Comaximal graph of commutative rings
Hamid Reza Maimani, Maryam Salimi, Asiyeh Sattari, Siamak Yassemi
Abstract
Let
R
be a commutative ring with identity. Let
Γ(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
Γ
2
(R)
of
Γ(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 $\Gamma_2(R)\setminus\J(R)$. In addition, it is shown that for two finite semi-local rings
R
and
S
, if
R
is reduced, then
Γ(R)≅Γ(S)
if and only if
R≅S
.