On the intersection graphs of modules and rings

TL;DR

This paper classifies modules and rings based on properties of their intersection graphs, providing structure theorems and characterizations for infinite graphs with finite degree maximal substructures, and explores graph invariants like clique and chromatic numbers.

Contribution

It offers new classifications and structure theorems for modules and rings with specific intersection graph properties, extending previous results and removing certain assumptions.

Findings

Rings with infinite intersection graphs and finite degree maximal left ideals are characterized.

Modules with infinite intersection graphs containing finite degree maximal submodules are structurally described.

In such modules, the clique number equals the chromatic number, and this equality also holds for the complement graph.

Abstract

We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in [A2]. We also generalize this result to modules, i.e. we get the structure theorem of modules for which their intersection graphs are infinite and contain maximal submodules of finite degree. Furthermore we omit the assumption of maximality of submodules and still get a satisfactory characterization of such modules. In addition we show that, if the intersection graph of a module is infinite but its clique number is finite, then the clique and chromatic numbers of the graph coincide. This fact was known earlier only in some particular cases. It appears that such equality holds also in the complement graph.