Strongly right alternative rings and Bol loops

TL;DR

This paper demonstrates that in finite strongly right alternative rings, both the units and quasiregular elements form Bol loops under their respective multiplications, partially answering questions posed by Goodaire.

Contribution

It establishes that the set of units and quasiregular elements in such rings are Bol loops, providing new insights into their algebraic structure.

Findings

Units form a Bol loop under ring multiplication

Quasiregular elements form a Bol loop under circle multiplication

Partially answers Goodaire's questions about these structures

Abstract

We partially answer two questions of Goodaire by showing that in a finite, strongly right alternative ring, the set of units (if the ring is with unity) is a Bol loop under ring multiplication, and the set of quasiregular elements is a Bol loop under "circle" multiplication.