A commutative version of the group ring

TL;DR

This paper introduces a commutative version of the group ring that translates group normal generation questions into ideal generation problems in commutative algebra, providing new insights and proofs.

Contribution

It constructs a commutative group ring and demonstrates its utility in analyzing group normal generation through ideal generation in commutative rings.

Findings

Provides an alternative proof for the normal generation of free product of cyclic groups

Establishes a correspondence between group normal generation and ideal generation in commutative rings

Introduces a new algebraic framework for studying group properties

Abstract

We construct a commutative version of the group ring and show that it allows one to translate questions about the normal generation of groups into questions about the generation of ideals in commutative rings. We demonstrate this with an alternative proof of a result about the normal generation of the free product of two cyclic groups.