are investigated and the algebraic and the
topological characterizations are given for graphical properties
of these graphs.