Co-maximal ideal graphs of matrix algebras

Abstract

Let R be a ring with unity. The co-maximal ideal graph of R, denoted by $\Gamma (R)$, is a graph whose vertices are all non-trivial left ideals of R, and two distinct vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. In this paper, some results on the co-maximal ideal graphs of matrix algebras are given. For instance, we determine the domination number, the clique number and a lower bound of the independence number of $\Gamma (M_n(\mathbb {F}_q))$, where $M_n(\mathbb {F}_q)$ is the ring of $n\times n$ matrices over the finite field $\mathbb {F}_q$. Furthermore, we characterize all rings (not necessarily commutative) whose domination numbers of their co-maximal ideal graphs are finite. Among other results, we show that if $\Gamma (R)\cong \Gamma (M_n(\mathbb {F}_q))$, where $n\ge 2$ is a positive integer and R is a ring, then $R\cong M_n(\mathbb {F}_q)$. Also, it is proved that if R and $R'$ are two finite reduced rings and $\Gamma (M_m(R))\cong \Gamma (M_n(R'))$, for some positive integers $m,n\ge 2$, then $m=n$ and $R\cong R'$.