Graphs of Commutatively Closed Sets

TL;DR

This paper explores the relationship between algebraic properties of rings and their associated graphs, introducing commutatively closed graphs and analyzing their properties, including diameter calculations for specific algebra classes.

Contribution

It introduces the concept of commutatively closed graphs and investigates their properties, especially focusing on diameter computations for artinian semisimple algebras.

Findings

Computed the diameter of artinian semisimple algebras.

Established properties of commutatively closed subsets in rings.

Linked algebraic structures with graph-theoretic characteristics.

Abstract

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. We introduce commutatively closed graphs and investigate properties of commutatively closed subsets of a ring with the help of graph theory. A particular attention is paid to constructions of subsets of a given diameter. In particular, we compute the diameter of artinian semisimple algebras.