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

Visweswaran, Subramanian (Saurashtra University)

Indonesian Journal of Combinatorics
Vol 8, No 1 (2024)

In this paper, with any atomic domain R which admits at least two maximal ideals, we associate an undirected graph denoted by ???(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 ???(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 ???(R) is connected. Also, in this paper, in some cases, a necessary and sufficient condition is determined so that ???(R) is connected.