Alternative algebras with the hyperbolic property

Abstract

We investigate the structure of an alternative finite dimensional $\Q$-algebra $\mathfrak{A}$ subject to the condition that for a $\mathbb{Z}$-order $\Gamma \subset \mathfrak{A}$, and thus for every $\mathbb{Z}$-order of $\mathfrak{A}$, the loop of units of $\U (Γ)$ 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.

arXiv commentThird author's Ph.D. (parcial); generalization of those results for the non-associative case, 9 pps. Conf.: Geometry, Topology, Algebra and Number Theory, Steklov Math. Inst., Moscow-Russia (Aug, 2010); XVIII Latin American Colloquium of Algebra, São Paulo-Brazil (July, 2009); Algebras, Representation and Applications, Ubatuba-Brazil (aug., 2007); Group and Group Rings, Bedlewo-Poland (2005)