Generating adjoint groups

TL;DR

This paper investigates the properties of adjoint groups of nil rings, proving limitations on their generation and constructing examples with specific torsion properties, advancing understanding of their algebraic structure.

Contribution

It provides new results on the generation of adjoint groups of nil rings, including non-bounded generation and examples with bounded torsion generators.

Findings

Adjoint groups of non-nilpotent Jacobson radical rings cannot be boundedly generated.

Constructed finitely generated, infinite-dimensional nil algebra with adjoint group generated by elements of bounded torsion.

Abstract

We prove two approximations of the open problem of whether the adjoint group of a non-nilpotent nil ring can be finitely generated: We show that the adjoint group of a non-nilpotent Jacobson radical cannot be boundedly generated, and on the other hand construct a finitely generated, infinite dimensional nil algebra whose adjoint group is generated by elements of bounded torsion.