When is the maximal graph of a non-quasi-local atomic domain connected?

Abstract

In this paper, with any atomic domain R which admits at least two maximal ideals, we associate an undirected graph denoted by 𝕄𝔾𝕀(R) whose vertex set is I(R)={Rπ | π∈ Irr(R)\J(R)} (where Irr(R) is the set of all irreducible elements of R and J(R) is the Jacobson radical of R) and distinct Rπ, Rπ' ∈ I(R) are adjacent if and only if Rπ + Rπ' ⊆ M for some maximal ideal M of R. We call 𝕄𝔾𝕀(R) as the maximal graph of R. We denote the set of all maximal ideals of R by Max(R). In this paper, some necessary (respectively, sufficient) conditions on Max(R) are provided such that 𝕄𝔾𝕀(R) is connected. Also, in this paper, in some cases, a necessary and sufficient condition is determined so that 𝕄𝔾𝕀(R) is connected.