A note on comaximal ideal graph of commutative rings

Abstract

Let R be a commutative ring with identity. The comaximal ideal graph G(R) of R is a simple graph with its vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I 1 and I 2 are adjacent if and only if I 1 + I 2 = R. In this paper, a dominating set of G(R) is constructed using elements of the center when R is a commutative Artinian ring. Also we prove that the domination number of G(R) is equal to the number of factors in the Artinian decomposition of R. Also, we characterize all commutative Artinian rings(non local rings) with identity for which G(R) is planar.

On the Comaximal (Ideal) Graph Associated With Amalgamated Algebra

“…The vertex set of C(R) consists of all proper ideals which are not contained in the Jacobson radical of R, and two distinct vertices I 1 and I 2 are adjacent if and only if I 1 + I 2 = R. Recently, in [13], some properties of the comaximal graph of an amalgamated algebra are investigated. The reader is referred to [8,9,14,15] for further detail on comaximal graphs.…”