When does an infinite ring have a finite compressed commuting graph?

TL;DR

This paper investigates the properties of infinite rings by analyzing their compressed commuting graphs, classifying those with finite graphs and finitely many subrings, revealing structural insights into their algebraic nature.

Contribution

It provides a complete classification of infinite unital rings with finite unital compressed commuting graphs and finitely many unital subrings, using semidirect products.

Findings

Infinite rings have infinite nonunital compressed commuting graphs.

Classified all infinite unital rings with finite unital compressed commuting graphs.

Identified conditions for rings to have finitely many unital subrings.

Abstract

We show that any infinite ring has an infinite nonunital compressed commuting graph. We classify all infinite unital rings with finite unital compressed commuting graph, using semidirect product of rings as our main tool. As a consequence we also classify infinite unital rings with only finitely many unital subrings.