Adjoint groups of $p$-nil rings and $p$-group automorphisms

TL;DR

This paper studies a special class of rings called $p$-nil rings, showing how their adjoint groups behave regularly and relate to automorphisms of $p$-groups, with applications in group theory.

Contribution

Introduces the class of $p$-nil rings and explores their adjoint groups, revealing their regular behavior and connection to $p$-group automorphisms.

Findings

$p$-nil rings have regular adjoint groups

Every $p$-ring contains a large $p$-nil ideal

Adjoint groups correspond to automorphism groups of $p$-groups

Abstract

We introduce a class of rings, namely the class of left or right $p$-nil rings, for which the adjoint groups behave regularly. Every $p$-ring is close to being left or right $p$-nil in the sense that it contains a large ideal belonging to this class. Also their adjoint groups occur naturally as groups of automorphisms of $p$-groups. These facts and some of their applications are investigated in this paper.