Co-maximal ideal graphs of matrix algebras

Abstract

Let R be a ring with unity. The co-maximal ideal graph of R, denoted by (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 (M n (F q )), where M n (F q ) is the ring of n × n matrices over the finite field 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 (R) ∼ = (M n (F q )), where n ≥ 2 is a positive integer and R is a ring, then R ∼ = M n (F q ). Also, it is proved that if R and R are two finite reduced rings and (M m (R)) ∼ = (M n (R )), for some positive integers m, n ≥ 2, then m = n and R ∼ = R .

More on the annihilator-ideal graph of a commutative ring

“…Papers in this field apply combinatorial methods to obtain algebraic results in ring theory (see for instance [1], [7], [13] and [15]). Moreover, for the most recent study in this direction see [6], [11] and [15]. Throughout this paper, R denotes a unitary commutative ring which is not an integral domain.…”