On groups with cubic polynomial conditions

TL;DR

This paper investigates conditions under which finitely generated subgroups of rings, satisfying specific cubic polynomial equations, generate subrings with finite rank, with particular focus on the case where the polynomial is (x-1)^3=0.

Contribution

It establishes a finite subset criterion ensuring subrings have finite Z-rank when elements satisfy certain cubic polynomial conditions.

Findings

Finite subset of G determines finite Z-rank of the subring

Special case where polynomial is (x-1)^3=0 analyzed

Conditions for finite Z-rank are characterized

Abstract

Given a finitely generated subgroup G of a ring R we provide a finite subset of G such that if each element of this set satisfies some cubic polynomial equation in one variable over the center Z of R then the subring generated by G has finite Z-rank. We specialize our considerations to the case where the polynomial equations are equal to (x-1)^3=0.