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

Abstract

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