Alternative algebras with the hyperbolic property

TL;DR

This paper studies the structure of certain finite dimensional alternative algebras over the rationals that lack free abelian subgroups of rank two in their unit groups, revealing radical properties and classifying related loops.

Contribution

It characterizes the radical of these algebras and classifies specific $RA$-loops with this property, advancing understanding of their algebraic structure.

Findings

Radical of such algebras coincides with the entire algebra.

Classified $RA$-loops with the hyperbolic property.

Open problem remains for group ring classifications.

Abstract

We investigate the structure of an alternative finite dimensional $\Q$-algebra $\mathfrak{A}$ subject to the condition that for a $\Z$-order $\Gamma \subset \mathfrak{A}$, and thus for every $\Z$-order of $\mathfrak{A}$, the loop of units of $\U (\Gamma)$ does not contain a free abelian subgroup of rank two. In particular, we prove that the radical of such an algebra associates with the whole algebra. We also classify $RA$-loops $L$ for which $\mathbb{Z}L$ has this property. The classification for group rings is still an open problem.